proceedings of the national seminar on present trends in algebra and its applications

Editors:

Prof. Dr Satyanarayana Bhavanari M.Tech., B.Ed., M.Sc., Ph.D.,

A.P. Scientist Awardee 2009, Fellow - A.P. Akademi of Sciences

Glory of India Award & International Achievers Award (Thailand 2011)

Top 100 Professionals - 2011 (IBC, England)

Dr Vijaya Kumari A. V. M.Sc., Ph.D.

Mr. Mohiddin Shaw Shaik M.Sc., M.Phil.,

Sponsored by UGC

Held at J.M.J College for Women, Tenali,

Guntur Dist, Andhra Pradesh, India

July 11-12, 2011

Proceedings of the

National Seminar on

Present Trends in

Algebra and its

Applications

Includes

Invited Lectures

Research Paper

Presentations

Abstracts

Proceedings of the National Seminar on Present Trends in Algebra and its Applications, J.M.J. College, Tenali, A.P., India., July 11-12, 2011.

(Editors: Prof. Dr Bhavanari Satyanarayana, Dr A.V. Vijaya Kumari, and Mr. Shaik Mohiddin Shaw)

First Published: July 2011

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The Editors of this Proceedings are extremely grateful to the authors of the talks articles printed in this book for their contribution and invaluable suggestions. This book is meant for educational and learning purposes. The author(s) of the papers in this book has/have taken all reasonable care to ensure that the contents of the book do not violate any existing copyrights or other intellectual property rights of any person in any manner whatsoever. In the event the author(s) has/have unable to track any source and if any copyright has been inadvertently infringed, please notify to the author in writing for corrective action.

PREFACEPREFACEPREFACEPREFACE

This Proceedings contains the substance of invited lectures and contributed research oral paper presentation at the Two-day National Seminar on “ Present Trends in Algebra and it’s Applications” (sponsored by UGC) organized in J.M.J. College, Tenali, Guntur District, Andhra Pradesh, India., (in collaboration with Department of Mathematics , Acharya Nagarjuna University, Guntur) from 11th to 12th July 2011. The main object of this seminar is to bring together many eminent professors, scholars, researchers well-versed in various fields of Mathematics like Graph Theory, Fuzzy Set Theory, Near-rings, Gamma Near-rings, Rings and Modules for exchange of ideas.

The editors express their gratitude to all the organizing committee members for their cooperation and help. Also special thanks to the invited speakers and paper presenters for sending their valuable talk scripts, papers and abstracts to include in the Proceedings of the National Seminar. The event certainly provides an opportunity for young researchers to strengthen their collaborative works of common interest. Finally, the editors would like to thank the Press and Electronic media for their extensive coverage of the news of the event.

Editors Prof. Dr Bhavanari Satyanarayana

Dr. A.V. Vijaya Kumari Mr. Mohiddin Shaw Shaik.

TWO-DAY NATIONAL SEMINAR ON “PRESENT TRENEDS IN ALGEBRA

AND ITS APPLICATIONS”. (UGC-SPONSORED)

JMJ College for Women , Tenali, A.P., India., 11th and 12th July 2011

(In Collaboration with Department of Mathematics, Acharya Nagarjuna University)

Chief Patron: Rev.Dr.Sr. Jacintha, Principal JMJ College for Women, Tenali.

Organizing Secretary:

Dr. A.V. Vijaya Kumari, H.O.D of Mathematics, JMJ College for Women.

Academic Secretary: Prof. Dr Bhavanari Satyanarayana,

M.Tech., B.Ed., M.Sc., Ph.D, A.P. Scientist Awardee – 2009

Glory of India Award (Bangkok-2011) Fellow, A.P. Academy of Scineces.

Advisory Committee:

Rev.Dr.Sr Mary Thomas, Correspondent, JMJ, College, Tenali. Rev.Dr.Sr. K. Mareelu, Vice – Principal, JMJ College for Women, Tenali.

Dr. T. V. Pradeep Kumar, Assistant Professor of Mathematics, ANU College of Engineering .

Organizing Committee:

Mrs. P. Sushma, Mobile: 9948681621 Ms. G. Rajya Lakshmi,

Ms. Ch. Radhika, Ms. B. Mary Swarna Latha

Ms. B. Bharathi. Faculty of Department of Mathematics, JMJ College for Women, Tenali

Sri Shaik Mohiddin Shaw (Project Fellow, UGC-MRP under the Principle Investgatorship

Prof. Dr. Bhavanari Satyanaryana, Department of Mathematics, ANU).

About the JMJ College for Women, Tenali, A.P., India.

Jesus May and Joseph (JMJ) College for Women is one of the minority institutions inspired by the teaching and the life of Jesus Christ in A.P. It awakens and inspires the students high ideals of social service and the prominent role they play as makers of home and society. The college was started on July 2, 1963 with 75 students. From this humble beginning the college has emerged as one of the best colleges in Acharya Nagarjuna University. At present the college offers 13 U.G. and 6 P.G. Courses. The college has well furnished class rooms, fully equipped laboratories, English language lab, Internet facility, Seminar room, Auditorium, Gym, Playground and excellent hostel with hygienic conditions. The College was inaugurated with the motto “including the excluded and giving the best to the least” to catering the needs of the ever changing society, with its extended vision and mission. A central government funded scheme Swadhar was inaugurated for the women in difficult circumstances in 2005. Anganwadi works training centre was introduced in 2003.

About the Department of Maths., Acharya Nagarjuna University. It is specialized in the fields of Rings and Modules, Nearrings, Gama-nearrings, Fuzzy Algebra and Graph Theory. It achieved four major research projects froms UGC. Prof. Davuluri Rama Kotaiah, Father of Department was the Vice-Chancellor of Acharya Nagarjuna University during the period 1988-91. Prof. Dr Bhavanari Satyanarayana, Academic Secretary of the present national seminar received INSA visiting Fellowship Award-2005, ANU best paper award-2006, AP Scientist-2009 award (from DST, New Delhi). Fellow-AP Academy of Sciences, 2010, Glory of India Award and International Achievers Award (Thailand, 2011). One of the TOP 100 PROFESSIONALS (Selected by IBC, Cambridge, England,2011). He was a selected scientist by UGC-HAS (Hungarian Academy of Sciences) 2003, and selected senior scientist by INSA-HAS-2005. He introduced Algebraic System Gama Near Ring in 1984. He is an author/editor for 36 books (including one book published by PHI, New Delhi; and five books published by VDM Verlag, Germany).

About the Seminar: As Mathematics is one of the most important subjects and plays a vital role in all branches of science, the Department of Mathematics of the college, in association with Department of Mathematics, Acharya Nagarjuna University, is organizing a National Seminar on “ALGEBRA AND ITS APPLICATIONS” on 11th and 12th July, 2011 so as to highlight its significance in every branch of science in the present scenario. The main objective of the seminar is to bring together the eminent academicians and the young researchers to share their latest ideas and help one another on the topics of their common interest in their research programs. The seminar also provides a forum for “Problem Solving Session for Mathematics, School Teachers” on the Second day from 2pm.

THEMES: The topics to be covered are: RINGS AND MODULES, NEAR-RINGS

AND GAMMA-NEAR-RINGS, FUZZY ALGEBRA, GRAPH THEORY, OTHER

RELATED AREAS.

Greetings to the National Seminar

19-06-2011 My best wishes for its success With Best Wishes and Regards

-----Joginder Singh, IPS (Retd.) Former Director, CBI, India, New Delhi 110077.

*********************** Dear Prof .Satyanarayana Greetings. I will not be able to attend the National Seminar at Tenali as I plan to stay here (USA) during those days . I wish the function a very grand successs with best wishes to you and family members Prof. PV Arunachalam , Former Vice-Chancellor, Dravidian University, Kuppam ------------------------------------------------------- Saturday, 25 June, 2011 1:59 AM Dear Sir, Thank you for the mail and information. I am very happy and pleased to know that you are academic secretary of the National Seminar on Algebra and its applications. I wish you and all the organizing commitee members a huge success. With warm regards, Dr Kedukodi Babushri Srinivas, Institute of Ring Theory, USA. ---------------------------------------------------------

Dt: 3-7-2011, MESSAGE: I am extremely happy to learn that a National Seminar on “Present trends in Algebra and its Applications” is going to conducted on July 11th and 12th, 2011 at a reputed institution JMJ College for women at Tenali. The topics selected for the seminar are note worthy and the themes under it are also interesting in the current affairs. We appreciate the Organizing Secretary Dr. A.V.Vijaya Kumari, HOD of Mathematics and the other members to have taken a great responsibility of conducting this seminar and benefiting the faculty of Mathematics.

Happy to note that the Academic Secretary , Prof. Bhavanari Satyanarayana chosen for this seminar is very rich academically. He has got many awards and rewards both at National and International levels. And also he has got much experience in conducting such seminars. It is proud to notice that a Galaxy of most learned invitees are to deliver their valuable lectures in this seminar.

It is observed that an opportunity is provided for the school teachers also by arranging one session exclusively and for which the members of our “Association for Improvement of Maths Education” are thankful to the organizers of the seminar.

As such we strongly hope and wish that the seminar will become a grand success. ………. Sri. Ch. V. Narasimha Rao, Director – A.I.M.Ed., VIJAYAWADA.

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari)

CONTENTS (Page No. 1)

Invited Talks

S.No Invited Speaker Title of the Talk Pages

01 Prof. Bhavanari Satyanarayana, Acharya Nagarjuna University

Γ-Near-Rings

(Key Note Address)

01–15

02 Prof. L. Nagamuni Reddy Professor of Mathematics (Retd), SVU, Tirupathi

Color preserving Automorphisms of Cayley

graphs

16–23

03 Prof. K. Suvarna Sri Krishna Devaraya University, Ananthapur

Some Properties of Derivations on Prime Rings.

24–30

04 Dr. Dasari Nagaraju Hindustan University, Chennai

Generalization of Dimension in Vector spaces to Rings

31-34

05 Prof. V. Sitaramaiah Pondicherry Engineering College Pondicherry.

Rearick’s isomorphism and a characterization of

Ψ- additive functions.

35-37

06 Dr. Kuncham Syam Prasad

Manipal University, Manipal Direct systems in N-groups 38-39

07 Dr. K.N.S. Kasi Viswanadham NIT, Warangal

Numerical Solution of some two point boundary value problems by collection method with B-SPlines

40-41

08 Prof. I.H. Nagaraja Rao Director, GVPCollege for P.G. Courses, Visakhapatnam

Congruent Related Graphs 42

09 Prof. Thota Srinivas Kakatiya University

Sandwich Near-Rings 43-47

10 Prof. K.L. N. Swamy Andhra University,Waltar

Boolean Algebra 48

11 Prof.S.A. Mariadoss St.Aloysius College, Mangalore, Karnataka

A graceful numbering of

a new class of graphs

48

12 Prof. K. Rama Krishna Prasad

S.V.U College of Engineering Tirupathi

Analysis of load carrying capacity in finite porous squeeze film bearing by Rapid Technique

48A

13 Prof. I.H. Nagaraja Rao Director, GVPCollege for P.G. Courses, Visakhapatnam

Congruent Related Graphs 128-134

14 Dr. Nanaji rao Andhra University, Visakhapatnam

Pseudo-complemented Almost Distributive Lattices

146-153

15 Prof. S.Sreenadh

SVUNiversity, Tirupathi Effects of Permeability, Elasticity on viscous flows in circular tube

158-162


CONTENTS (Page No. 2)

15 minutes talks

S.No Speaker Title of the Presentation Pages 01 Mohiddin Shaw Shaik A Dimension of Modules

Over Associative Rings. 49-54

02 Shakeera Shaik “Ideal Mapping in Gamma Rings”

55-59

03 K.S. Bala Murugan Uniform and Essential Ideals in Associative Rings

60-62

04 Jagadeesha .B Interval valued C-Prime Fuzzy Ideals of Near-rings

63-66

05 Dr.Pradeep Kumar T.V Some types of Prime ideals in Gamma Near – Rings

156-157

Survey Article – Oral Presentation (Un referred) S.No Speaker Title of the Presentation Pages

01 Satya Sri Bhavanari MBBS IV year, Zhijiyang University,

Republic of China

Golden Ratio and Human Body

123-125

Full Papers – Oral Presentation (Un referred)

S.No Authors Title of the Paper for

Oral Presentation

Pages

01 S.A. Mariadoss, St.Aloysius College, Mangalore, Karnataka Presenter: S.A. Mariadoss

A graceful numbering of a new class of graphs

67-72

02 L. Madhavi, Yogi Vemana University, Kadapa, A.P. Presenter: L. Madhavi

Enumeration of Hamilton Cycles and Triangles in an Arithmetical Graph Associated with Euler

Totient Function Φ

73-77

03 D. Bharathi, Sri Venkateswara University, Titupathi, A.P, India.

Presenter: D. Bharathi

Prime Right Alternative Rings

78-81

04 T. Nagaiah, P. Narasimha swamy Kakatiya University, Warangal Presenter: T.Nagaiah

A note on Anti Fuzzy Ideals in Near Subtraction Semi Groups.

82-90

05 K. Suvarna and K. Madhusudhan Reddy, Sri Krishna Devaraya University, Anantapur. Presenter: K. Madhusudhan Reddy

Rings with [x, yn] – [xn

, y] in the center

91-96

06 G. Shobhalatha, P. Sreenivasulu Reddy and K. Hari Babu, Gates institute of technology Gooty, Anantapur Presenter: K. Hari Babu

Cancellative Left (Right) Regular Semigroups

97-100

07 P. Prathapa Reddy & K. Suvarna Sri Krishna Devaraya University, Anantapur Presenter: P. Prathapa Reddy

Some Results on Weakly Periodic Rings

101-104


CONTENTS (Page No. 3) 08 G. Shobhalatha & P. Sreenivasulu Reddy

Sri Krishna Devaraya University,Anantapur Presenter: P. Sreenivasulu Reddy

Regular semigroups satisfying the identity abc = cb

105-111

09 K. Suvarna and D.S. Irfana Sri Krishna Devaraya University, Anantapur. Presenter: D.S. Irfana

The ideal generated by sets contained in nucleus

112-113

10 Davuluri Nagamani Chundi Ranganayakulu Engg College, Guntur Presenter: D. Nagamani

Shortest Path Problem (An Application of Graph

Theory)

126-127

11 K.V.S. Sarma and I.H.N. Rao Visakhapatnam Presenter: K.V.S. Sarma

On Lower Difference Graphs

135-136

12 G. Jaya Chandra Reddy, C.Eswara

Reddy and K. Rama Krishna Prasad S.V.U. Tirupathi Presenter: Prof. Rama Krishna Prasad

Analysis of load carrying capacity in finite porous squeeze film bearings by

rapid technique

137-145

ABSTRACTS – ORAL PRESENTATION

S.No Authors Title of theAbstract

Presentation

Pages

1 AR. Meenakshi and N. Jeyabalan Karpagam University, Coimbatore Presenter: N. Jeyabalan

Fuzzy Homomorphism, Flags and Cosets of Incline Algebra

114

2 B. Krishnaveni and G. Ganesan

Adikavi Nannaya University, Rajahmundry, A.P Presenter: B. Krishnaveni

Reduction of the Region of Ambiguity in Rough Sets under Fuzziness

114

3 N.V. Ramana Murty Andhra Loyola College,Vijayawada Presenter: N.V. Ramana Murthy

Pure Fuzzy Subgroups

115

4 P.Venu Gopala Rao, Andhra Loyola College Vijayawada. Presenter: P. Venu Gopal

Fuzzy ideals of Seminearrings

116

5 I.H.N. Rao and K.V.S. Sarma

Visakhapatnam

Presenter: K.V.S. Sarma

On Lower Difference Graphs

117

6 Manoj Kumar Patel IT-BHU, Varanasi-221005, UP

Presenter: Manoj Kumar Patel

On Semi Projective Modules

117

7 Varun Kumar I T, Banaras Hindu University Varanasi-

Presenter: Varun Kumar

A note on finite injective modules

117

8 T.V. Pradeep Kumar and N.V. Nagendram ANU College of Engineering Presenter: N.V. Nagendram

Regular Delta Near Rings

118


CONTENTS (Page No. 4) 9 S. Sreenadh, A. Raga Pallavi, T.

Savitha and Ch.Badari Narayana S.V.University, Tirupathi, A.P Presenter: Ch Badari Narayana

Flow of a Jeffery fluid through an artery with multiple stenoses

119

10 D.Venkateswarlu Naidu,

S.Sreenadh and Vishwamohan S.V.U. Tirupathi, A.P. Presenter: D.V.Naidu

Peristaltic Transport of a Power-Law Fluid in Contact with a Newtonial Fluid in an Inclined Porous Channel

119

11 A.Parandama, S. Sreenadh, and

A.N.S. Srinivas S.V.U. Tirupathi, A.P Presenter: A. Parandama

Peristalic transport of Power-law fluid in contact with a Jeffrey fluid in a channel with permeable walls

119

12 S. Sreenadh,R.Madhan

Kumar,P.Devaki and E.Sudhakara S.V.U. Tirupathi, A.P Presenter: E.Sudhakara

Non-Linear analysis of poiseullie flow of a Jeffrey fluid between two parallel plates

120

13 P.Hari Prabhakaran,S.Sreenath

and R.Saravana S.V.U. Tirupathi, A.P

Presenter: Saravana

Effects of induced magnetic field on peristaltic flow of a fourth grade fluid in an inclined planar channel filled with porous material

120

14 M.Arokiasamy Andhra Layola College, Vijayawada

Presenter: M.Arokiasamy

Graph theory and its influence in various fields of knowledge

120

15 Satyanarayana Bhavanari,

Godloza. L, Babu Prasad.M and

Syam Prasad.K

Presenter: Babu Prasad M

Ideals and direct Products of zero square near rings

121

16 A. Anjaneyulu V.S.R.& N.V.R College, Tenali Presenter: Dr.A.Anjaneyulu

Pseudo Symmetric Ideals of a semigroup

122

17 D. Madhusudhana Rao V.S.R.& N.V.R College, Tenali Presenter: D. Madhusudhana Rao

N(A) - Semigroups 154

18 A. Gangadhara Rao V.S.R.& N.V.R College, Tenali Presenter: A. Gangadhara Rao.

Pseudo Integral Semigroups 154

19 B.Re.Victor Babu and

K. Rajyalakshmi,

Presenter: K. Rajyalakshmi

Second Order Response Surface Model with Neighbor Effects

155

20 Mr Mallikharjun Bhavanari

VIT University, Vellore, Tamilnadu Presenter; Mallikharjun Bhavanari

Some Concepts of Graph Theory Applied in Electronics

157

21 At the end: Invitation, Progremme

sheets and some other paper

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P.,

(July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 01-15

1

“ΓΓΓΓ-NEAR-RINGS”

--------------------------------------------------------------------------------------------------------------------------------

The concept Γ-ring, a generalization of a ring was introduced by Nobusawa [1] and generalized by Barnes [1]. A generalization of both the concepts near-ring and the

gamma-ring, namely Γ-near-ring was introduced by Satyanarayana [1] and later studied by several authors like Booth [ 1, 2, 3 ], Booth & Greonewald [ 1, 2, 3 ], Jun , Sapanci & Ozturk [ 1 ], Satyanarayana [ 1, 2, 3, 4 ], Satyanarayana & Syam Prasad [ 1 ], Selvaraj & George [ 1, 2 ] , Syam Prasad [ 1 ], Syam Prasad & Satyanarayana [ 1 ], Mustafa, & Mehmet Ali [ 1 ]

1. Fundamental Definitions & Results

1.1 Definition (Satyanarayana [1]): Let (M, +) be a group (not necessarily Abelian) and Γ

be a non-empty set. Then M is said to be a ΓΓΓΓ-near-ring if there exists a mapping

M × Γ × M → M (the image of (a, α, b) is denoted by aαb), satisfying the following conditions:

(i) (a + b)αc = aαc + bαc; and

(ii) (aαb)βc = aα(bβc) for all a, b, c ∈ M and α, β ∈ Γ.

M is said to be a zero-symmetric Γ-near-ring if aαo = o for all a ∈ M and

α ∈ Γ, where o is the additive identity in M.

A natural Example of Γ-near-ring is given below:

1.2 Example (Satyanarayana [3]): Let (G, +) be a group and X be a non-empty set. Let

M = {f / f:X → G}. Then M is a group under point wise addition.

If G is non-abelian, then (M, +) is non - abelian. To see this, let a, b ∈ Γ such that

a + b ≠ b + a. Now define fa, fb from x to G by fa(x) = b for all x ∈ X. fa, fb ∈ M and

fa + fb ≠ fb + fa. Thus if, G is non-abelian, then M is also non-abelian.

Let Γ be the set of all mappings of G into X. If f1, f2 ∈ M and g ∈ Γ, then,

obviously, f1gf2 ∈ M. For all f1, f2, f3 ∈ M and g1, g2 ∈ Γ, it is clear that i) (f1gf2)g2f3 = f1g1(f2g2f3); and ii) (f1+f2)g1f3 = f1g1f3 + f2g1f3.

But f1g1(f2 +f3) need not be equal to f1g1f2 + f1g1f3. To see this, fix o ≠ z ∈ G and u ∈ X.

Define Gu: G → X by gu(x) = u for all x ∈ G and fz:X → G by fz(x) = z for all x ∈ X.

Now for any two elements f2, f3 ∈ M, consider

fzgu(f2+ f3) and fzguf2 + fzguf3. For all x ∈ X [fzgu(f2+ f3)] (x) = fz[gu(f2(x) + f3(x))] = fz(u) = z and

Keynote Address _____________________________________________________________

Prof. Dr Bhavanari Satyanarayana AP Scientist Awardee (by Govt. of India, 2009)

Fellow, AP Akademy of Sciences, 2011 Glory of India Awardee, and

International Achievers Awardee (Thailand, March 2011) TOP 100 PROFESSIONALS (IBC, England, 2011)

Acharya Nagarjuana University, Nagarjuna Nagar 522 510 .



2

[fzguf2 + fzguf3](x) = fzguf2(x) + fzguf3(x) = fz(u) + fz(u) = z + z. Since z ≠ o, we have

z ≠ z + z and hence fzgu(f2+ f3) ≠ fzguf2 + fzguf3. Now we have the following:

If (G, +) is non-abelian and X is a non-empty set then M = { f / f: X → G } is a non-abelian group under pointwise addition and there exists a mapping

M × Γ × M → M

Where Γ = { g / g: G → X } satisfying the following conditions: i) (f1gf2)g2f3 = f1g1(f2g2f3); and ii) (f1+f2)g1f3 = f1g1f3 + f2g1f3

for all f1, f2,f3 ∈ M and for all g1, g2 ∈ Γ. Therefore M is a Γ-near-ring.

1.3 Definition (Satyanarayana [3]): Let M be a Γ-near-ring. Then a normal subgroup I of (M, +) is called

(i) a left ideal if aα(b + i) - aαb ∈ I for all a, b ∈ M, α ∈ Γ and i ∈ I;

(ii) a right ideal if iαa ∈ I for all a ∈ M, α ∈ Γ, i ∈ I; and (iii) an ideal if it is both a left and a right ideal.

1.4 Definition: (Satyanarayana [3]): An ideal A of M is said to be prime if B and C are

ideals of M such that BΓC ⊆ A implies B ⊆ A or C ⊆ A.

1.5 Definition (Satyanarayana [3]): Let M1 and M2 be Γ-near-rings. A group

homomorphism f of (M1, +) into (M2, +) is said to be a ΓΓΓΓ-homomorphism if

f(xαy) = f(x)αf(y) for all x, y ∈ M and α ∈ Γ.

We say that f is a Γ-isomorphism f is one-one and onto.

For an ideal I of a Γ-near-ring, the quotient Γ-near-ring M/I defined as usual.

1.6 Theorem (Satyanarayana [3]): Let I be an ideal of M and f, the canonical group

epimorphism of M onto M/I. Then f is a Γ-homomorphism of M onto M/I with kernal I.

Conversely if f is a Γ-epimorphism of M1 onto M2 and I is the kernal of f then M1/I is isomorphic to M2.

1.7 Theorem (Satyanarayana [3]): Let f be a Γ-homomorphism of M1 onto M2 with Kernal I and J*, a non-empty subset of M2. Then J* is an ideal of M2 if and only if f-1(J*) = J is an ideal of M1 containing I. In this case we have M1/J, M2/J* and

(M1/I)/ (J/I) are Γ-isomorphic.

1.8 Example (Satyanarayana [3]): Let G be non-trivial group and X be a non-empty set.

If M is the set of all mappings from X into G and Γ be the set of all mappings from G into

X, then M is a Γ-near-ring. Let y be a non-zero fixed element of G. Define φ: X → G by

φ(x) = y for every x ∈ X. Then o ≠ φ ∈ M, where o is the additive identity in M and

φgo = φ ≠ o for any g ∈ Γ. Therefore M is a Γ-near-ring, which is not zero symmetric.

1.9 Notation: For any two subsets A, B of M the set {aαb | a∈A, α∈Γ, b∈B} is denoted

by either AB or AΓB. {x∈A| x∉B} is denoted by A \ B. For any subset X of M, the smallest ideal containing X is denoted by <X>. If X = {a} then <X> is denoted by <a>.



3

2. The f-Prime Radical in ΓΓΓΓ-Nearrings

Satyanarayana [3] introduced the concepts of f-prime ideal and f-prime radical in Γ-near-rings, and obtained a characterization of f-prime radical in terms of f-strongly nilpotent elements.

Throughout this section f stands for a mapping from M into the set of all ideals of M, satisfying the following conditions:

(i) a ∈ f(a);

(ii) x ∈ f(a) + A, A is an ideal ⇒ f(x) ⊆ f(a) + A

Such type of mappings may be called as ideal mappings. A natural example for this is

given here. Let M be a Γ-near-ring and Q ⊆ M. Define, for each a ∈ M, f(a) = <{a} U Q>, the ideal generated by the union of Q and {a}. Then f satisfies the above two conditions, and hence f is an ideal mapping.

2.1 Definition (Satyanarayana [4]): A subset H of M is said to be

(i). an m-system if, for every h1, h2 ∈ H there exist h11 ∈ <h1 > and h1

2 ∈ < h2>, α ∈ Γ

such that h11αh1

2 ∈ H; (ii). an f-system if H contains an m-system H*, called a kernal of H, such that, for every

h ∈ H, f(h) ∩ H* ≠ φ. In this case we write that H(H*) is an f-system.

2.2 Definition (Satyanarayana [4]): An ideal A of M is said to be

(i) Prime if B and C are ideals of M such that BΓC ⊆ A ⇒ B ⊆ A or C ⊆ A. (ii) f-prime if M\A is an f-system.

2.3 Note (Satyanarayana [4]) The following statements are clear. (i) A is a prime ideal if and only if M\A is an m-system; (ii) Every m-system is an f-system.

(iii) A is a prime ideal ⇒ M\A is an m-system ⇒ M\A is an f-system ⇒ A is f-prime (iv) Every f-prime ideal need not be a prime ideal.

2.4 Example (Satyanarayana [4]): Let N1 be a near-ring with a non-nilpotent element x.

Let N2, N3 be near-rings. Consider M = N1 ⊕ N2 ⊕ N3 , the near ring which is the direct

sum of N1, N2, N3. Write Γ = {.}, where “.” is the product in M. Now, M is a Γ-near-ring

and Ii = Ni, 1 ≤ i ≤ 3 are ideals of M. Write S* = {x, x2, x3, ...} and f(a) = <{a, x}> for all

a ∈ M. Now S* is an m-system, S* ⊆ M\I2 and M\I2 is an f-system with kernal S*.

Therefore I2 is an f-prime ideal. But I2 is not a prime ideal because I1 ⊄ I2, I3 ⊄ I2 and I1

I3 ⊆ I2 . Hence, in general, every f-prime ideal need not be a prime ideal.

2.5 Definitions (Satyanarayana [4]): (i) A subset H of M is said to be nilpotent if

Hn = {0} (that is, HΓ HΓ...H = {0} for some integer n ≥ 2.

(ii) An element a ∈ M is said to be nilpotent if {a}n = 0, that is, (a Γ)n-1a = {0} for some

n ≥ 2. (iii) A subset H of M is said to be nil if every element of H is nilpotent.

(iv) An element a ∈ M is said to be f-nilpotent (resp. f-nil) if f(a) is nilpotent (resp. nil). (v) A subset H of M is said to be f-nil if every element of H is f-nilpotent.



4

2.6 Remark (Satyanarayana [4]) Let a ∈ M and H ⊆ M. Then the following holds:

(i) a is f-nilpotent ⇒ a is f-nil ⇒ a is nilpotent;

(ii) H is f-nilpotent ⇒ H is f-nil ⇒ H is nil;

(iii) H is f-nilpotent ⇒ H is nilpotent ⇒ H is nil.

2.7 Examples (Satyanarayana [4]) (i) Let N be a near-ring with x, y ∈ N such that x is

nilpotent and y is not nilpotent. Now, M = N is a Γ -near-ring with Γ = {.}. Write

f(a) = <{a, y}> for all a ∈ M. Now, y ∈ f(a), y is not nilpotent and so f(a) is not nil. So x is not f-nil but it is nilpotent. (ii) If Q is an ideal of N which is nil but not nilpotent, then define f(a) = <({a}U Q)> for

all a ∈ M. For any q ∈ Q, we have f(q) = Q and so Q is f-nil but not f-nilpotent.

2.8 Lemma (Satyanarayana [4]) Let P be an ideal of M. Then, for any two subsets A

and B of M, we have (A +P)Γ(B + P) ⊆ AΓB + P.

2.9 Lemma (Satyanarayana [4]) Let S (S*) be an f-system in M and let A be an ideal in M which does not meet S. Then A is contained in a maximal ideal P which does not meet S. Let ideal P necessarily be an f-prime ideal.

2.10 Definition (Satyanarayana [4]) The f-radical (denoted by f-rad (A)) of an ideal A is defined to be the set of all elements a of M with the property that every f-system which contains ‘a’ contains an element of A.

2.11 Theorem (Satyanarayana [4]) The f-radical of an ideal A is the intersection of all f-prime ideals containing A.

2.12 Definition (Satyanarayana [4]) Let A be an ideal of M. An element a in M is said to be strongly nilpotent modulo A if, for every sequence x1, x2 , . . . of elements of M such

that x1 = a and xi = xi-11 αi-1 x

*i-1 ∈ <xi-1>, there exists an integer k such that xs ∈ A for

s ≥ k. An element a ∈ M is said to be strongly nilpotent if it is strongly nilpotent modulo (0).

An element x ∈ M is said to be f-strongly nilpotent modulo A if every element of f(x) is strongly nilpotent modulo A.

x ∈M is said to be strongly nilpotent if every element of f(x) is strongly nilpotent. It is clear that every f-strongly nilpotent element is strongly nilpotent. The following example establishes that the converse is not true.

2.13 Example (Satyanarayana [4]) Let N be a near-ring such that (0) does not equal the

prime radical of N ≠ N. Let x ∈ N \ (prime radical of N). We consider M = N as a

Γ-near-ring with Γ = {.}. Write f(a) = <{a, x}> for every element a ∈ N. Now by a known result, we get that x is not strongly nilpotent.

Since x ∈ f(a) for all a, we have that no element of N is f-strongly nilpotent, where all elements of the prime radical of N are strongly nilpotent.

2.14 Lemma (Satyanarayana [4]) Let a1 , a2 , . . . be a sequence of elements of M with

ai = a1i-1α i-1a

*i-1, for some α i-1 ∈ Γ and a1

i-1, a*

i-1 ∈ <ai-1>.

Then {ai | i ≥ 1} is an m-sequence.

2.15 Theorem (Satyanarayana [4]) f-rad M = {x ∈ M| x is f-strongly nilpotent} U {0}.



5

2.16 Theorem (Satyanarayana [4]) If A is an ideal of M, then

f-rad (A) = {x ∈ M | x is f-strongly nilpotent modulo A} U A. Some aspects of radical theory (Jocobson radical type, etc) were studied by Booth [ 1, 2, 3] and Booth & Gronewald [ 1, 2, 3].

3. Fuzzyness in ΓΓΓΓ-Near-Rings

The concept of Fuzzy ideal of a near-ring was introduced by Abou-Zaid [1] and later studied it was studied by Datta & Biswas [1]. Jun, Sapanci and Ozturk [1] intoruced

the concept of “fuzzy ideal” in Γ-near-rings and studied some fundamental properties.

Henceforth, M stands for a zero-symmetric Γ-near-ring.

3.1 Definition: Let µ: M → [0, 1]. Then µ is said to be a fuzzy ideal of M if it satisfies the following conditions:

(i) µ(x + y) ≥ min{µ(x), µ(y)};

(ii) µ(-x) = µ(x);

(iii) µ(x) = µ(y + x – y);

(iv) µ(xαy) ≥ µ(x); and

(v) µ{(xα(y + z) – xαy} ≥ µ(z) for all x, y, z ∈ M and α ∈ Γ.

3.2 Proposition (Jun, Sapanci & Ozturk [1]): Let µ be a fuzzy subset of M. Then the

level subsets µt = { x ∈ M / µ(x) ≥ t }, t ∈ im µ, are ideals of M if and only if µ is a fuzzy ideal of M.

3.3 Note (Satyanarayana & Syam Prasad [1]):

(i) If µ is a fuzzy ideal of M then µ(x + y) = µ(y + x) for all x, y ∈ M.

(ii) If µ is fuzzy ideal of M then µ(o) ≥ µ(x) for all x ∈ M.

Verification: (i) Put z = x + y. Now µ(x + y) = µ(z) = µ( -x + z + x) (since µ is a

fuzzy ideal) = µ( -x + x + y + x) = µ(y + x).

(ii) Clearly o = oαx for all α ∈ Γ and x ∈ M.

This implies µ(o) = µ(oαx). Consider µ(o). Now

µ(o) = µ {oα(o + x) – oαo} ≥ µ(x) (since µ is a fuzzy ideal of M).

Therefore µ(o) ≥ µ(x) for all x ∈ M.

3.4 Lemma (Satyanarayana & Syam Prasad [1]): Let µ be a fuzzy ideal of M. If

µ(x – y) = µ(o) then µ(x) = µ(y) for all x, y ∈ M.

3.5 Proposition (Th. 2.2 of Syam Prasad & Satyanarayana [1]): Let I be an ideal of a

Γ-near-ring M and t < s in [0, 1]. Then the fuzzy subset µ defined by



6

∈ =)(µ

otherwiset

I x if sx is a fuzzy ideal of M.

3.6 Definition: Let X and Y be two non-empty sets and f: X → Y. Let µ and σ be fuzzy

subsets of X and Y respectively. Then f(µ), the image of µ under f, is a fuzzy subset of Y

defined by (f(µ))(y)

=

≠= =

φ

φ

(y)f if 0

(y)f if µ(x)sup

1-

-1

y f(x)

and f-1(σ), the pre-image of σ under f, is a fuzzy subset of X defined by

(f -1(σ))(x) = σ(f(x)) for all x ∈ X.

3.7 Lemma (Syam Prasad [1]): Let M and M1 be two Γ-near-rings and f: M → M1 be a

Γ-near-ring homomorphism. If f is surjective and µ is a fuzzy ideal of M, then so is f(µ).

If σ is a fuzzy ideal of M1 then f -1(σ) is a fuzzy ideal in M.

3.8 Proposition (Syam Prasad [1]): Let M and M1 be two Γ-near-rings, h: M → M1 be an

Γ-epimorphism and µ, σ be fuzzy ideals of M and M1 respectively; then

(i) h(h-1(σ)) = σ;

(ii) h-1(h(µ)) ⊇ µ; and

(iii) h-1(h(µ)) = µ if µ is constant on ker h.

3.9 Definition: Let µ and σ be two fuzzy subsets of M. Then the fuzzy subset σoτ of M, defined by

(σoτ)(x) = z yx

supα=

{min (σ(y), τ(z))}

if x is expressible as a product x = yαz for some α ∈ Γ.

= 0, otherwise, for all x, y, z ∈ M.

4. Fuzzy Cosets in ΓΓΓΓ-Near-rings

4.1 Definition (Def. 2.1 of Satyanarayana & Syam Prasad [1]): Let µ be a fuzzy ideal of

a Γ-near-ring M and m ∈ M. Then a fuzzy subset m + µ defined by

(m + µ)(m1) = µ(m1 – m) for all m1 ∈ M, is called a fuzzy coset of the fuzzy ideal µ.

4.2 Proposition (Lemma 2.2 (i) of Satyanarayana & Syam Prasad [1]): If µ is a fuzzy

ideal of M. Then x + µ = y + µ if and only if µ(x – y) = µ(0).

4.3 Corollary (Lemma 2.2 (ii) of Satyanarayana & Syam Prasad [1]): If x + µ = y + µ

then µ(x) = µ(y).

4.4 Proposition (Lemma 2.2 (v) of Satyanarayana & Syam Prasad [1]): Every fuzzy coset

of a fuzzy ideal µ of M is constant on every coset of ordinary ideal Mµ where



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Mµ = { x ∈ M / µ(x) = µ(0) }.

4.5 Corollary (Lemma 2.2 (vi) of Satyanarayana & Syam Prasad [1]): If z ∈ Mµ then

(x + µ)(z) = µ(x).

4.6 Theorem (Th. 2.4 of Satyanarayana & Syam Prasad [1]): Let µ be a fuzzy ideal of M.

Then the set of fuzzy cosets of µ is a Γ-near-ring with respect to the operations defined by

(x + µ) + (y + µ) = (x + y) + µ ; and

(x + µ)α(y + µ) = xαy + µ for all x, y ∈ M and α ∈ Γ.

4.7 Proposition (Lemma 2.6 of Satyanarayana & Syam Prasad [1]): Let µ be a fuzzy

ideal of M; the fuzzy subset θµ of M/µ, is defined by θµ(x + µ) = µ(x) for all x ∈ M, is a

fuzzy ideal of M/µ.

4.8 Theorem (Th. 3.3 of Satyanarayana & Syam Prasad [1]): If µ is a fuzzy ideal of M

then the map θ: M → M/µ defined by θ(x) = x + µ, x ∈ M, is a Γ-near-ring homorphism

with kernal Mµ ={ x ∈ M / µ(x) = µ(0) }.

4.9 Theorem (Th. 3.3 of Satyanarayana & Syam Prasad [1]): The Γ-near-ring M/µ is

isomorphic to the Γ-near-ring M/Mµ. The isomorphic correspondence is given by

x + µ : → x + Mµ.

4.10 Lemma (Lemma 3.5 of Satyanarayana & Syam Prasad [1]): Let µ and σ be two

fuzzy ideals of M such that σ ⊇ µ and σ(0) = µ(0). Then the fuzzy subset θσ of M/µ

defined by θσ(x + µ) = σ(x) for all x ∈ M is a fuzzy ideal of M/µ such that θσ ⊇ θµ.

4.11 Notation: The fuzzy ideal θσ of M/µ is denoted by σ/µ.

4.12 Lemma (Lemma 3.7 of Satyanarayana & Syam Prasad [1]): Let µ be a fuzzy ideal

of M and θ be a fuzzy ideal of M/µ such that θ ⊇ θµ. Then the fuzzy subset σθ of M

defined by σθ(x) = θ(x + µ) for all x ∈ M is a fuzzy ideal of M such that σθ ⊇ µ.

4.13 Theorem (Th. 3.9 of Satyanarayana & Syam Prasad [1]): Let µ be a fuzzy ideal of M. There exist an order preserving bijective correspondence between the set P of all

fuzzy ideal of σ of M such that σ ⊇ µ and σ(0) = µ(0) and the set θ of all fuzzy ideal θ of

M/µ such that θ ⊇ θµ.

4.14 Proposition (Th. 3.11 of Satyanarayana & Syam Prasad [1]): Let h: M → M1 be an

epimorphism and σ is a fuzzy ideal of M1 such that µ = h-1(σ). Then the map

ψ: M/µ → M1/σ defined by ψ(x + µ) = h(x) + σ is a Γ-near-ring isomorphism.

5. Fuzzy Prime ideals of ΓΓΓΓ-near-rings

5.1 Definition (Def. 2.1 of Syam Prasad & Satyanarayana [1]): A fuzzy ideal µ of M is



8

said to be a fuzzy prime ideal of M if µ is a not a constant function; and for any two

fuzzy ideals σ and τ of M, σοτ ⊆ µ implies that either σ ⊆ µ or τ ⊆ µ.

5.2 Theorem (Th. 2.3 of Syam Prasad & Satyanarayana [1]): If µ is a fuzzy prime ideal

of a M then Mµ = {x ∈ M / µ(x) = µ(0)} is a prime ideal of M.

5.3 Proposition (Syam Prasad [1]): Let I be an ideal of M and α ∈ [0, 1). Let µ be a

fuzzy subset of M, defined by µ(x) = ∈

otherwise s

I x if 1. Then µ is a fuzzy prime ideal of

M if I is a prime ideal of M.

5.4 Corollary (Syam Prasad [1]): Let I be an ideal of M. Then λI is a fuzzy prime ideal of M if and only if I is a prime ideal of M.

5.5 Lemma (Lemma 2.6 of Syam Prasad & Satyanarayana [1]): If µ is a fuzzy prime

ideal of M, then µ(0) = 1.

5.6 Proposition (Th. 2.7 of Syam Prasad & Satyanarayana [1]): If µ is a fuzzy prime ideal

of M, then |Im µ| = 2.

6. Completely Prime ideals in Gamma Near-rings.

Throughout this section we consider only zero-symmetric right near-rings, and M denotes

a Γ-near-ring. 6.1 (Definition 2.1 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let γ ∈ Γ. A γ-ideal I of M is said to be

(i) γ-completely prime if a, b ∈ M, aγb ∈ I ⇒ a ∈ I or b ∈ I.

(ii) γ-completely semi-prime if a ∈ M, aγa ∈ I ⇒ a ∈ I. 6.2 ( Note 2.1 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let M be a Γ-near-ring and γ ∈ Γ. Write N = M. Now (N, +, *γ) is a near-ring. Let

I be a γ-ideal of M.

(i) I is a γ-completely prime γ-ideal of M if and only if I is a completely prime ideal of the near-ring (N, +, *γ).

(ii) I is a γ-completely semi-prime γ-ideal of M if and only if I is a completely semi-prime ideal of the near-ring (N, +, *γ).



9

6.3 (Remark 2.3 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Every γ-completely prime γ-ideal of M is a γ-completely semi-prime γ-ideal of M. 6.4 (Corollary 2.4 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let M be a Γ-near-ring, γ ∈ Γ and A be a γ-ideal of M. Then A is

γ-completely semi-prime γ-ideal if and only if A is the intersection of γ-completely prime

γ-ideals of M containing A. 6.5(Definition 2.5 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let A be a proper ideal of M. The intersection of all γ-completely prime γ-

ideals of M containing A of M, is called as the γ-completely prime radical of A and it is

denoted by C-γ-rad(A). The γ-completely prime radical of M is defined as the γ-

completely prime radical of the zero ideal, and it is denoted by C-γ-rad(M). 6.6 (Note 2.6 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): From Theorem 1.16, and Theorem 6.4 we conclude the following:

(i) An ideal A of a near-ring is completely semi-prime ⇔ A = C-rad(A).

(ii) A γ-ideal A of a Γ-near-ring M is γ-completely semi-prime ⇔ A = C-γ-rad(A). 6.7(Definitions 2.7 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): (i). A γ-ideal P of a Γ-near-ring M is said to be a γ-prime γ-ideal of M (with

respect to γ ∈ Γ) if AγB ⊆ P for any two γ-ideals A, B of M implies A ⊆ P or B ⊆ P.

(ii). A γ-ideal S of a Γ-near-ring M is said to be a γ-semi-prime γ-ideal of M (with

respect to γ ∈ Γ) if AγA ⊆ S for any γ-ideal A of M implies A ⊆ S. 6.8 (Note 2.8 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let P be an γ-ideal of a Γ-near-ring M and γ ∈ Γ. Then we have the following:

(i). P is a γ-prime γ-ideal of the Γ-near-ring M ⇔ P is a prime ideal of the near-ring

(M, +, *γ).

(ii). P is a γ-semi-prime γ-ideal of the Γ-near-ring M ⇔ P is semi-prime ideal of the near-

ring (M, +, *γ).

(iii).Suppose that S is a γ-ideal of M. Then (by Theorem1.15) we have that S is

γ-semi-prime γ-ideal of M ⇔ S is the intersection of all γ - prime ideals P of M containing S. The following corollary follows from Theorem 1.17.



10

6.9 (Corollary 2.9 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]):Corollary: A γ-ideal P of a Γ-near-ring M is γ-prime and γ-completely semi-

prime ⇔ it is γ-completely prime. 6.10 (Definitions 2.10 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let I be a γ-ideal of a Γ-near-ring M for γ ∈ Γ.

I is called a minimal γ-prime (γ-Completely Prime, respectively) γ-ideal of M if it is

minimal in the set of all γ-prime (γ-Completely Prime, respectively) γ-ideals containing I. The following corollary follows from Theorem 1.18. 6.11 (Corollary 2.11 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let P be a γ-ideal of a Γ-near-ring M for γ ∈ Γ. Every minimal γ-prime

γ-ideal P of a γ-completely semi-prime γ-ideal I is a γ-completely prime γ-ideal. More

over P is a minimal γ-completely prime γ-ideal of I. The following corollary follows from Theorem 1.19. 6.12 (Corollary 2.12 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let γ ∈ Γ. If I is γ-completely semi-prime γ-ideal of M, then I is the

intersection of all minimal γ-completely prime γ-ideals of I. 6.13 (Corollary 2.13 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let γ ∈ Γ and P be a γ-ideal of M. If P is a γ-prime γ-ideal and I is a

γ-completely semi-prime γ-ideal, then P is a minimal γ-prime γ-ideal of I if and only if P

is a minimal γ-completely prime γ-ideal of I.

Let γ ∈ Γ. By applying the Corollary 1.21 to the near-ring (M, +, *γ) we get the following.

6.14 (Corollary 2.14 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let γ ∈ Γ. If I is a γ-completely semi-prime γ-ideal of M, then I is the

intersection of all γ-completely prime γ-ideals of M containing I (that is, I = ∩ {P / P is a

γ-completely prime γ-ideal of M such that I ⊆ M} = C-γ-rad(I)).

6.15 (Example 2.15 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Let us consider the Example 2.11 of Satyanarayana [6]. In this example, (G, +) is the Klein four group where G = {0, a, b, c}. We define multiplication on G as follows:

. 0 a b c

0 0 0 0 0

a a a a a

b 0 a b c

c a 0 c b

This (G, +, .) is a near-ring which is not zero symmetric. The ideal {0, a} is only the nontrivial ideal and also it is completely prime.



11

(i) Write M = G, the Klein four group and G = {0, a, b, c}. Define multiplication on G as

above. If we write Γ = {.}, then M is a Γ-near-ring, which is not a zero symmetric

Γ-near-ring (because aγ0 = a.0 ≠ 0). It is clear that for γ ∈ Γ, the γ-ideal {0, a} of M is

only the nontrivial γ-completely prime γ-ideal. The γ-ideal (0) of M is γ-completely

semi-prime γ-ideal, but not γ-completely prime γ-ideal (because cγa = c.a = 0 and

a ≠ 0 ≠ c). Hence the γ-completely semi-prime γ-ideal (0) can not be written as the

intersection of its minimal γ-completely prime γ-ideals.

From this example 6.15, we can conclude that if M is not a zero symmetric Γ-near-ring, then the corollary 6.14 need not be true.

6.16 (Notation 2.16 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]): Notation: Let A be a γ-ideal of M. The intersection of all γ-prime ideals

containing A is called the γ-prime radical of A and it is denoted by P-γ-rad(A).

The γ-prime radical of M is defined as the γ-prime radical of the zero ideal (0).

So P-γ-rad(M) = P-γ-rad(0).

6.17 (Theorem 2.17 of Satyanarayana, Pradeep Kumar, Sreenadh and Eswaraiah Setty [1]):) Let A be an ideal of M. Then

(i). P-γ-rad(A) is a γ-semi-prime γ-ideal.

(ii). The γ-prime radical of M is a γ-semi-prime γ-ideal.

7. Completely Semi-prime ideals in Gamma Near-rings.

7.1 (2.1 Definition of Pradeep Kumar, Satyanarayana, Syam Prasad and Mohiddin shaw [1]) : An ideal A of M is said to be semi-prime if B is an ideal of M such that

BΓB ⊆ A implies B ⊆ A.

7.2 (2.1 Definition of Pradeep Kumar, Satyanarayana, Syam Prasad and Mohiddin shaw [1]): An ideal I of M is said to be completely semi-prime ideal of M if it satisfies

the following condition: aΓa ⊆ I ⇒ a∈ I .

7.3 (2.2 Definitions of Pradeep Kumar, Satyanarayana, Syam Prasad and Mohiddin shaw [1]: (i) An element a in M is said to be a nilpotent element, if there exists a positive

integer n such that (aΓ)na = aΓaΓa...Γa = 0. (ii) An ideal A of M is said to be a nilpotent ideal, if there exists a positive integer n such

that (AΓ)nA = AΓAΓA...ΓA = 0. We denote the sum of all nilpotent ideals of M by SN(M).

7.4 (2.3 Lemma of Pradeep Kumar, Satyanarayana, Syam Prasad and Mohiddin shaw [1]): (i) If J ⊆ M and J2 ⊆ I , I is completely semi-prime ideal, then J ⊆ I (in particular every completely semi-prime ideal is a semi-prime ideal).

(ii) If I is completely semi-prime ideal of M, then aΓb ⊆ I ⇒ bΓa ⊆ I.

(ii) Suppose I is completely semiprime ideal of M and a, b ∈ M such that aΓb ⊆ I.

7.5 (2.4 Lemma of Pradeep Kumar, Satyanarayana, Syam Prasad and Mohiddin shaw [1]): (i) If a ∈ M and I is an ideal of M, then (I : a) = {x ∈ M / xΓa ⊆ I} is a left ideal of M.

(ii). If I is a completely semi-prime ideal of M, then (I : a) is an ideal of M.



12

7.6 (2.5 Theorem of Pradeep Kumar, Satyanarayana, Syam Prasad and Mohiddin shaw [1]): If S is a semi-prime ideal of M, then the following are equivalent:

(i) If xΓx ⊆ S, then <x>Γ<x> ⊆ S. (ii) S is completely semi-prime ideal of M.

(iii) If xΓy ⊆ S, then <x>Γ<y> ⊆ S.

.8. Mislaneous concepts on Γ-near-rings

Selvaraj & George [1] introduced the notion of strongly regular 2-primal Γ-near-rings

and studied some characterizations of 2-primal and strongly 2-primal Γ-near-rings.

Selvaraj & George [2] gave some characterizations of left strongly regular Γ-near-rings.

Also proved that in a weakly left duo Γ-near-rings N, N is left weakly π-regular if and

only if N is left strongly π-regular.

Mustafa Uckun and Mehmet Ali Ozturk [1] studied the notion of symmetric bi

Γ-Derivations, symmetric bi generalization Γ-Derivations in Γ-near-rings.

8.1 Definition: Let M be a Γ-near-ring and D(⋅, ⋅) a symmetric bi-additive mapping of

M. D(⋅, ⋅) is said to be a symmetric bi-ΓΓΓΓ-derivation if D(xγy, z) = D(x, z)γy + xγD(y, z)

for all x, y, z ∈ M and γ ∈ Γ. Then, for any y ∈ M, a mapping x|→ D(x, y) is a

Γ-derivation.

Considering M as a 2-torsion free 3-prime left gamma-near-ring with multiplicative centre C, Mustafa Uckun and Mehmet Ali Ozturk [1] studied the trace of symmetric bi-gamma-derivations (also symmetric bi-generalized gamma-derivations) on M.

8.2 Theorems (Mustafa Uckun and Mehmet Ali Ozturk [1]): Let D(.,.) be a non-zero symmetric bi-gamma-derivation of M and F(.,.) a symmetric bi-additive mapping of M. Let d and f be traces of D(.,.) and F(.,.), respectively. In this case

(1) If d(M) is a subset of C, then M is a commutative ring.

(2) If d(y), d(y) + d(y) are elements of C(D(x,z)) for all x, y, z in M, then M is a commutative ring.

(3) If F(.,.) is a non-zero symmetric bi-generalized gamma-derivation of M associated with D(.,.) and f(M) is a subset of C, then M is a commutative ring.

(4) If F(.,.) is a non-zero symmetric bi-generalized gamma-derivation of M associated with D(.,.) and f(y), f(y) + f(y) are elements of C(D(x,z)) for all x, y, z in M, then M is a commutative ring.

Acknowledgements The author thank the authorities of JMJ College, Tenali for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra and its Applications (Sponsored by UGC) in the College. He also thanks Dr A. V. Vijaya Kumari for inviting me to present this talk at the National Seminar.



13

References Barnes W. E. [ 1 ] “On the Γ-rings of Nobusawa”, Pacific J. Math., 18 (1966) 411-422.

Booth G. L [ 1 ] “A Note on Γ-near-rings” Stud. Sci. Math. Hunger, 23 (1988) 471-475.

[ 2 ] “Radicals of Γ-near-rings” Publ. math Debrecen, 37 (1990) 223-230.

[ 3 ] “Radicals of Γ-near-rings” Questiones Mathematicae 14 (1991) 117-127.

Booth G. L & Greonewald N. J [ 1 ] “On Radicals of Gamma Near-rings”, Math. Japanica 35 (2) (1990) 417 - 425.

[ 2 ] “Equiprime Γ-near-rings”, Q.M. 14 (1991) 411-417.

[ 3 ] “Matrix Γ-near-rings” Math. Japanica 38 (5) (1993) 973 - 979

Dutta T. K. & Biswas B. K [ 1 ] “Fuzzy Ideals of Near-rings”, Bull. Cal. Math. Soc. 89 (1997) 447-456.

Jun Y. B., Sapanci M., & Ozturk M. A. [ 1 ] “Fuzzy Ideals of Gamma Near-rings”, Tr. J of Mathematics, 22 (1998) 449-459.

Nobusawa N [ 1 ] “On a generalization of the ring theory”, Osaka J. Maths, 1 (1964) 81-89.

Pilz G [ 1 ] Near-rings, North Holland Publ. Co., 1983.

Pradeep Kumar T V. [ 1 ] “Contributions to Near-ring theory - III”, Doctoral Thesis, Acharya Nagarjuna University, 2006.

Pradeep Kumar T.V., Satyanarayana Bhavanari, Syam Prasad K., and Mohiddin Shaw [1] Some results on Completely Semi-prime Ideals in Gamma Near-rings, Proceedings of the National Seminar on Present Trends in Mathematics and its Applications (Editors: Dr Eswaraiah Setty Sreeramula, Prof. Dr Bhavanari Satyanarayana, and Syam Prasad Kuncham), Nov.11-12, 2010, PP 101-105.

Ramakotaiah Davuluri [1] “Theory of Near-rings”, Ph.D. Diss., Andhra univ.,1968.

Salah Abou-Zaid [ 1 ] “On fuzzy subnear-rings and ideals”, Fuzzy Sets and Systems, 44 (1991) 139-146. [2] Sambasivarao.V and Satyanarayana.Bh. “The Prime radical in near-rings”, Indian J.

Pure and Appl. Math. 15(4) (1984) 361-364.

Satyanarayana Bhavanari. [ 1 ] "A Note on Γ-rings", Proceedings of the Japan Academy 59-A (1983) 382-83. [2] “A Note on g-prime Radical in Gamma rings”, Quaestiones Mathematicae, 12 (4) (1989) 415-423. [3] “Contributions to Near-ring Theory”, Doctoral Dissertation, Acharya Nagarjuna University, 1984. Published by VDM Verlag Dr Mullar, Germany, 2009 (ISBN: 978-3-639-22417-7). [4] "The f-prime radical in Γ-near-rings", South-East Asian Bulletin of Mathematics



14

23 (1999) 507-511.

[5] "A Note on Γ-near-rings", Indian J. Mathematics (B.N. Prasad Birth Centenary commemoration volume) 41(1999) 427- 433.

[6] "Modules over Gamma Nearrings" Acharya Nagarjuna International Journal of Mathematics and Information Technology, 01 (2004) 109-120. [7] “A Note on Completely Semi-prime Ideals in Near- rings”, International Journal of Computational Mathematical Ideas, Vol.1, No.3 (2009) 107-112.

Satyanarayana Bhavanari and Mohiddin Shaw Sk [1] “Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978-3-639-23197-7). Satyanarayana Bhavanari and Nagaraju D [1] “Dimension and Graph Theoretic Aspects of Rings (Monograph)” VDM Verlag Dr Muller, Germany, 2011. (ISBN: 978-3-639-30558-6) Satyanarayana Bh, Nagaraju D, Balamurugan K. S & Godloza L [ 1 ] “Finite Dimension in Associative Rings”, Kyungpook Mathematical Journal, 48 (2008), 37-43.

Satyanarayana Bhavanari, Pradeep Kumar T.V., Seenadh Sridharamalle and Eswaraiah Setty Sriramula [1] “On Completely prime and Completely Semi-Prime Ideals in gamma-Near-rings”,

International Journal of Computational Mathematical Ideas, Vol.2, No.1 & 2, 2010.

PP 22-27.

Satyanarayana Bhavanari, Pradeep Kumar T.V. and Srinivasa Rao M [1] “On Prime left ideals in Γ-rings”, Indian J. Pure & Appl. Mathematics 31 (2000)

687-693.

Satyanarayana Bhavanari and Rama Prasad J.L [1] “Prime Fuzzy Submodules”, VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978- 3-639-24355-0).

Satyanaryana Bhavanari & Richard Wiegandt [1] "On the f-prime Radical of Near-rings", in the book Nearrings and Nearfields (Edited by H. Kiechel, A. Kreuzer & M.J. Thomsen) (Proc. 18th International Conference on Nearrings and Nearfields, Universitat Bundeswar, Hamburg,

Germany, July 27-Aug 03, 2003) Springer Verlag, Netherlands, 2005, pp 293-299.

Satyanarayana Bh. & Syam Prasad K. [1] "On Fuzzy Cosets of Gamma Nearrings", Turkish J. Mathematics 29 (2005) 11-22. [2] “Discrete Mathematics and Graph Theory”, Printice Hall of Inida, New Delhi, 2009. (ISBN:978-81-203-3842-5). Satyanarayana Bh., Syam Prasad K., Pradeep Kumar T. V., and Srinivas T. [1] “Some Results on Fuzzy Cosets and Homomorphisms of N-groups”, East Asian Math. J. 23 (2007) 23-36.

Selvaraj C. & George R.



15

[1] “On Strongly 2-Primal Γ-near-rings”, submitted.

[2] “On Strongly Regular Γ-near-rings”, submitted.

Syam Prasad K. [1] “Contributions to Near-ring theory - II”, Doctoral Thesis, Acharya Nagarjuna University, 2000.

Syam Prasad K. & Satyanarayana Bh. [ 1 ] "Fuzzy Prime Ideal of a Gamma Nearing", Soochow J. Mathematics 31 (2005) 121-129.

Uçkun, Mustafa, & Öztürk, Mehmet Ali,

[ 1 ] “On Trace of Symmetric Bi-Gamma-Derivations in Gamma-Near-Rings”, Houston Journal of Mathematics, 33 (2) (2007) 323-339.

Some more References:

C.Selvaraj and R.George, [1] “On Strongly prime gamma-near rings”, Tamkang Journal of Mathematics, Vol. 39, 1(2008) PP 33 – 43.

C.Selvaraj and R.George and G.L. Booth, [1] “On Strongly Equi prime gamma-near rings”, Bulletin of the Institute of Mathematics, Academia Sincia (New series) Vol.4 (2009) No.1, PP 35 – 46.

G.F. Birkenmeir, G.L. Booth and N.J. Groenewald [1] “Lattices of radicals of Near-rings”, Communications in Algebra, 29:8, PP 3593-3604.

Mustafa ASCI [1] “Γ-(σ, τ) Derivations on Gamma Near-rings”, International Mathematical Forum, Vol.2, No.3 (2007) PP 97-102.

K.Bave and J.W Park [1] “Γ-Near- Fields and their Characterization by Quasi-ideals”, International Mathematical Forum, Vol.5, No.3 (2010), PP 109-116.

Yong Bae Jun, Kyung Ho Kim and M.A.Ozturk [1] “Fuzzy Maximal Ideals of Gamma Near rings”, Turk. J. Math 25(2001) 457-463.

Yong Bae Jun, Kyung Ho Kim and yong Uk Cho [1] “On gamma derivation in Gamma Nearrings”, Soochow Journal of Mathematics, Vol.29, No.3 (2003) PP 275-282.

Young UK Cho, T. Tamizh Chelvam and N. Meenakumari [1] “P(R, M) Γ-Near-rings”, J. Korea Soc. Math. Educ Ser B. Pure Appl. Math. Vol.13, No.2, (2006) PP 113 -120.

Yong UK Cho, M.A. Oztuturk and Young Bae Jun [1] “Intuitionstic Fuzzy Theory of Ideals in Γ-Near-rings”, International Journal of pure and Applied Mathematics (IJPAM) Vol.17, No.2 (2004).

Yong UK Cho and Young Bae Jun [1] “Γ-Derivations in Prime and Semi-Prime Γ-Near-rings”, Indian Journal of Pure and applied Mathematics 33(10) (2002), PP 1489 – 1494.

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July 11-12,

2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 16-23 Page: 16

COLOR PRESERVING

AUTOMORPHISMS OF

CAYLEY GRAPHS

1. Graph Theory Notions. Definition 1.1: A graph G is an ordered pair (V,E) consisting of a finite set V of elements called

vertices and a set E of unordered pairs of elements of V called edges.

The edge e, determined by the pair of vertices u,v is denoted by e = (u,v),or e = uv. In this case

we say that u and v are the end vertices of e and u and v are incident with the edge e and the edge e is

incident with u and v. Also u and v are said to be adjacent. If the presence of the graph is to be

stressed, the vertex set V and the edge set E of G(V,E) are also denoted by V(G) and E(G)

respectively.

Definition 1.2: The number of edges incident with a vertex v is called the degree of v and it is

denoted by d(v).

The minimum and the maximum degrees of a vertex of a graph are respectively denoted by δ

and ∆.

The number of vertices of a graph G is denoted υ(G), or υ and the number of edges of G is

denoted by ε (G).

Definition 1.3: A vertex v of a graph G which is not adjacent with any other vertex of G is called an

isolated vertex of G.

Definition1.4: For any vertex v of G, if (v,v) is an edge, then it is called a loop. Definition 1.5: Two or more edges are said to be parallel or multiple if both edges have the same

end vertices.

Definition 1.6: A graph G is said to be simple if it has no loops and no multiple edges.

Definition 1.7: A subgraph of a graph G(V,E) is a graph H(U,F) with U ⊆ V and F ⊆ E.

i. If U = V, then H is called a spanning subgraph of G, ii. If H is a subgraph of G, then G is called a super graph of H.

Definition 1.8: i. A walk in a graph G is a finite non – null sequence v0 e1v1e2 . . .vR-1ekvR, whose

terms are alternatively vertices and edges such that for 1≤ i ≤ k, the end vertices of ei are vi – 1 and vi

and it is called a (vo,vk) – walk, vo is called the origin and vk, the terminus of the walk.

ii. A walk is called trail if all the edges of the walk are district.

iii. A trial in which the vertices are also district is called a path. The path determined by the

vertices v1,v2, . . . ,vk is denoted by v1 v2 . . . vk and it is said to be of length k – 1.

iv. A path whose origin and terminus are the same is called a cycle. A cycle v1v2 . . . vkv1 of

length k is denoted by (v1v2 . . . vkv1).

Definition 1.1.9: A graph G is said to be a complete graph if every vertex is adjacent to every other

vertex of G.

Definition 1.10: A graph G is said to be connected if there is atleast one path between every pair of

vertices of G. Otherwise G is said to be disconnected.

A disconnected graph is partitioned into a finite number of connected graphs called its

components.

INVITED TALK

____________________________________________________________________________________

Prof. Dr L.Nagamuni Reddy,

Former Professor of Mathematics, S.V. University, Tirupati – 517 502.



Definition 1.11: A bipartite graph is a graph G whose vertex set can be partitioned into two disjoint

subsets X and Y such that each edge has one end in X and the other in Y.

Definition 1.12: A graph is said to be k – regular, if d(v) = k, for some positive integer k and for

every v in V(G).

A regular graph is one, that is k – regular for some positive integer k.

Definition 1.13: i. A tour, of G is a walk that traverse each edge of G at least once.

ii. An Euler tour is a tour, which travererses each edge exactly one.

iii. A graph is Eulerian if it contains an Euler tour.

Definition 1.14: i. A path that contains every vertex of G is called a Hamilton path. ii. A Hamilton cycle of G is a cycle that contains every vertex of G.

iii. A graph is Hamiltomian if it contains Hamilton path.

2 Cayley Graphs

The theory of groups provides an interesting and powerful abstract approach to the study of

symmetries of various graphs. A new class of graphs, namely, Cayley graphs can be constructed by

making use of a graph.

Definition 2.1: Let (X, °) be a group. A subset S of X is called a symmetric subset if s ε S implies

that s-1 ε S. Definition 2.2: Let (X, °) be a group and S, a symmetric subset of X not containing the identity

element e of X. The graph G whose vertex set is X the edge set E = {(g,gs)/ s ε S}is called the

Cayley graph of X corresponding to the set S and it is denoted by G(X,S).

Remark 2.3: The edge set E = {(g,gs)/s ε S} = {(g,h)/g-1 h ε S}. Observe that this is a directed

graph, By slightly modifying the definition of adjacent in the Definition 1.2.2, namely, E = {(g,h)/g-

1h ε S, or h g-1 ε S}, one can make G(X,S) an undirected graph. Example2.4: Consider the group (S3.°) where the elements of S3 are

=

=

=

=

=

123

321,

231

321,

213

321,

132

321,

321

321δγβαi and

=

312

321θ . The subset S = {α, β}

is a symmetric subset of S3 since α-1 = β and β-1

= α The cayley graph G(S3, S) is given below.

Since iα = αλ, α α = β, α β = i, γα = θ and γ β = δ, there are an edge between i and α, α and β,

α and i, γ and θ and γ and δ respectively. Similarly we can find the other edges and find G(S3,S).

Remark 2.5: Observe that this graph is disconnected and it has two components namely I and II

Example 2.6: Consider the same group S3 and the symmetric subset S1= {α,β,γ}. (Here γ-1 = γ εS

1).

G (S3, S) II I



The cayley graph G(S3,S1) is given below.

Since α, β Є S1, the two components of G(S3,S)are in G (S3, S

1). In addition G (S3, S1)has some

more edges contributed by γ ε S1 the edges.

Since β γ = θ, θ γ = β, there are an edges between β and θ and θ and β. Similarly other edges

can be drawn.

Remark 2.7: Observe that the above graph is connected and the set S1 is a generating set of S3. So

one gets a doubt whether there is any connection between there facts, namely, S1 is a generating set of

S3 and G(S3,S1) is connected. The answer is in the affirmative and we establish this fact in the next

theorem.

Theorem 2.8: Let (X, ·) be a graph and S, a symmetric subset of X. The Cayley graph G(X,S) is

connected if and only if S is a set of generators of X.

Proof: Let the symmetric sub set S of group (X, °) be a set of generators of X. Let g, h be any two

vertices of the Caycle graph G(X,S). Then g, h ε X.

Let k = g-1h. Then (X, ·) being a group, k ε X. Since S is a set of generators of X,

r ε X. Since S is set of generators of X,

k = s1.s2 . . . sm for some s1, s2 . . . , sm ε S.

Put gs1 = g1, g1s2 = g2, . . ., gm – 1sm= gm. Then

gm = gs1s2 . . . sm = gk = h

and (g,g1), (g1,g2), . . . (gm – 1, gm) are respectively (g,gs1), (g1,g1s2), . . ., (gm – 1, gm – 1 sm), which are

edges in G(X,S) with sm = h, since s1,s2, . . . sm ε S. So there is a (g,h) – path, namely, g g1 g2 . . . gm –

1 h in G(X,S). Hence G(X,S) is connected.

Conversely assume that G(X, S) is connected. Let g ε X. Choose an element s ε S. Since

S ⊆ X, k ε X, the vertex set of G(X,S). As G(X,S) is connected and g, s ε X, the vertex set of G(X,S),

there is a (s,g) – path, say,

g0 g1 g2 . . . gm, where gm = g and g0 = s.

Since (gi, gi+1) is an edge in G(X,S),

gi+1 = gi si+1 for some si+1 ε S, 0 ≤ i ≤ m – 1.

G (S3, S1)



So

g = gm = gm – 1 sm = gm – 2 sm – 1 sm = . . .

= s0 s1 s2 . . . sm, s, si ε S, 1 ≤ i ≤ m – 1.

So S generates X.

Theorem 2.9: Let (X, °) be a group and S, a symmetric subset of X. The Cayley graph G(X,S) is |S| -

regular. Moreover the number of edges in G(X,S) is 2

|||| SX.

Proof: Let g be any vertex of the graph G(X,S). By the definition of G(X,S), for every s ε S, (g, gs) is

an edge of G(X,S). Further, since X is a group, if s,t ε S and s ≠ t then g s ≠ gt, so the number of edges

through each vertex of G(X,S) is |S| and thus G(X,S) is |S| - regular.

Since each vertex has degree |S|, the sum of the degree of all the vertices is |X| |S|. However

each edge is associated with two vertices so that number of edges in G(X,S) is 2

|||| SX.

3. Automorphisms of graphs

Definition 3.1: The graphs G[V, E ] and G´ [V �,E΄ ] are said to be isomorphic if there exists a

bijection φ : V→ V' such that φ preserves adjacency and non adjacency. Further φ is said to be a

graph isomorphism of G[V, E] onto G´ [V´, E´].

Definition3.2: An automorphism φ of a graph G[V,E] is a bijection φ : V→ V

which preserves adjacency and non adjacency. (Preserving of non adjacency is not required if G is a

finite graph).

Theorem 3.3: The set A(G) of all automorphisms of a given graph G[V, E] forms group.

Proof: Let α1 and α2 є A(G),the set of all automorphisms of G[V, E] .Since α1 and α 2 are bijections

on V, α1α2 is also a bijection. Let u and v be any two adjecent vertices, then α2(u) and α 2(v) are

adjecent. This implies α 1α2 (u). And α1 α 2(v) are adjecent. Thus α1 α 2 preserves adjacency. Similarly

we can prove that α 1α2 preserves non-adjacency and thus α 1α 2 is in A(G). Associative property

follows from the associative property of composition of mappings and the identity mapping will serve

as the identity element. Let α be any automorphism of G. Since α is a bijection on V, its inverse α-1

(wich is given by α-1(y) = x <=> α(x) = y ), is also bijection on V. To see that α-1 is also an

automorphism, let u, v be any two adjecent vertices. Since α is bijection on V, α(x) = u and α(y) = v,

for some x, y in V. Hence α(x) and α(y) are adjecent. This implies that x and y are adjecent, because α

preserves non-adjacency. That is α-1(u) and α-1 (v) are adjecent. Similarly we can prove that α-1

preserves non-adjacency and hence α-1 is in A(G). Therefore A(G) is a group. The group A(G) is

usually called the automorphism group.

The automorphism group A(G) can be viewed as a permutation group of the vertex set V and

we shall denote this permutation group byAo(G) and call it as the vertex group of G. Infact A(G) and

Ao(G) are one and the same as for as vertex set V is concerned.

Definition 3.4: A graph G is said to be vertex transitive if its automorphism group A(G) acts

transitively on V(G), that is for every pair of vertices v, u ε h there is α ε A(G) such that α (v) = u.

Theorem 3.5: If S isa generating set of X then the Cayley graph G(X, S) is vertex transitive.

Proof: For each g in X, let us define a mapping λg: X→ X such that λg(x) = gx, for every x in X.

Clearly λg is a permutation on X. Let us consider the set L(X)={ λg/ g є X}. It is easy to see that L(X)

is a group with respect to composition of mappings.



Let λk be any element of L(X). Then λk is a permutation on the vertex set X. Further if ‘g’ and

‘h’ are any two adjecent vertices in G(X,S), then g-1h є S or (kg)-1(kh) є S. This implies that λk(g)

and λk(h) are adjecent. Thus λk preserves adjacency. Similarly we can prove that λk-1 is also preserves

adjacency. On the other hand if ‘g’ and ‘h’ are non-adjecent, then λk(g) and λk(h) are also non

adjecent, other wise λk-1 (λk(g)) and λk

-1 (λk(h)) are adjecent, or g and h are adjcent, contrary to the

choice g and h. Thus λk preserves non-adjacency. Therefore λk is an automorphism of the graph

G(X,S) and hence L(X) is a sub group of the automorphism group A(G).

Finally we shall see that L(X) acts transitively on V(G) = X. For this, let g and h are any two

vertices in X. Let us take hg-1 = k, then λk(g) = kg = hg-1 g = h . Hence L(X) acts transitively on V(G).

Thus A(G) has a subgroup L(X) acting transitively on its vertex set V(G) , so that G(X,S) is vertex

transitive.

Corollary 3.6: The automorphism group of Cayley graph has a sub group that acts transitively on its

vertex set V(G).

Proof: In the proof of the theorem 2.3.5, we have observed that the sub-group L(X) acts transitively on V(G). Further it is easy to see that the mapping G → λg is bijection X to L(X). So |L(X)| = |V(G)| and hence L(X) acts transtively on V(G). Theorem 3.7: Let π be an automorphism of the group X such that π (S) = S, then regarded as a

permutation of V(G) = X , π is a graph automorphism of the graph G(X,S) fixing the vertex ‘e’,

where ‘e’ is the identity element in X.

Proof: Let π is a group automorphism of X such that π (S) = S. Clearly π is permutation on X and

fixes the identity element e . Further, if ‘g’ and ‘h’ are any two adjecent vertices in the graph G(X, S),

then g-1hєS or [π (g)]-1 π (h) є S. This implies that π (g) and π (h) are adjecent in G(X,S) and hence π

preserves adjacency. Similarly we can show that π preserves non-adjacency. Therefore π is graph

automorphism.

Remark i : In the theorem 2.3.5 we have seen that L(X) is a sub group of A(G).However

L(X) may not be the whole of A(G). To see this, let us consider the cayley

graph G(S3, S), where S = {(1,2), (1,3), (2,3)}.

By the theorem 3.7, each group automorphism of S3 is also a graph automorphism and fixes the

identity element e in S3. Since all the elements in S form a conjugate class in S3, each group

automorphism π of S3 fixes S that is π (S) = S (refer p. 105 [l]).Thus the stabilizer A(G)e contains at

least 6 elements, because the group of automorphism of S3 contains at least 6 elements. Hence A(G)

is not regular on S3. But from the corollary 3.6, we have L(X) is regular.Thus L(X) is not the whole of

A(G) in all cases.

Remark ii: The converse of the theorem 2.3.6 need not be true in general. However the following

partial converse is true.

Theorem 3.8: Let G be a loopless connected graph. Then its automorphism group A(G) has a sub-

group K acting transitively on V(G) iff G is a Cayley graph G(X, S) for some subset S of K

generating X.

Proof: Let G be a loop less connected graph with vertex set V(G) = {v1,v2 ,. . . vn} such that its

automorphism group A(G) has a subgroup K acting regularly on V(G). Then | K | = n and for each vi,

1≤ i ≤ n , there is a unique ki in K such that ki(v1) = vi , Clearly k1 is the identity element.

Let us consider the set S = {ki / (v1,vi) є E(G)}. Since G has no loops S does not contain the

identity element k1.First we shall show that S is symmetric, that is,S = S-1. From the definition of S, we

have



ki є S => (v1, vi) є E(G) => (vi, v1) є E(G) =>(ki(vl), v1) є E(G) => ki-1( ki (v1), v1) є E(G)

(since ki є A(G)) => (v1, ki-1(v1)) є E(G) => ki

-1 є S Thus S = S-1.

Let us construct the Cayley graph H = G(X, S), where X = K. Define a mappingφ : V(H) →

V(G) that is φ : K → V such that

φ (ki)= vi , i = 1, 2,…, n.

Since v1, v2, ...vn and k1, k2...kn are all distinct, φ is a bijection. Now we shall show that φ is

a graph isomorphism from H to G. For this we have to show that φ preserves adjacency and non

adjacency. To see that φ preserves adjacency, let (ki, kj) є Є(H). Then

(ki, kj) є E(H) => ki-1kj = krє S => (v1,vr)є E(G) => ki(v1, vr) є E(G) (since ki є A(G) =>

(vi, ki (vr)) є E(G) => (vi, kikr(v1) ) є E(G) => (vi, kj(v1) ) є E(G) (since ki-1kj = kr) =>(vi, vj)) є E(G)

Thus φ preserves adjacency.

To prove that φ preserves non-adjacency, it is enough if we prove φ -1 preserves adjacency.

Let, vi , vj є V(G). Then

(vi ,vj) є E(G) => (ki(v1), vj) є E(G) => (v1, ki-1(vj)) є E(G) => (v1, vr) є E(G) where vr = ki

-1 (vj)

But

ki-1,(vj) = vr => ki

-1kj(v1) = kr(vl).

Because K is acting transitively on V(G) and (v1, vr) є E(G), gives us ki-1kj = kr .є S. So (ki , kj) є

E(H) or (φ -1 (Vi), φ-1(vj)) є E(H). Thus, φ -1 preserves adjacency and hence φ is graph isomorphism

of H to G. Since G is connected, the graph H= G(X, S) is also connected and hence by the theorem

2.2, S generates X.

The converse follows from the corollary 3.6.

4. COLOR PRESERVING AUTOMORPHISIMS OF CAYLEY GRAPHS

In this section we introduce the concept of Cayley graph and some of its properties. Definition 4.1: Let X is a group and S a symmetric sub set of X not containing the identity e. Then

the Cayley graph D has the vertex set V=X and the edge

set E={(g, gh)/h є S}. In the Cayley graph D(X,S), (g, k) is an edge if k = gh, for some h in S. Here h

(= g-1k) is called the color of the edge (g, k).We shall make this concept clear by the following

example.

Example 4.2: Let X = S3 = {e, α , α2, τ, ατ, α2 τ}, where α =(l,2,3) and τ=(1,2), and

S={ α , τ }. Then the diagram of graph D(X,S) is shown below. Observe that each edge is indicated by

its corresponding color.



Definition 4.3: An automorphism of G(X,S) is said to be color preserving, if it preserves the color of

the edges, that is , if e is an edge of G(X,S) with color s ε S and α is an automorphism of D such that α

(e) also has color s, then α is called a color preserving automorphism.

Theorem 4.4: An automorphism α of G(X,S) is color preserving if and only if α(gh) = α (g)h, for any

h є S and g є X.

Proof: Suppose α is a color preserving automorphism of G(X,S). Let ‘g’ be any vertex of G(X, S),

and ‘h’ be any element of S. Then (g, gh) is an edge in G(X,S) with color h. Since α is an

automorphism of G (X,S), (α(g), α(gh)) is an edge of G(X,S). But α is color preserving implies that

α (gh) = α (g) h. Hence α (gh) = α (g)h, for any g є X and h є S.

On the other hand let us assume that α is an automorphism of G(X,S) such that

α(gh) = α(g)h,for any gєX and h є S. For any edge (g, k) of D(X , S) with color h , we have k = gh and

α(g)h = α (gh) = α (k). So the edge (α(g), α(k)) is also has the color ‘h’ , and hence α is color

preserving.

The proof the following theorem is immediate from the theorem 4.4.

Theorem 4.5: The set of color preserving automorphisms forms a group with respect to composition

of mappings.

We now prove the main theorem on color preserving maps of the graph G(X, S).

Theorem 4.6: The group of color preserving automorphisms of a Cayley graph G(X, S) is isomorphic

to X, where S is a set of generators of X.

Proof: Let C(G) be the group of all color preserving automorphisms of G(X, S). For each g є X, let

us define αg: X→X such that

αg(x) = gx, for every xє X . . . (1).

Clearly αg is bijection. Let us define L(X)={αg / g є X} and consider the mapping

φ : X → L(X) given by φ (g) = αg, for every g є X. One can easily see that φ is a bijection. Further

for g, h є X and x є X αgh(x) = ghx = gαh(x) = αg(αh(x)) So αgh = αg αh. This gives

φ (gh) = αgαh = φ (g) φ (h), showing that φ is homomorphism and hence an isomorphism. That is,

X ≅ L(X) . . . (2).

Now we shall prove that C(G) = L(X). Let αg є L(X), then αg is bijection on X . Further, if

(gi, gj) be any edge in G(X, S), then gi-1gj є S or gi

-1g-1ggj є S. This implies (g gi )-1( g gj) є S, so

that (g gi, g gj) is an edge in G(X, S). Thus (αg(gi),αg(gj)) is an edge in G(X, S) and hence αg preserves

adjacency. Similarly we can prove that αg preserves non-adjacency. There fore α is an automorphism

of G(X, S). More over , for any h є S and x є X, αg(xh) = gxh = αg(x)h. So αg is color preserving, or

αg є C(G). Thus

L(X) ⊆ C(G) . . . (3).

On the other hand if α є C(G), we shall show that α = αg, for some g є X. For this let us take

α(g1) = g . . . (4).

where g1 is the identity element in X. First we shall show that

α(gi) = ggi, for all gi є X.

To see this, let gi є X. Suppose assume that gi є S. Then (g1, gi) є Є(G), the edge set of

G(X, S), because g1-1gi = gi є S. But α(g1, gi) = (α(g1), α(gi) )= (g, gj), where gj = α (gi). The edge

(g, gj) has color g-1 gj. Since α is color preserving, we have gi = g-1gj or, gj = ggi or, α(gi) = ggi, which

is as required. Next let gs є (X – S). Since S is a set of generators of X , we can write

gs = h1h2 ...hm, for some h1 , h2,..., hm є S or gs = g1h1h2...hm, since g1 is identity element of X.



Put g1h1 = g2, g2h2 = g3, ...., gmhm = gm+1. Then gs = gm+1 .Now we shall show that α(gm+1) =

ggm+1 . By the equation(4), we have α(g1)=ggl, α(g2) = α(g1h1) = α(g1)h1 = gg1h1 = gg2, α(g3) = α(g2h2)

= α(g2)h2 = gg2h2 = gg3. Proceeding in this way, we get α(gm+1) = ggm+I. That is α(gs) = ggs.

Thus,

α(gi) = ggi, for every gi є X, where g = α(g1) .

By the equation (1), we have

α(gi) =αg(gi), for gi є X or α = αg, where g = α(g1)

So α є L(X) or C(G) ⊆ L(X) . . . (5).

From (3) and (5) we get ,L(X) = C(G). Thus by (2), X ≅ C(G).

REFERENCES

1. Bhattacharya, P.; Jain, S.K.; and S.R. Naugpaul, Abstract Algebra 2/e Cambridge University Press, (1997).

2. Biggs,N.L.; Algebraic Graph Theory Cambridge University Press (1974)

3. Bondy J.A.; and U.S.R. Murthy, Graph Theory and related topics Acad. Press, New York (1979)

4. Frucht, R.; Graphs of degree Three with a given abstract group, Canad.J.Math.1 (1949), 365-378.

5. Frank. Harary.; Graph Theory, Addison wesley, Reading (1969).

6. Herstein. I.N.; Topics in algebra, 2/e, Vikas Publishing House Private Limited, New Delhi. (1975).

7. Imrich, W.; on Graphical regular representations of groups, colloq. Math. Soc. Janos Bolyai (1973), 905-923.

8. Imrich.W.; and Watkins on automorphism of groups of cayley graphs, period. Math.Hungar.7 (1976), 243-258.

9. Konig,D.; Theorie der endlichen und unendlichen Graphen, Leipzig (1936) Reprinted chelsea, Newyork (1950)

10. krishnamoorthy, V.; and K.R. Parthasarathy, Co-spectral graphs and digraphs with given automorphism group, J.combnin.Theory Ser.B19 (1975) 204-213

11. Parthasarathy K.R.; Basic graph Theory, Tata Me.Graw-Hill Publishing Company Limited (1994)

12. Reinaldo E. Giudici, Arora A. Olivieri., On Quadratic Modulo 2n Cayley graphs,Universidad

Simon Bolivar. Departmento de mathematics Apertedo postal 89000.Caracas1080-A.Venizula (1996).

13. Reinaldo.E., Giudici and Pedro Berrizbeitia counting pure K-cycles in sequences of Cayley- graphs, Departmento de mathematics. Universidad Simon Bolivar, Apertedo postal 89000. Caracas 1080-A. Venizula (1994).



24

SOME PROPERTIES OF DERIVATIONS ON PRIME RINGS ------------------------------------------------------------------------------------------------------------ INTRODUCTION:

In 1957, Posner [15] studied derivations in prime rings and proved that if d is a nonzero derivation of a prime ring R such that ad(a) – d(a)a is in the center of R, then R is commutative. Herstein [9] proved that if n > 1 and x – x

n is in the center of a ring R for all x in R, then R is commutative. Using this result, Bell and Daif [4] studied the derivations and commutativity in prime rings.

In 1978, Herstein [11] proved that if R is a prime ring with char. ≠ 2 and R

admits a nonzero derivation d such that [d(x), d(y)] = 0 for all x,y∈R, then R is commutative. Macdonald [14] established some group – theoretic results in terms of inner derivations. Bell and Kappe [5] studied the analogous results for rings in which derivations satisfy certain algebraic conditions. In this talk, we discuss these results for rings with left derivations. We prove that a mapping d on a semiprime ring R is a left derivation if and only if it is a central derivation. It is shown that if a left derivation d acts as a homomorphism or an antihomomorphism on a nonzero right ideal U of a prime ring R or [d(x),x]=0, then d=0. Also, we prove that if [d(x),d(y)] = [x,y] or [d(x),d(y)] = 0, then R is commutative. Elementary properties of derivations:

An additive map d from a ring R to R is called a derivation on R if d(xy) = d(x)y +

xd(y) for all x, y ∈ R. An additive mapping d : R → R is a left derivation if d(xy)=

xd(y) + yd(x) hold for all x,y in R. A mapping d :R → R is called central if d(x) ∈

Z for all x in R.The center Z of R is defined as Z = {z ∈ R/[z,R] = 0}. A ring R is called prime if xay = 0 for all x, y in R implies x = 0 or y = 0. A ring R is called semiprime if xax = 0 implies x = 0 for all x in R. A mapping f from a ring R into

ring S is a homomorphism if for all a ,b ∈ R, f (a + b ) = f(a) + f(b) and f ( a b ) =

f(a). f(b). A mapping f : R → S is an antihomomorphism if for all a, b ∈ R,

f( a + b)= f(a) + f(b) and f( a b ) = f(b). f(a).

Posner, Herstein , Felzenszwalb , Daif and Bell have investigated the properties of prime or semiprime rings with derivations. We use the following elementary identities: In any ring R, [xy,z] = x[y,z] + [x,z]y and [x,yz] = y[x,z] + [x,y]z hold for

all x, y, z ∈R. If R is any ring and d any derivation, we have

INVITED TALK __________________________________________________

Prof. Dr K. SUVARNA

Sri KrishnaDevaraya University, Anantapur-515055, A.P., India.



25

d(Z) ⊆ Z, d([x,y]) = [x,d(y)] + [d(x),y] and d(x

n) = x

n-1d(x) + x

n-2 d(x)x+……+ d(x) x

n-1,

in particular, if [x,d(x)] = 0, then d(xn) = nx

n-1d(x), for all x, y ∈ R.

(2)

The following are the important properties of derivations in rings [3,8,11,12,15] :

Theorem 1 : Let R be any ring, d a derivation of R such that d3 ≠ 0. Then A, the

subring generated by all d(r), r∈R, contains a nonzero ideal of R.

Theorem 2 : Let d be a derivation of a prime ring R and a be an element of R. If

ad(x) = 0 for all x ∈ R, then either a = 0 or d is zero. Theorem 3 : Let R be a prime ring, and let p, q, r be elements of R such that paqar = 0 for all a in R. Then one, atleast, of p,q,r is zero.

Theorem 4 : Let R be a prime ring with an idempotent e ≠ 0,1. If d is a derivation

of R such that d(e+ex-exe) = 0 for all x∈R, then d = 0.

Theorem 5 : Let R be a prime ring of char. ≠ 2 and d1,d2 derivation of R such that the iterate d1d2 is also a derivation, then one atleast of d1,d2 is zero. Theorem 6 : Let R be a prime ring and d a derivation of R such that ad(a) – d(a)a

= 0 for all a ∈ R. Then R is commutative, or d is zero. Corollary 1 : Let R be a prime ring with nontrivial idempotents. If d is a derivation

of R such that d(xn) = 0, n = n(x) ≥ 1, for all x ∈ R, then d = 0.

Theorem 7 : A finite field admits no nonzero derivation. (3)

Theorem 8 : Let R be a prime ring and d ≠ 0 a derivation of R. Suppose that d(xn)

= 0 for all x ∈ R, where n > 1 is a fixed integer. Then R satisfies a generalized polynomial identity (GPI) f ( X ) = Xn d( a ) - d( a ) Xn. Theorem 9 : Let n > 1, and let R be a prime ring. If R admits a nonzero derivation

d such that d(xn) ∈ Z for all x ∈ R, then R is infinite and commutative.

Theorem 10 : Let R be a prime ring and let d ≠ 0 be a derivation of R. Suppose that

a ∈ R such that ad(x) = d(x)a for all x ∈ R. Then (i) if R is of char. ≠ 2, a must be

in Z, the center of R. (ii) If R is of char. 2, then a2 ∈ Z. Left derivations on prime rings: Now we prove some properties of left derivations on prime rings. We use the following results of Bell and Kappe [5] :

Lemma 1 : (i) Let U be a subring of a ring R, and let d be a derivation of R which

acts as a homomorphism on U. Then d(x)x (y-d(y)) = 0 for all x,y ∈ U. (ii) Let V be a right ideal of R and d a derivation of R acting as an

antihomomorphism on V. Then d(x)y [r,d(x)] = 0 for all x,y ∈ V and r ∈ R. Theorem 11 : Let R be a semiprime ring. If d is a derivation of R which is either an endomorphism or an antiendomorphism, then d = 0. (4)



26

Theorem 12 : Let R be a prime ring and U a nonzero right ideal of R. If d is a derivation of R which acts as a homomorphism or an antihomomorphism on U, then d = 0 on R. Now using the above results we prove the following : Theorem 13 : A mapping d on a semiprime ring R is a left derivation if and only if it is a central derivation.

Proof : Let R be a semiprime ring and d : R → R a mapping on R. It is clear that if d is a central derivation, then d is a left derivation. So, let us suppose that d is a left derivation. Then d(xy

2) = xd(y

2) + y

2d(x) = 2xyd(y) + y

2d(x), (1)

for all x,y ∈ R. Also d((xy)y) = xyd(y) + yd(xy) = xyd(y) + yxd(y) + y

2d(x). (2)

From (1) and (2) , xyd(y) = yxd(y).

⇒ [x,y]d(y) = 0, (3)

for all x,y ∈ R.

We replace x by zx in (3) and using (3) again, we get [z,y]xd(y) =0. By interchanging x and z in the last equation, then [x,y]zd(y) = 0, (4) for all x,y,z in R

On the other hand, a linearization of (3) leads to [x,y+u]d(y+u) = 0.

⇒ [x,u]d(y) = -[x,y]d(u) = [y,x]d(u)⋅ (5) We replace z by d(u)z[x,u] in (4) and using (5), then 0 = [x,y]d(u) z [x,u]d(y) = -[x,y]d(u) z [x,y] d(u),

i.e., [x,y] d(u) z [x,y] d(u) = 0. (6)

Since R is semiprime, by (6) we get [x,y]d(u) = 0 for all x,y,u ∈ R.

By [8], d(u) ∈ Z for all u ∈ R. Hence d(xy) = xd(y) + yd(x). This shows that d is a left derivation on R which maps R into its center. � As a consequence, we get the following : Corollary 2 : Let R be a prime ring. If R admits a nonzero left derivation d, then R is commutative. � Theorem 14 : Let R be a prime ring and U a nonzero right ideal of R . Suppose d :

R → R is a left derivation of R. If d acts as a homomorphism on U, then d = 0 on R. If d acts as an antihomomorphism on U, then d = 0 on R. Proof : (i) If d acts as a homomorphism on U, then we have d(x)d(y) = d(xy) = xd(y) + yd(x), (7)

for all x,y ∈ U. We replace x =xy in equation (7), then

d(xy) d(y) = xyd(y) + yd(xy) for all x,y ∈ U.



27

From equation (7) implies that xd(y)d(y) = xyd(y).

⇒ x(d(y)-y)d(y) = 0, (8)

for all x,y ∈ U. By replacing x = xr in equation (8), then we get

xr (d(y)-y)d(y) = 0, for all x,y ∈ U and r ∈ R.

⇒ xR (d(y)-y) d(y) ={0}, for all x,y ∈ U. The primeness of R forces either x = 0 or (d(y)-y) d(y) = 0.

Since U is a nonzero ideal of R, we have (d(y)-y) d(y) = 0.

⇒ d(y2) = yd(y).

Since d is a left derivation, we get yd(y) = 0. By linearizing y by x+y, then yd(x) + xd(y) = 0, (9)

for all x,y ∈ U. We replace x by yx in equation (9), then yxd(y) = 0, (10)

for all x,y ∈ U. By substituting x by sx in equation (10), then we get

ysx d(y) = 0 for all x,y ∈ U and s ∈ R.

Thus for each y ∈ U, the primeness of R forces that either y = 0 or xd(y) = 0. But y = 0 also implies that xd(y) = 0, (11)

for all x,y ∈ U. If we replace x by xr in equation (11), then we get

xrd(y) = 0, for all x,y ∈ U and r ∈ R. Hence xRd(y) = 0 which implies d(y) = 0, (12)

for all y ∈ U. We replace y by sy in equation (12), then we get yd(s) = 0, (13)

for all y ∈ U and s ∈ R.

Again replacing y by yr in equation (13), then we get

yrd(s) = 0, for all y ∈ U and r,s ∈ R. Hence y R d(s) = {0}. Since R is prime and U a nonzero right ideal of R, we have d = 0 on R. (ii) If d acts as an antihomomorphism on U. By our hypothesis, we have d(xy) = d(y)d(x) = xd(y) + yd(x), (14)

for all x,y ∈ U. By substituting xy for y in equation (14), then



28

d(xy) d(x) = d(x(xy)) = xd(xy) + xyd(x), (15)

for all x,y ∈ U. We multiply equation (14) on the right hand side by d(x) and using d is an antihomomorphism on U, then we get d(xy) d(x) = xd(xy) + yd(x)d(x), (16)

for all x,y ∈ U. By combining equations (15) and (16), we get xyd(x) = yd(x)d(x) (17) In equation (17), we replace y by ry, then xryd(x) = ryd(x)d(x), (18)

for all x,y ∈ U and r ∈ R. We multiply equation (17) on left by r and combining with equation (18), then [r,x] yd(x) = 0, (19)

for all x,y ∈ U and r ∈ R. In equation (19) we replace y by sy, then

[r,x]syd(x) = 0, for all x,y ∈ U and r,s ∈ R.

Hence [r,x]Ryd(x) = {0} for all x,y ∈ U and r ∈ R. Thus, for each x ∈ U, the

primeness of R forces that either [r,x] = 0 or yd(x) = 0. Let A = {x∈U/yd(x) = 0,

for all y ∈ U} and B = { x∈U/[r,x] = 0, for all r ∈ R}. Then clearly A and B are additive subgroups of U, whose union is U. By Brauer’s trick, we have yd(x) = 0

for all x,y ∈ U or [r,x] = 0 for all x ∈ U and r∈ R. If [r,x] =0, we replace x by sx,

then [r,sx] = 0 which implies [r,s]x = 0 for all x ∈ U and r,s ∈ R. Therefore

[r,s]Rx = {0}. The primeness of R forces either x = 0 or [r,s] = 0. But U ≠ {0},

then we have [r,s] = 0 for all r,s ∈ R, that is R is commutative. So d(xy) = yd(x) +

xd(y) for all x,y ∈ U which implies that d is a derivation which acts as an a antihomomorphism on U. Hence by Theorem 12 we have d = 0 on R. Thus we have remaining possibility that yd(x) = 0, (20)

for all x,y ∈ U

If we replace y by yr in equation (20), then yrd(x) = 0, for all x,y ∈ U and r ∈ R. Hence yRd(x) = 0 which implies that d(x) = 0, (21)

for all x ∈ U. By substituting sx for x in equation (21), then we obtain d(sx) = 0 which implies that xd(s) = 0, (22)

for all x ∈ U and s ∈ R.

We replace x by xr in equation (22), then xrd(s) = 0, for all x ∈ U and r,s ∈ R. Hence xRd(s) = {0}. Since R is prime and U a nonzero right ideal of R, we have d = 0 on R. �



29

Theorem 15 : Let R be a prime ring with char.≠2, U a nonzero right ideal of R and d be a left derivation of R. If U is noncommutative such that [d(x),x] = 0 for all x

∈ U, then d = 0. Proof : By linearizing the equation

[x,d(x)] = 0 which gives [x,y]d(x) = 0, for all x,y ∈ U. (23) we replace y by yz in equation (23) and using this equation, then

[x,y]zd(x) = 0, for all x, y, z ∈ U. (24)

By writing z by zr, r ∈ R, in equation (24), then we obtain [x,y]zrd(x) = 0 for all

x, y, z ∈ U and r ∈ R.

If we interchange z and r, then we get [x,y]rzd(x) = 0, for all x, y, z ∈ U and

r ∈ R. By primeness property, either [x,y] = 0 or d(x) = 0. Since U is noncommutative, we have d = 0. �

Theorem 16 : Let R be a prime ring with char.≠ 2, U a right ideal of R and d be a

nonzero left derivation of R. If [d(x),d(y)] = [x,y] for all x, y∈U, then [d(x),x] = 0 and hence R is commutative. Proof : By taking xy instead of y in the hypothesis, then we get [x,xy] = [d(x), d(xy)], whence x[x,y] = [d(x), d(x)y + xd(y)]

= x[d(x),d(y)] + [d(x),x]d(y) + [d(x),y]d(x) and so, [d(x),x] d(y) + [d(x),y]d(x) = 0, (25)

for all x, y ∈ U

we replace y by cy = yc, where c ∈ Z and using (25), we arrive at [d(x),x] yd(c) =

0, for all x, y ∈ U. Since 0 ≠ d(c) ∈ Z and U is a nonzero right ideal of R, we have

[d(x),x] = 0, for all x ∈ U. By using the similar procedure as in Theorem 15. Then we get either [x,y] = 0 or d(x) = 0. Since d is nonzero, we have [x,y] = 0. Hence R is commutative. �

Theorem 17 : Let R be a prime ring with char.≠ 2, U a right ideal of R and d be a

nonzero left derivation of R. If [d(x),d(y)] = 0 for all x, y ∈ U, then R is commutative. Proof : By taking xy instead of y in the hypothesis, then we get [d(x),d(xy)] = 0.

⇒ [d(x),x]d(y) + [d(x),y]d(x) = 0, (26)

for all x, y ∈ U. The proof is now completed by using Theorem 16 from the equation (25). � Bresar [6] defined generalized derivation of rings. He investigated the properties and the structure of prime rings. Hvala [13] initiated the algebraic study



30

of generalized derivation and extended some results concerning derivation to generalized derivation. In [7] Bresar and Vukman introduced the notion of orthogonality for a pair d, g of derivations on a semiprime ring, and they gave several necessary and sufficient conditions for d and g to be orthogonal. In 2004, Argac, Nakajima and Albas [1] extended these results to orthogonal generalized derivations. In 2010, Atteya [2] studied some results on generalized derivations of a semiprime ring R and proved that R contains a nonzero central ideal. Yen [16] studied nonassociative rings with a special derivation. The studies of Posner, Herstein , Bell, Kappe, Martindale, Daif , Hvala , Bresar , Vukman ,Yen and Atteya have opened many avenues for further work.

References:

[1] N.Argac,A.Nakajima,and E.Albas, On orthogonal generalized derivations of semiprime rings, Turk. J. Math. 28(2004), 185-194. [2] M.J.Atteya, On generalized derivations of semiprime rings, Inter. J. Algebra. 4 (2010), 591-598. [3] H.E.Bell, On the commutativity of prime rings with derivation, Quaestiones Mathematicae.22(1999),329-335. [4] H.E.Bell, and M.N.Daif, On derivations and commutativity in prime rings, Acta.Math.Hungar.66(1995),337–343. [5] H.E.Bell, and L.C.Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta. Math. Hungar. 53(1989), 339 – 346] [6] M.Bresar, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991), 89-93. [7] M.Bresar, and J.Vukman, Orthogonal derivation and extension of a theorem of Posner, Radovi Mathematicki 5 (1989), 237-246. [8] B.Felzenszwalb, Derivations in prime rings, Proceedings of the American Mathematical Society.84(1982),16-20. [9] I.N.Herstein, A generalization of a theorem of Jacobson, Amer. J. Math. 73 (1951), 756-762. [10] I.N.Herstein, Rings with involution, Univ. of Chicago press, Chicago, (1976). [11] I.N.Herstein, A note on derivations, Canad. Math. Bull. 21 (1978), 369-370. [12] I.N.Herstein,A note on derivations II, Canad. Math. Bull. 22 (1979), 509-511. [13] B.Hvala, Generalized derivations in prime rings, Comm. Algebra. 26 (1998), 1147-1166. [14] I.D.Macdonald, Some groups elements defined by commutators, Math. Scientist, 4(1979), 129-131 [15] E.C.Posner, Derivations in prime rings, Proc. Amer.Math.Soc.8 (1957),1093- 1100. [16] C.T.Yen, Nonassociative rings with a special derivation, Tamkang J.Math. 26 (1995).



31

“Generalization of Dimension

in Vector spaces to Rings”

------------------------------------------------------------------------------------------------------------ It is well known that the dimension of a Vector Space is defined as the number of

elements in the basis. A. W. Goldie (University of Leeds) [2] generalized the dimension concept to modules over rings. A Module M is said to have finite Goldie dimension (FGD, in short) if M does not contain a direct sum of infinite number of non-zero submodules. Goldie proved a structure theorem for modules which states that “a module with FGD contains uniform submodules U1, U2, …, Un whose sum is direct and essential in M”. The number n obtained here is independent of the choice of U1, U2, …, Un and it is called as Goldie dimension of M. The concept Goldie dimension in Modules was studied by several authors like Satyanarayana, Syam Prasad, Nagaraju (refer [3, 9]).

If we consider ring as a module over itself, then the existing literature tells about dimension theory for ideals (i.e., two sided ideals) in case of commutative rings; and left (or right) ideals in case of associative (but not commutative) rings. So at present we can understand the structure theorem for associative rings in terms of one sided ideals only (that is, if R has FGD with respect to left (right) ideals, then there exist n uniform left (or right) ideals of R whose sum is direct and essential in R). This result cannot say about the structure theorem for associative rings in terms of two sided ideals. To fill this gap, Satyanarayana, Nagaraju, Balamurugan & Godloza [5] started studying the concepts: complement, essential, uniform, finite dimension with respect to two sided ideals of R. Finite Dimension with respect to two sided Ideals

1 Definition (Def.4.1 of Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): (i) We say that R has Finite Dimension on Ideals (FDI, in short) if R does not contain a direct sum of infinite number of non-zero (two sided) ideals of R.

(ii) Let (0) ≠ K ⊴ R. We say that K has Finite Dimension on Ideals of R (FDIR, in short) if K does not contain a direct sum of infinite number of non-zero ideals of R. It is clear that if R has FDI, then every ideal K of R has FDIR. 2 Theorem (Th. 4.2 of Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): K has

FDIR ⇔ for any strictly increasing sequence H1, H2, … of ideals of R contained in K,

there is an integer i such that Hk ≤e Hk+1 for every k ≥ i. 3 Lemma (Lemma 4.3 of Satyanarayana, Nagaraju, Balamurugan & Godloza [5]):

Suppose R has FDI and (0) ≠ K ⊴ R. Then K contains a uniform ideal.

INVITED TALK (20 Min) _________________________________________________________

Dr Dasari Nagaraju, M.Sc., M.Phil., Ph.D.,

HITS, Hindustan University, Padur,

OMR, Near Kelambakkam, Chennai-603 103.


(July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP:

32

4 Theorem (Th. 4.4 of Satyanarayana, Nagaraju, Balamurugan & Godloza [5]):

Let 0 ≠ H ⊴ R. Suppose R has FDI. (i) (Existence) There exist uniform ideals U1, U2, … Un whose sum is direct and essential in H;

(ii) If Vi, 1 ≤ i ≤ k are uniform ideals of R, such that vi ⊆ H and the sum of vi’s is

direct, then k ≤ n.

(iii) (Uniqueness) if Vi, 1 ≤ i ≤ k are uniform ideals of R whose sum is direct and essential in H, then k = n. 5 Definition (Def. 4.6 of Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): The number n of the above Theorem is independent of the choice of the uniform ideals. This number n is called the dimension of R, and is denoted by dim R. 6 Theorem (Th 1.2 of Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): Suppose R has FDI.

(i). If H ⊴ R, K ⊴ R and H ⊆ K, then dim H ≤ dim K;

(ii) If (0) ≠ Ai is an ideal of R for all i, 1 ≤ i ≤ t whose sum is direct, and

Ai ⊆ H, 1 ≤ i ≤ t, then dim H ≥ t;

(iii) H is uniform ⇔ dim H = 1;

(iv) If H is a non-zero ideal of R, then dim H ≥ 1;

(v) If Ii, 1 ≤ i ≤ k are uniform ideals of R whose sum is direct, then k ≤ dim R.

Moreover dim H = max{k / there exist uniform ideals Ii, 1 ≤ i ≤ k of R whose sum is

direct, Ii ⊆ H, 1 ≤ i ≤ k}; (vi). If n = dim R, then the number of summands in any decomposition of a given ideal I of R as a direct sum of non-zero ideals of R is at most n.; and

(vii) If f: R → S is an isomorphism and R has FDI, then S has FDI and dim R = dim S. 7 Result (Result 1.3 of Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): If H and K

are ideals of R with H ∩ K = (0), then dim (K + H) = dim K + dim H. Using the above Result and the principle of mathematical induction, we get the following Corollary.

8 Corollary (Cor. 1.4 of Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): (i). If I1, I2, …, In are ideals of R whose sum is direct, then

dim(I1 ⊕ I2 ⊕ … ⊕ In) = dim I1 + dim I2 + … + dim In.

(ii). Suppose Ri, 1 ≤ i ≤ n are rings and R = i

n

iR

1=⊕ is the direct sum of the rings

Ri, 1 ≤ i ≤ n. Then each Ri has FDI if and only if R has FDI. If R has FDI, then

dim R = ∑=

n

i

iR1

dim .



33

9 Theorem (Th. 2.1 of Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): If R has

FDI with dim R = n and H ⊴ R, then the following conditions are equivalent:

(i). H ≤e R; (ii). dim H = dim R; and (iii). H contains a direct sum of n uniform ideals. 10 Proposition (Prop 2.3 of Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): If R has FDI and if an ideal H of R has no proper essential extensions, then R/H has

FDI and dim(R/H) ≤ dim R. 11 Proposition (Prop. 2.4 of Satyanarayana, Nagaraju, Godloza & Sreenadh [6]):

Suppose R has FDI and K ⊴ R.

(i) K is a complement ideal ⇔ K has no proper essential extensions; and

(ii) If K is a complement, then R/K has FDI, and dim(R/K) ≤ dim R. 12 Theorem (Th. 2.3 of Satyanarayana, Nagaraju, Babu Prasad & Mohiddin Shaw [7]):

Let K be an ideal of R and π: R → R/K be the canonical epimorphism. Then the following three conditions are equivalent: (i) K is a complement; (ii) For any ideal K1 of R containing K, we have that K1 is a complement

in R ⇔ π(K1) is complement in R/K; and

(iii) For any essential ideal S of R, π(S) is essential in R/K.

Dimension of the Quotient Ring R/K

13 Lemma (Lemma 3.1 of Satyanarayana, Nagaraju, Babu Prasad & Mohiddin Shaw [7]): Let R be a Ring with FDI. If A is an ideal of R such that dim(R/A) = 1 and A is not essential in R, then dim(R/A) = dim R - dim A.

It is well known that if V is a finite dimensional vector space and W is a subspace

of V, then dim(V/W) = dim V - dim W. This dimension condition may not hold for a general ideal W of a Ring V where “dim” denotes the “finite dimension”. For this, observe the following examples.

14 Examples: Write R = ℤ, the ring of integers. Since every ideal of ℤ is essential in ℤ,

it follows that ℤ is uniform and so dim R = 1.

(i) Write K = 6ℤ. Now K is an uniform ideal of R. So dim K = 1 and

dim R - dim K = 1 - 1 = 0. Now R/K = ℤ/6ℤ ≅ ℤ6 ≅ ℤ2 + ℤ3 and so dim(R/K) = 2.

Thus dim(R/K) = 2 ≠ 0 = dim R - dim K.

(ii) Let p, q be distinct primes and consider H, the ideal of ℤ generated by the product of

these primes (that is, H = pqℤ). Now H is uniform ideal and so dim H = 1. It is known

that ℤ/H = ℤpq ≅ ℤp ⊕ ℤq, and ℤp, ℤq are uniform ideals. So dim(ℤ/H) = 2.



34

Thus dim (ℤ/H) = 2 ≠ 0 = 1 - 1 = dim ℤ - dim H.

Hence, there arise a type of ideals K which satisfy the condition dim(R/K) = dimR–dimK.

15 Theorem (Th. 3.3 of Satyanarayana, Nagaraju, Babu Prasad & Mohiddin Shaw [7]): If R has FDI and K is a complement ideal, then dim(R/K) = dim R – dim K.

References

[1]. Faechini Alberto (1998) “Module Theory” (Progress in Mathematics, Vol.167), Birkhauser Verlag, Switzerland.

[2]. A. W. Goldie (1972) "The Structure of Noetherian Rings", Lectures on Rings and Modules, Springer-Verlag, New York.

[3]. Bh. Satyanarayana "A note on E-direct and S-inverse Systems", Proc. of the Japan

Academy, 64-A (1988) 292-295. [3A] Satyanarayana Bhavanari “Contributions to Near-ring Theory”, Doctoral Dissertation, Acharya Nagarjuna University, 1984. Published by VDM Verlag Dr Mullar, Germany, 2009 (ISBN: 978-3-639-22417-7). [4]. Satyanarayana Bhavanari and Mohiddin Shaw Shaik "Fuzzy Dimension of Modules

over Rings (Monograph)", VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-639-23197-7).

[4A] Satyanarayana Bhavanari and Nagaraju D “Dimension and Graph Theoretic Aspects of Rings (Monograph)” VDM Verlag Dr Muller, Germany, 2011. (ISBN: 978-3-639-30558-6) [5]. Satyanarayana Bh., Nagaraju D., Balamurugan K. S., & Godloza L. "Finite

Dimension in Associative Rings", Kyungpook Mathematical Journal, 48 (2008) 37-43. (SOUTH KOREA).

[6]. Satyanarayana Bhavanari, Nagaraju Dasari, Godloza Lungisile, & S. Sreenadh “Some Dimension Conditions in Rings with Finite Dimension”, The PMU Journal of

Humanities and Sciences 1 (2010) 69-75 (INDIA). [7]. Satyanarayana Bhavanari, Nagaraju Dasari, Babu Prasad Munagala & Mohiddin

Shaw Shaik. “On the Dimension of the Quotient Ring R/K where K is a complement”, International Journal of Contemporary Advanced Mathematics, 1(2)(2010)16-22 (MALAYSIA).

[7A] Satyanarayana Bhavanari and Rama Prasad J.L “Prime Fuzzy Submodules”,

VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978-3-639-24355-0). [8]. Satyanarayana Bhavanari and K. Syam Prasad (2009) “Discrete Mathematics and

Graph Theory”, Printice Hall of India, New Delhi (ISBN: 978-81-203-3842-5). [9].Satyanarayana Bhavanari, K. Syam Prasad and D. Nagaraju (2006) "A Theorem on

Modules with Finite Goldie Dimension", Soochow J. Maths 32(2), pp 311-315.

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July 11-

12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 35-36 Page-35

“REARICK’S ISOMORPHISM AND A CHARACTERIZATION OF ΨΨΨΨ-ADDITIVE FUNCTIONS” --------------------------------------------------------------------------------------------------------------------- An arithmetic function is a complex-valued function defined on the set of positive integers +Z . The set of arithmetic functions will be denoted by F. The classical Dirichlet convolution denoted by D is defined by

|

( )( ) ( ) ( / ) ,d n

f D g n f d g n d=∑ (1)

for , .f g F and n +∈ ∈Z The sum on the right hand side of (1) is taken over all

positive divisors d of .n ,n +∈Z

Let 1F denote the set of arithmetic functions f with (1)f real

1{ : (1) 0}.P f F f′ = ∈ > (2)

It is well-known that an arithmetic function f which is not identically zero

is said to be multiplicative if ( ) ( ) ( ),f mn f m f n= for all ,m n +∈Z with ( , ) 1;m n = here

as usual, the symbol (a, b) denotes the greatest common divisor of a and b. The set of multiplicative functions will be denoted by .M

In 1968, David Rearick (cf. [7], Theorem 9) among other things proved that the groups 1( , ), ( , )P D F′ + and ( , )M D are all isomorphic. In fact, Rearick (cf. [7],

Theorems 2 and 3) showed that the logarithmic operator 1:( , ) ( , )L P D F′ → + defined

by (1) log (1),Lf f= (3)

And for n >1, 1

|

( ) ( ) ( / ) log ,d n

Lf n f d f n d d−=∑ (4)

is an isomorphism, where 1f − is the inverse of f with respect to the Dirichlet

convolution D so that 1

|

( ) ( / ) ( ),d n

f d f n d e n− =∑

for all ,n +∈Z where 1, 1,

( )0, 1.

if ne n

if n

==

> (5)

Rearick (cf. [7], Theorem 9) also observed that the results mentioned above in the case of Dirichlet convolution can be extended to the unitary convolution. That is, Rearick proved that the groups 1( , ), ( , )P U F′ + and ( , )M U are all isomorphic (A divisor

d of n is called a unitary divisor if (d,n/d) =1 and write d||n; the unitary convolution U is obtained by replacing D by U and d|n by d||n in (1)).

In this talk we discuss possible extension of these results to ΨΨΨΨ-convolution

INVITED TALK _____________________________________________________

Prof. Dr V.SITARAMAIAH Department of Mathematics, Pondicherry Engineering College, Puducherry


12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 35-36 Page-36

introduced by D.H.Lehmer(cf.[3]). Placing mild condition on ΨΨΨΨ it can be shown that the logarithmic operator 1:( , ) ( , )L P Fψ′ → + defined by

(1) log (1),Lf f= (6)

and for n>1, 1

( , )

( ) ( ) ( ) ( ),x y n

Lf n f x f y h xψ

−

=

= ∑ (7)

is an isomorphism if h is a ΨΨΨΨ-additive function and non vanishing on {1}.+ −Z

It is interesting to note that when ΨΨΨΨ is a Lehmer-Narkiewicz convolution ,the converse of the above result is also true. That is, if h(1)=0 and L given in (6) and (7)

is an isomorphism , then h is ΨΨΨΨ -additive and is non-vanishing on {1}.+ −Z

(These results are to appear in Annales Univ.Sci.Budapest.,sect.Comp., vol.31 (2011) in a paper jointly written with G.Rajmohan)

REFERENCES [1] E.Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math.Z., 74(1960),66-80. [2] Donald Gold Smith, A generalized convolution for arithmetic functions, Duke

Math.J., 38(1971)279-283. [3] D.H.Lehmer, Arithmetic of double series, Trans.Amer.Math.soc.,33(1931),945-957. [4] W.Narkiewicz, On a class of arithmetical convolutions, Colloq.Math.,10(1963),945-957.

[5] J.L.Nicolas and V.Sitaramaiah , Existence of unity in Lehmer’s ΨΨΨΨ - product ring –II, Indian J.Pure and Appl.Math.,33(10)(2002), 1503-1514.

[6] G.Rajmohan and V.Sitaramaiah, On regular ΨΨΨΨ- convolutions –II , Annales Univ.Sci.Budapest.,sect.Comp.,27(2007),111-136. [7] D.Rearick, Operators on algebras of arithmetic functions, Duke

Math.J.,35(1968),761-766.

[8] V.Sitaramaiah , On the ΨΨΨΨ-product of D.H.Lehmer, Indian J.Pure and

Appl.Math.,16(1985),904-1008.

[9] V.Sitaramaiah, On the existence of unity in Lehmer’s ΨΨΨΨ-product ring, Indian

J.Pure and Appl.Math.,20(1989),1154-1190.

[10] V.Sitaramaiah and M.V.Subbarao, On a class of ΨΨΨΨ-convolutions preserving multiplicativity, Indian J.Pure and Appl.Math.,22(1991),819-832.

[11] V.Sitaramaiah and M.V.Subbarao, On a class of ΨΨΨΨ-convolutions preserving multiplicativity-II, Indian J.Pure and Appl.Math.,25(1994),1233-1242.

[12] V.Sitaramaiah and M.V.Subbarao , On regular ΨΨΨΨ-convolutions ,J.Indian math.

Soc., 64(1997),131-150. [13] R.Vaidyanathaswamy, The theory of the multiplicative arithmetic functions, Trans.Amer.Math.,soc.,33(1931),579-662.



37

Direct Systems

in N-Groups

-------------------------------------------------------------------------------------------------------------------------------- The concept of E-direct systems in modules introduced by Satyanarayana [1]. The extension of this idea to N-groups introduced in Satyanarayana & Syam Prasad [1]. The aim of the lecture is to present some interesting results on Direct and E-direct systems in N-groups.

Direct Systems:

Definition: A non empty family {Gi}i∈1 of proper ideals of G is said to be a direct system if for any finite number of elements i1, i2, … , ik of I, there is an element iO in I such

that 0i

G ⊇ 1i

G + … + ki

G .

Theorem: (Satyanarayana & Syam Prasad [1]) The following are equivalent:

(i) G has ACCI

(ii) For any ideal J of G, the condition: every direct system of ideals of G

which are contained in J is bounded above by an ideal J* of G where J* ⊊ J.

(iii) Every ideal J of G is finitely generated.

E–Direct System

Definition: A family {Gi}i∈I of ideal of G is said to be an E-direct system if for any finite

number of elements i1, i2, …, ik of I, there is an element i0 ∈ I such that

0iG ⊇

1iG +

2iG + … +

kiG and

0iG is non-essential in G.

Example: Write Gi = Z2 (integers modulo 2) with usual addition for i ∈ ℕ. Consider the nearring N = Z2, the set of integers with usual addition and multiplication of integers

modulo 2. Then each Gi is an N-group. Consider the direct product G = Σ*Gi of N-groups

{Gi / i ∈ ℕ}. Let πi be the projection map from G to Gi for i ∈ ℕ. We know that the

projection maps πi’s are N-epimorphisms. For any non-empty subset J of ℕ, write

GJ = {x ∈ G / πi(x) = 0 for every i ∉ J where πi is the projection map}. (Verification follows).

INVITED TALK ___________________________________________________________

Dr Kuncham Syam Prasad Manipal University, Manipal

Email: [email protected] [email protected]



38

Theorem: (Satyanarayana & Syam Prasad [1]): If G has FGD then every E-direct system of non-zero ideals of G is bounded above by a non-essential ideal of G.

Theorem: (Satyanarayana & Syam Prasad [1]): If every E-direct system of non-zero ideal of G is bounded above by a non-essential ideal of G then G has FGD.

Theorem: (Satyanarayana & Syam Prasad [1]): For an N-group G, the following two conditions are equivalent: (i) G has FGD; and (ii) Every E-direct system of non-zero ideal of G is bounded above by a non-essential ideal of G.

Acknowledgement: I take this opportunity to thank Dr A V Vijaya Kumari (Organizing

Secretry) and Prof. Dr Bhavanari Satyanarayana (Academic Secretary,) the organizers of the

National Seminar on Algebra and its applications (Sponsored by UGC).

References:

ANDERSON F.W. & FULLER K. R. [1] “Rings and Categories of Modules”, Springer Verlag, New York, 1974. PILZ G. [1] “Nearrings”, North Holland, New York, 1983. REDDY Y.V. & SATYANARAYANA BHAVANARI [2] “A Note on N-groups”, Indian J. Pure & Appl. Math. 19 (1988) 842-845. SATYANARAYANA BHAVANARI. [1] “A Note on E-direct and S-inverse systems”, Proc. of the Japan Academy, 64-A (1988) 292-295. [2] “Contributions to Near-ring Theory”, (Ph.D., Dissertation submitted to Acharya Nagarjuna University, 1984), VDM Verlag Dr Muller, Germany, 2010. (ISBN: 978-3-639-22417-7). SATYANARAYANA BHAVANARI & SYAM PRASAD KUNCHAM [1] "A Result on E-direct systems in N-groups", Indian J. Pure & Appl. Math. 29 (1998) 285 – 287. [2] “On Direct and Inverse Systems in N-Groups” ,Indian Journal of Mathematics, 42 (2) 183-192, 2000. [3]. “Discrete Mathematics and Graph Theory”, Prentice Hall India Pvt. Ltd. 2009. SYAM PRASAD KUNCHAM and SATYANARAYANA BHAVANARI [1] Dimension of N-groups and Fuzzy Ideals in Gamma Nearrings, VDM Verlag Dr. Muller, Germany, 2011 (ISBN: 978-3-639-36838-3)

THE HIGHEST FORM OF PURE THOUGHT IS IN MATHEMATICS. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

LIFE IS GOOD FOR ONLY TWO THINGS:

DISCOVERING MATHEMATICS AND TEACHING MATHEMATICS



39

NUMERICAL SOLUTION OF SOME TWO POINT BOUNDARY VALUE PROBLEMS BY COLLOCATION METHOD WITH B-SPLINES

------------------------------------------------------------------------------------------------------------

The Finite Element Method involves variational methods like Rayliegh Ritz method, Least Squares method, Petrov-Galerkin method, Galerkin method, Collocation method etc.,. In Finite Element method the approximate solution can be written as a linear combination of a set of functions which constitutes a basis for the approximation space under consideration.

In finite element methods, most of the researchers use Lagrange and Hermite functions as basis functions. They are C0 and C1 functions. When mth order B-splines are used as basis functions, they are Cm−1 functions. Due to this, the approximate solution can be obtained with minimum number of intervals in the space variable domain.

When the given differential operator is self-adjoint and positive definite, then only one can use the Rayliegh Ritz method to find the approximate solution to the given differential equation. In Petrov-Galerkin method, the residual of approximation is made orthogonal to some weight functions. Here the set of weight functions must be a linearly independent set; otherwise the system of equations provided by the method will not be linearly independent and hence is not solvable. The Least Squares method can be treated as a special case of Petrov-Galerkin method. In Galerkin method, the residual of approximation is made orthogonal to the basis functions. When we use Galerkin method, a weak form of approximate solution for a given differential equation exists and is unique under appropriate conditions irrespective of properties of a given differential operator. Further, the weak solution also tends to the classical solution of the given differential equation provided sufficient attention is given to the boundary conditions.

The collocation method seeks an approximate solution by requiring the residual of the equation to be identically zero at N selected points(Collocated points), where N is the number of basis functions in the basis, in the domain of the given differential equation. The selection of the points is crucial in obtaining a well conditioned system of equations and ultimately in obtaining an accurate solution. Once the collocated points are chosen appropriately in the domain of the given differential equation, the collocation method is the easiest among the variational methods of Finite Element Method. In view of this, we intend to present the use of collocation method to solve some two point boundary value problems, namely singular perturbation problems, coupled system of second order boundary value problems with cublic B-splines and fourth order boundary value problems with quintic B-splines.

The solution of a nonlinear boundary value problem has been obtained as the limit of solutions of a sequence of linear problems generated by quasilinearization technique. Several numerical examples of linear and nonlinear boundary value problems have been considered for testing the efficiency of proposed collocation methods.

INVITED TALK ----------------------------------------------

Dr K.N.S. Kasi Viswanadham National Institute of Technology Warangal - 506 004, AP, (INDIA)



40

DEFINITION OF CUBIC B-SPLINES The existence of cubic B-spline interpolate s(x) to a function f(x) in a closed interval [a, b] for spaced knots

a = x0 < x1 < … < xn-1 < xn = b is established by constructing it. The construction of s(x) is done with the help of cubic B-splines. Introduce six additional knots x-3, x-2, x-1, xn+1, xn+2, xn+3 such that

x-3 < x-2 < x-1 < x0 and xn < xn+1 < xn+2 < xn+3. Now the cubic B- splines Bi(x), are defined by

∈

′

−= +−

+

−=

+∑otherwise0

],[if)(

)(

)( 22

2

2

3

ii

i

ir r

r

ixxx

x

xx

xB π

where

≤

≥−=− +

xx

xxxxxx

r

rrr

if0

if)()(

33

and ))()()()(()( 2112 ++−− −−−−−= iiiii xxxxxxxxxxxπ .

It can be shown that the set {B-1(x), B0(x), B1(x), …, Bn(x), Bn+1(x)} forms a basis

for the space S3(π) of cubic polynomial splines. The cubic B-splines are the unique non zero splines of smallest compact support with knots at

x-3 < x-2 < x-1 < x0 < x1 < …< xn < xn+1 < xn+2 < xn+3.

DEFINITION OF QUINTIC B-SPLINES The existence of the quintic spline interpolate s(x) to a function in a closed

interval [a,b] for spaced knots (need not be evenly spaced) bxxxxxa nn =<<<<<= −1210 L is

established by constructing it. The construction of s(x) is done with the help of the quintic B-splines. Introduce ten additional knots x-5, x-4, x-3, x-2, x-1, xn+1, xn+2, xn+3, xn+4 and xn+5 such that

x-5 < x-4 < x-3 < x-2 < x-1 < x0 and xn < xn+1 < xn+2 < xn+3 < xn+4 < xn+5.

Now the quintic B-splines )(xBi are defined by

wiseother0

],[,)(

)()( 33

3

3

5

=

∈′

−= +−

+

−=

+∑ ii

i

ir r

ri xxx

x

xxxB

π

where

xx

xxxxxx

r

rrr

≤=

≥−=− +

if 0

if,)()( 55

and

∏+

−=

−=3

3

)()(i

ir

rxxxπ .

Here the set { })(),(),(,),(),(),( 21012 xBxBxBxBxBxB nnn ++−− L forms a basis for the space

)(5 πS of fifth degree polynomial splines. The quintic B-splines are the unique non zero

splines of smallest compact support with knots at x-5 < x-4 < x-3 < x-2 < x-1 < x0 < … < xn < xn+1 < xn+2 < xn+3 < xn+4 < x n+5 .



41

1) Linear singular perturbation two-point boundary value problem Consider the linear singular perturbation two point boundary value problem

εy″(x) + a(x) y′(x) + b(x) y(x) = c(x), 0<x<1 with y(0)=y0 and y(1)=y1

where ε is a small positive parameter (0<ε <<1) and y0, y1 are given constants, a(x), b(x) and c(x) are assumed to be continuously differentiable functions in [0,1]. Further, we assume that a(x) ≥ M >0 throughout the interval [0,1], where M is some positive constant. This assumption merely implies that the boundary layer will be in the neighbourhood of x=0. The numerical methods produce good results only when we take a step size h ≤ ε. This is very costly and time consuming process. Hence the researchers are concentrating on developing the methods, which can work with a reasonable step size h. Here we developed a collocation method with cubic B-splines along with the equidistribution of error principle to solve the above singular perturbation problem.

2) Coupled system of linear second order boundary value problems Consider a system of second order linear boundary value problems of the type

)()()()()()()( 1252423121110 xfuxauxauxauxauxauxa =+′+′′++′+′′ a < x < b

)()()()()()()( 2252423121110 xfuxbuxbuxbuxbuxbuxb =+′+′′++′+′′ a < x < b

subject to u1(a) = u10, u1(b) = u11, u2(a) = u20 and u2(b) = u21 where u10, u11, u20, u21 are finite real constants and a0(x), a1(x),…, a5(x), b0(x), b1(x), …, b5(x) are all continuous on [a, b]. We developed a collocation method with cubic B-splines to solve the above coupled system of linear second order boundary value problems.

3) Linear fourth order boundary value problems Consider the general linear fourth order boundary value problem a0(x)y(4) + a1(x)y''' + a2(x)y'' + a3(x)y' + a4(x)y = b(x), a < x < b subject to the boundary conditions y(a) = A0, y(b) = B0, y'(a) = A1 and y'(b) = B1 where a0(x), a1(x), a2(x), a3(x), a4(x), b(x) are continuous functions on [a,b] and A0, B0, A1, B1 are constants. We developed a collocation method with quintic B-splines to solve the above linear fourth order boundary value problems.

References : 1. J.N.Reddy, An introduction to Finite Element Method, TMH, 3rd Edition, 2005. 2. P.M.Prenter, Splines and Variational Method, John Wiley and Sons, 1989. 3. Carl de Boor, A practical guide to Splines, Springer Verlag, 1978. 4. R.E.Bellman and R.E.Kalaba, Quasilinearization and Nonlinear Boundary Value

Problems, American Elsevier, 1965. LIVE AS IF YOU WERE TO DIE TOMMORROW.

LEARN AS IF YOU WERE TO LIVE FOR EVER ……….. JATI PITA MAHATMA GANDHI

-----------------------------------------------------------------------------------------------------

WAKE UP THE DIVINITY WITH IN YOU …SWAMY VIVEKANANDA


12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) 42

CONGRUENT RELATED GRAPHS

--------------------------------------------------------------------------------------------------------------------- ABSTRACT: In artificial intelligence and information organization & retrieval to maintain secrecy of information some coding has to be done. In defence problems in order to keep secrecy confidential, we exhibit relationship between pairs of soldiers. Now,we associate a graph and define adjacency suitably. The type of graph introduced here is very useful in the above areas. The elementary study of this concept is being done by Saradhi. This graph is named as Number – theoretic graph.

This talk is mainly devoted to find the size (number of edges) of certain graphs of this type. The extended work forms as part of the dissertation work of Mr. K.V.S. Sarma under my guidance.

INVITED TALK

--------------------------------------------------------

Prof Dr I.H.Nagaraja Rao

Director & Sr. Prof., G.V.P. College for P.G. Courses,

Visakhapatnam

Not Gold, But only Men can make

A people Great and strong

Men who for truth and Honors Sake

Stand fast and suffer Long

Brave Men who works while others sleep

Who dare while others fly

Build the Nations Pillars deep,

And Lift the Country to the Sky.

They alone live who live for others,They alone live who live for others,They alone live who live for others,They alone live who live for others,

the rest are more dead than alive.the rest are more dead than alive.the rest are more dead than alive.the rest are more dead than alive. DO YOUR DUTY, DONOT THINK FOR THE FRUITS THERE OFF.

If Hard Work is your Weapon Success becomes your Slave.

(ISBN: 978-3-639-36838-3)



43

SANDWICH NEAR-RINGS -------------------------------------------------------------------------------------------

ABSTRACT: Let X be a topological space, G an additive topological group and α

a continuous function from G into X. Denote by N(X, G, α) the near-ring of all continuous functions from X into G, where addition of functions is pointwise and

the product fg of two functions f, g ∈ N(X, G, α) is defined by fg = foαog, that is

fg is just the composition of the functions f, α, g. The near-ring N(X, G, α) is

known as “sandwich near-ring” with sandwich function α. Let N(Y, H, β) be another sandwich near-ring. In this paper the following theorem is proved. Theorem: Let X and Y be non discrete, first countable, normal topological spaces. Let G, H be metric spaces. Further suppose that G and H are topological groups in

which every closed sphere is pathwise connected. Then for each isomorphism φ

from N(X, G, α) onto N(Y, H, β) there exists a unique homeomorphism h from

R(α) onto R(β) and a unique topological isomorphism t from G onto H such that the following diagram commutes.

R(α) ----------------->G ----------------->R(α)

R(β)-----------------> H ----------------->R(β) Section-1-Pre Requisites Definition 1.1: An algebraic system (N, +, • ) is called a near-ring (right if it satisfies the following, conditions: i) (N, +) is a group (not necessarily abelian) ii) (N, • ) is a semigroup

iii) (a + b) • c = a • c + b • c, for all a, b, c ∈ N Definition 1.2: Let X be a topological space, G an additive topological group and

α a continuous function from G into X. Denote by N(X, G, α) the near-ring of all continuous functions from X into G, where addition of functions is pointwise and

φ (f)

f

β

t h h

Key words: Near-ring, sandwich near-ring, topological group 2000 subject classification: Primary: 16Y 30, Secondary: 22 XX

INVITED TALK ----------------------------------------------------------------------

Prof. Dr Srinivas Thota Kakatiya University,

Warangal, (A.P.) – 506009 E-mail: [email protected]

α



44

the product fg of two functions f, g ∈ N (X, G, α) is defined by fg = foαog that is,

fg is just the composition of the functions f, α, g. The near-ring N(X, G, α) is

known as “sandwich near-ring” with sandwich function α.

Let N (X, G, α) and N(Y, H, β) be two sandwich near-rings, Suppose that there exists a homeomorphism h from X onto Y and topological isomorphism t

from G onto H such that. h o α = β o t.

Then the mapping φ from N (X, G, α) into N(Y, H, β) defined by

φ(f) = tofoh-1 for each f ∈ N (X, G, α), is an isomorphism from N(X, G, α) onto

N(Y, H, β) and this isomorphism is known as natural isomorphism. K.D. Magill Jr [ 2 ] has introduced the above concept and proved an isomorphism theorem which states that if the spaces, groups and sandwich

functions all satisfy appropriate conditions, then N(X, G, α) and N (Y, H, β) are isomorphic if and only if there exists a homemorphism h from X onto Y and a topological isomorphism t from G onto H such that the following diagram

commutes for each f ∈ N(X, G, α). X ----------------->G -----------------> X Y -----------------> H ----------------->Y

Magill has proved that the isomorphism from N(X, G, α) onto N(Y, H, β) is natural and he obtained the above result for two different types of hypotheses [1, 2]. The aim of this paper is to get the same result but with another hypothesis.

Notation: For any point z ∈ G, the symbol <z> will denote the constant function

in N(X, G, α) which maps all of X into the point z. The following two results are given by Magill.

Lemma 1, 3: [2] : Let φ be an isomorphism from N(X, G, α) onto N(Y, H, β). Then there exists a unique isomorphism t (not necessarily a topological

isomorphism) from G onto H such that φ (<z>) = <t(z)> for each z ∈ G. Definition 1.4: A collection of functions from X to G is said to “distinguish

points” if for distinct points x and y in X, f(x) ≠ f(y) for some f in the collection.

Lemma 1.5: [2]: - Suppose both N(X, G, α) and N(Y, H, β) distinguish points and

that φ is an isomorphism from the former onto the latter. Then there exists a unique

bijection h from R(α) onto R(β) and a unique algebraic isomorphism t from G onto H such that the following diagram commutes.

φ (f) β

t h

f

h

α



45

R(α) ----------------->G ----------------->R(α)

R(β)-----------------> H ----------------->R(β) Diagram 1.5.1

Section-2-Main Results Lemma 2.6: Let X be a normal topological space and G be a pathwise connected

topological space, such that G contains at least two points. Then N(X, G, α) distinguish points.

Proof: Let K be a closed sub set of X and p ∉ K. then by normality there exists a

continuous function f: X -� [0, 1] such that f(p) = 1 and f(x) = 0, ∀ x ∈ K.

Since G is pathwise connected, there exists a function ψ: [0, 1] -� G such that

ψ (0) = z, ψ (1) = w, w, z ∈ G. Put ψ = ψ 0 f. Then ψ (x) = z, ∀ x ∈ K and

ψ (p) = w. Thus ψ (K) = z and ψ (p) = w. In particular taking K to be a singleton

set, we observe that N(X, G, α) distinguish pints. Theorem 2.7: - Let X and Y be normal topological spaces. Let G and H be

pathwise connected topological groups. Let α and β be non constant continuous

functions from G into X and H into Y respectively. Then for each isomorphism φ

from N(X, G, α) onto N(Y, H, β), there exists a unique homeomorphism h from

R(α) onto R(β) and a unique algebraic isomorphism t from G onto H such that the Diagram 1.5.1. commutes.

Proof: N(X, G, α) and N(Y, H, β) distinguish points is clear by Lemma 2.6 and we observe that Lemma 1.3 and Lemma 1.5 holds in this case also. Hence there exists

a unique bijection h from R(α) onto R(β) and a unique algebraic isomorphism t from G onto H such that the Diagram 1.5.1 commutes. Now we have to show that h is a homeomorphism.

Define A(p, f) = {x∈X : α(f (x)) = p}, ∀ p ∈ X, f ∈ N (X, G, α)

B(q, g) = {y∈Y : β(g (y)) = q}, ∀ q ∈ Y, g ∈ N(Y, H, β). Then we have

h[R (α) ∩ A (p, f)] = R (β) ∩ B (h (p), φ(f)) and

h-1 [R (β) ∩ B (q, g)] = R(α) ∩ A (h-1 (q), φ-1 (g)). (These are observed using the fact that the diagram 1.5.1 is commutative).

Now we will show that ℑ = {A (p, f): p ∈ X, f ∈ N (X, G, α)} is a basis for the closed subsets of X, and the analogous family will be a basis for the closed subsets

φ (f) β

t h h

α f



46

of Y. Let K be any closed subset of X and p ∉ K. Since α is not constant, there

exists z, w ∈ G such that α (z) ≠ α(w). As in Lemma 2.6, there exists a continuous

function fp: X � G such that fp (x) = z, ∀ x ∈ K and fp (p) = w.

Then K ⊂ A (α (z), fp). Since for all p ∉ K, there exists such fp, we have that

K = Kp∉

∩ A (α(z), fp). Therefore ℑ is a basis for the closed subsets of X and hence

h is a homeomorphism from R (α) onto R (β). Lemma 2.8: - Let G be a metric space in which every closed sphere is pathwise

connected. Then given any sequence { }∞

=1nnx in G converging to x0 in G, there

exists a continuous function f from [0, 1] to G such that f(1/n) = xn and f(0) = x0

Proof: Let { }∞

=1nnx be a sequence in G converging to x0 in G. Let εI = d (xi, xi+1),

i = 1,2…. Since every closed sphere in G is pathwise connected, there exists a continuous function f1 : [1/2, 1] � G such that f1 (1) = x1, f1 (1/2) = x2 and the

range of f1 entirely lies in the closed sphere 1Sε (x1) with centre at x1 and radius ε1.

Again there exists a continuous function f2 : [1/3, 1/2] � G such that f2 (1/3) = x3

and f2 (1/2) = x2 and the range of f2 lies in 2εS (x2). Thus by recursively we can

define a function fn : [(1/n+1), (1/n)] � G by fn (1/n+1) = xn+1 and fn (1/n) = xn

Define f: [0, 1] � G by f(t) = fn (t), if (1/n+1) ≤ t ≤ 1/n f(0) = x0 That is f: [0, 1] � G such that f(1/n) = xn and f(0) = x0 . Now to show that f is continuous.

Let Sδ(x0) be a sphere with centre at x0 and radius as δ. Since { }∞

=1nnx � x0,

there exists an no such that xn ∈ S δ/4(x0), ∀ n ≥ n0. Choose ε > 0 such that

0 < ε < 1/n0. Consider 0 < t < ε, then there exists an n ≥ n0 such that

(1/n+1) ≤ t < 1/n. then since f(t) = fn (t) in [(1/n+1), 1/n] and by construction the

range of fn (t) lies in the closed sphere nεS (xn), where εn = d(xn, xn+1), n ≥ n0.

Now we will show that nε

S (xn) ⊂ Sδ (x0). Let y ∈ nε

S (xn). Then d (y. xn) < εn.

Also d(xn, xn+1) ≤ d (xn, x0) + d (xn+1, x0) ≤ (δ/4) + (δ/4) = (δ/2).

So d(y, x0) ≤ d(y, xn) + d(xn, x0) < (ε/n) + (δ/4) < (δ/2) + (δ/4) < δ.

Hence nε

S (xn) ⊂ Sδ (x0). This is true for every t lying between 0 and ε. Therefore

f-1 (Sδ (x0)) ⊃ (0, ε). Hence f is continuous at 0 and by above construction f is obviously continuous at the other points. Now we give the main result of this paper. Theorem 2.9: - Let X and Y be non discrete, first countable, normal topological spaces. Let G, H be metric spaces. Further suppose that G and H are topological groups in which every closed sphere is pathwise connected. Then for each

isomorphism φ from N(X, G, α) onto N(Y, H, β) there exists a unique



47

homeomorphism h from R(α) onto R(β) and a unique topological isomorphism t from G onto H such that the Diagram 1.5.1 commutes. Proof: This result can be proved as in [2]. However for the sake of continuity we give detailed proof. By Theorem 2.7 there exists a unique homeomorphism h from

R(α) onto R(β) and a unique algebraic isomorphism t from G onto H such that Diagram 1.5.1 commutes. It remains to show that t is a homeomorphism.

Let { }∞

=1nnx be a sequence in G converging to x0 in G. Sinc X is not discrete

there exists at least one point p which is non isolated. Also X is normal implies that X is Hausdorff. Now since X is first countable and Hausdorff there exists a

sequence { }∞

=1nnp of distinct points converging to p in X. Let A = { }∞

=1nnp ∪ {p}.

Define a mapping h: A � [0, 1] by h(pn) = 1/n, h(p) = 0. Now, since X is normal and by the application of Teietz extension theorem h can be extended to whole of X, which we denote by the same h. Hence h: X � [0, 1] such that h(pn) = 1/n and h(p) = 0. Next, by lemma 2.8 there exists g: [0, 1] � G such that g (1/n) = xn and g (0) = x0 . Put f = g 0 h. Then f: X � G such that f (pn) = xn and f(p) = x0. Now since the Diagram 1.5.1. commutes we have

φ(f) 0 h = t 0 f and β 0 t = h 0 α.

Lim t (xn) = Lim t (f(pn)) = Lim φ (f) (h (pn)) = φ (f) (h (p)) = t (f (p)) = t (x0). Hence t is continuous and in the same manner we can show that t-1 is continuous, so that t is a homeomorphism. ACKNOWLEDGEMENT The author is very much grateful to Prof. Magill and Prof. A . Radhakrishna. REFERENCES 1)K.D. Magill Jr.: Semigroups and nearrings of contiuous functions, General topology and its relations to Modern Analysis and Algebra ii. Proc. Third Prague, Top, Symp. 1971 (Academic, 1972) 283-288. 2) K.D. Magill Jr.: Isomorphisms of sandwich near-rings of continuous functions, Bol, Un. Math. Ital (B) 5 (1986), 209-222. 3) G. Pilz: Near-rings, North-Holland Math studies 23, New York, 1983. 4) G.F. Simmons: Introduction to topology and Modern analysis, International student edition, Mc. Graw-Hill International book company. 5) S. Willard: General Topology, Addison Wesley, Reading, mass, 1970 MATHEMATICS is one of the great creations of Human

mind. It is the basic language for all exact sciences. A significant future in recent times is the use of high-level mathematics in Engineering Science, Computer Science,

Economics, Finance, Biology, Medicine and so on.

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July

11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) P: 48A

ANALYSIS OF LOAD CARRYING CAPACITY IN FINITE POROUS SQUEEZE FILM BEARINGS BY RAPID TECHNIQUE ----------------------------------------------------------------------------------------------------------------

ABSTRACT: Porous bearings have an advantage over conventional sleeve bearings, in which no external supply of lubricant is necessary for their satisfactory operation for long periods. These can operate as self-acting bearings; squeeze film bearings or externally pressurized bearings. Squeeze film is a phenomenon of two lubricated surfaces approaching each other with a normal velocity. The thin film of lubricant between the two surfaces acts as a cushion and prevents the surfaces making instantaneous contact. Generally, the pressure distribution in this fluid film is determined by using numerical methods. In the present analysis, a modified Reynolds equation has been developed by combining the Reynolds equation and the Laplace equation for the finite porous squeeze film bearing. The load carrying capacity is studied by solving this modified Reynolds equation using Rapid Technique. The results are shown graphically for selected parameter values. The results indicate, as the L/D ratio, and or eccentricity ratio increases, load carrying capacity increases but as the permeability parameter increases load carrying capacity decreases. For highly porous bearings, the squeezing effects are not significant. The obtained results are compared with the short bearing and solid bearing analysis. ----------------------------------------------------------------------------------------------------------------

A Fuzzy Logic based search technique for Digital libraries ---------------------------------------------------------------------------------------------------------------- Abstract: The conventional search techniques for searching a database follow Boolean logic. When a database search is made with a given set of keywords, a minor variation in the keyword due to difference in spelling standard, typographical error or difference in inflection of the same stem, results in failure in getting records from the database. This situation demands a special kind of search technique, which should take into account approximation along with exactness. Fuzzy sets are more appropriate to represent imprecise information and matching based on it better suited to process it. As fuzzy logic is an extension of conventional logic it handles the concept of partial truth along with true and false. Fuzzy logic more closely follows the way humans think and helps to handle real world complexities more efficiently. This paper presents a new information retrieval support technique used for database search, using fuzzy based matching and is expected to significantly enhance searching benefits over conventional approaches.

INVITED TALK ------------------------------------------------ Prof. K. Rama Krishna Prasad

S.V.U College of Engineering

Tirupathi

INVITED TALK ------------------------------------------------

Prof. Dr MV Ramana Murthy, Osmania University, Hyderabad

Cell:0-9441187914. E-mail: [email protected]



A DIMENSION OF MODULES OVER

ASSOCIATIVE RINGS

-------------------------------------------------------------------------------------------- Abstract: The aim of the talk is to provide information regarding the generalization of

the concept ‘dimension’ of vector spaces to modules over associative rings. This paper explains a fundamental structure theorem related to the dimension of modules that was proved in 1957 by a well known algebraist Prof. A.W. Goldie (University of Leeds). This paper is very much useful for the beginners in research to understand the algebraic techniques how definitions and other new notions go on, and how logic is generated to form some important structure theorems in algebra.

Introduction

In recent decades interest has arisen in algebraic systems with binary operations addition and multiplication. 'Ring' is one of such systems. A ring (or an associative ring) is an algebraic system (R, +, .) satisfying the conditions:

i) (R, +) is an Abelian group; ii) (R, .) is a semi-group; and

iii) a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c ∈ R.

Moreover, if there exists an element 1 ∈ R such that 1a = a = a1 for all a ∈ R, then we say that R is a ring with identity. Ring theory became an important part of Algebra. We know that the algebraic structure ‘vector space’ is normally defined over a field. A similar structure defined over an associative ring is called as ‘module’. Goldie [1] introduced the concept of Finite Goldie Dimension (FGD, in short) in modules. A module M is said to have FGD if M contains no infinite direct sum of non-zero submodules. Goldie proved a structure theorem for modules which states that “a module with FGD contains uniform submodules U1, U2, …, Un whose sum is direct and essential in M”. The number n obtained here is independent of the choice of U1, U2, …, Un and it is called as Goldie Dimension of M. Later this dimension theory was studied/developed by the authors like: Reddy, Satyanarayana, Syam Prasad & Nagaraju.

Modules

In this section, we discuss some fundamental definitions and examples on modules which are used in the next section. 1.1 Definition: Let R be an associative ring. An Abelian group (M, +) is said to be a

module over R if there exists a mapping (called scalar multiplication) f : R × M → M (the image of (r, m) is denoted by rm) satisfying the following three conditions:

(i) r(a+b) = ra + rb; (ii) (r+s)a = ra + sa; and

(iii) r(sa) = (rs)a for all a, b ∈ M and r, s ∈ R.

15 Min. Talk

-----------------

Mr. Mohiddin Shaw



Moreover, if R is a ring with identity 1; and 1m = m for all m ∈ M, then M is called a unital R–module or unitary module.

1.2 Note: In the definition 1.1, we write the elements of R on left to the elements of M. So we call this module as a left R-module or left module.

Throughout our discussion, M stands for a non-zero left module.

1.3 Examples: (i) Every ring R is a module over itself.

(ii) Every Abelian group is a module over the ring of integers ℤ. (iii) Every vector space over a field F, is a module over the ring F. 1.4 Definition: Let M and M1 be R-modules. A mapping f : M → M1 is said to be an R-module homomorphism (or module homomorphism) if f satisfies the following conditions:

(i). f(x + y) = f(x) + f(y); and (ii) f(rx) = rf(x) for all x,y ∈ M and r ∈ R.

1.5 Definition: Let M be an R-Module. A non-empty subset N of M is called an R-submodule (or a submodule) of M if

(i) For any x, y ∈ N ⇒ x - y ∈ N; and (ii) x ∈ N and r ∈ R ⇒ rx ∈ N.

1.6 Note: (i). For any module M, the sets {0} and M are submodules of M. These two submodules are called trivial submodules of M. (ii). For a non-empty subset X of M, we define the submodule of M generated by X as the intersection of all submodules of M containing X. (iii). The submodule of M generated by X is denoted by <X> or (X). We write <x> (instead of <{x}> or (x)).

1.7 Definitions: Let M be an R-module and M1, M2, …, Ms submodules of M. We

say that M is the direct sum of Mi, 1 ≤ i ≤ s if every element m ∈ M can be

expressed in unique manner as m = m1 + m2 + … + ms with mi ∈ Mi, 1 ≤ i ≤ s.

A Module M is said to be cyclic if there exists an element a ∈ M such that

M = {ra / r ∈ R}. An R-module M is said to be finitely generated if there exist

elements aj ∈ M, 1 ≤ j ≤ n such that M = {r1a1 + … + rnan / rj ∈ R, for 1 ≤ j ≤ n}. It is clear that every cyclic module is finitely generated.

1.8 Definition: (i) If K, A are submodules of M, and K is a maximal submodule of

M with respect to the property K ∩ A = (0), then K is said to be a complement of A (or a complement submodule in M).

(ii) A non-zero submodule K of M is called essential (or large) in M (or M is an

essential extension of K) if A is a submodule of M such that K ∩ A = (0), imply A = (0).

(iii) If A is essential submodule of M and A ≠ M, then we say that M is a proper essential extension of A.

1.9 Note: (i) If K is essential in M, then we denote this fact by K ≤e M.



(ii) Let M be a module, A a submodule of M and B be a maximal among the

submodules of M with respect to the property that A ∩ B = (0). Then A ⊕ B ≤e M, and B is a complement of A. (iii) If C is a submodule of M which is maximal with

respect to the property: C ⊇ A and C∩B = (0), then C ⊕ B ≤e M. Moreover C is a complement of B containing A.

1.10 Result: (i). The intersection of finite number of essential submodules is essential;

(ii). If I, J, K are submodules of M such that I ≤e J, and J ≤e K, then I ≤e K;

(iii). I ≤e J ⇒ (I∩K) ≤e (J ∩ K);

(iv). If I ⊆ J ⊆ K, then I ≤e K ⇔ I ≤e J, and J ≤e K; and

(v). Suppose H, K are two modules and f : K → H is a module isomorphism.

If A is a submodule of K, then A ≤e K ⇔ f(A) ≤e H.

1.11 Lemma: Let G1, G2, … Gn, H1, H2, … Hn be submodules of M such that the

sum G1 + G2 + … + Gn is direct and Hi ⊆ Gi for 1 ≤ i ≤ n. Then

(H1 + H2 + … + Hn) ≤e (G1 + G2 + … + Gn) ⇔ each Hi ≤e Gi for 1 ≤ i ≤ n.

1.12 Note: Consider submodules A, B, C of M as in Note 1.9(ii) & (iii).

Here A⊕B ≤e M and A⊕B ⊆ C⊕B ⊆ M. Using Result 1.10(iv), we get that

A⊕B ≤e C⊕B. By Lemma 1.11, it follows that A ≤e C. Note that C is a complement submodule which is also an essential extension of A.

Finite Goldie Dimension In Modules

It is well known that the dimension of a vector space is defined as the number of elements in its basis. One can define a basis of a vector space as a maximal set of linearly independent vectors or a minimal set of vectors which spans the space. The former when generalized to modules over rings, becomes the concept of “Goldie Dimension”. 2.1 Definition: A module M is said to have Finite Goldie Dimension (written as FGD) if M does not contain a direct sum of infinite number of non-zero submodules. 2.2 Remark: The following two conditions are equivalent:

(i) M has FGD; and

(ii) For any strictly increasing sequence H0 ⊆ H1 ⊆ … of submodules of M,

there exists an integer i such that Hk ≤e Hk+1 for every k ≥ i.

2.3 Note: (i) If M has DCC (Descending Chain Condition) on its submodules, then M has FGD.

(ii). If M has ACC (Ascending Chain Condition) on its submodules, then M has FGD.

(iii) Let M be a module with FGD and H be a non-zero submodule of M. Then H has FGD.



2.4 Definition: A non-zero submodule K of M is said to be an uniform submodule if every non-zero submodule of K is essential in K.

2.5 Theorem: (i) K is an uniform submodule ⇔ K1 , K2 are two submodules of K

such that K1 ∩K2 = (0) ⇒ K1 = (0) or K2 = (0).

(ii) Let K and H be two submodules and f : K → H be module isomorphism. If U

is submodule of K, then U is uniform in K ⇔ f(U) is uniform in H.

(iii) Let H and K be two submodules of M such that H∩K = (0). For a submodule

U of H, we have that U is uniform ⇔ (U + K)/K is uniform in M/K.

2.6 Lemma: Let M be a non-zero module with FGD. Then every non-zero submodule of M contains a uniform submodule.

2.7 Remark: Let M be a module and K a uniform submodule of M.

If L is a submodule of M such that L ⊆ K, then either L = (0) or L is uniform.

2.8 Theorem: (Goldie [ 1 ]): Let M be a module with finite Goldie dimension. (i) (Existence) There exist uniform submodules U1, U2, …, Un whose sum is

direct and essential in M.

(ii) If J is a submodule of M such that J ∩ Ui ≠ (0) for all i (1 ≤ i ≤ n), then J is essential in M.

(iii) (Uniqueness) If there exist uniform submodules V1, V2, …, Vk whose sum is direct and essential in M, then k = n. 2.9 Definition: Let M be a module with FGD. Then by Theorem 2.8, there exist

uniform submodules Ui, 1 ≤ i ≤ n whose sum is direct and essential in M. The number ‘n’ is independent of the choice of the uniform submodules. This number n is called the Goldie dimension of M, and it is denoted by dim M.

2.10 Definition: (i) The elements a1, a2, …, an ∈ M are said to be linearly independent

if the sum of the submodules generated by ai’s (for 1 ≤ i ≤ n) is direct.

(ii) A non-zero element a ∈ M is said to be uniform element (or u–element) if (a) is an uniform submodule.

2.11 Definition: The elements a1, a2, …, an ∈ M are said to be u-linearly independent

elements if (i) each ai is an u-element; and (ii) a1, a2, …, an are linearly independent.

If a1, a2, …, an ∈ M are linearly independent (u-linearly independent), then the set {a1, a2, …, an} is called a linearly independent set (u-linearly independent set). An u-linearly independent set containing the maximum number of u-linearly independent elements is called a basis for M.



Minimal Elements

3.1 Definition: An element x ∈ M is said to be a minimal element if the submodule generated by x is minimal in the set of all non-zero submodules of M. 3.2 Theorem: If M has DCC on its submodules, then every nonzero submodule of M contains a minimal element.

3.3 Note: There are modules which do not satisfy DCC on its submodules, but contains a minimal element. For this we observe the following example.

3.4 Example: Write M = ΖΖΖΖ ⊕ ΖΖΖΖ6. Now M is a module over the ring R = ΖΖΖΖ. Clearly

M have no DCC on its submodules. Consider a = (0, 2) ∈ M. Now the submodule

generated by a, that is, ΖΖΖΖa = {(0, 0), (0, 2), (0, 4)} is a minimal element in the set of all non-zero submodules of M. Hence a is a minimal element. 3.5 Theorem: Every minimal element is an u-element.

3.6 Note: (i) The converse of Theorem 3.5 is not true. For this observe the example 3.7 (given below). (ii) If M is a vector space over a field R, then every non-zero element is a minimal element as well as an u-element. 3.7 Example: Write M = ΖΖΖΖ as a module over the ring R = ΖΖΖΖ. Since ΖΖΖΖ is a uniform module, and 1 is a generator, we have that 1 is an u-element. But 2ΖΖΖΖ is a proper submodule of 1.ΖΖΖΖ = ΖΖΖΖ = M. Hence 1 cannot be a minimal element. Thus 1 is an u-element but not a minimal element. 3.9 Lemma: If x is an u-element of a module M with DCC on submodules, then there

exist minimal element y ∈ Rx such that Ry ≤e Rx. 3.10 Theorem: If M has DCC on its submodules, then there exist linearly independent minimal elements x1, x2, ….., xn in M where n = dim M, and the sum <x1> + …….+ <xn> is direct and essential in M. Also B = {x1, x2,….,xn} forms a basis for M.

Acknowledgements The author acknowledge Dr A V Vijaya Kumari, Organizing Secretary, and Prof. Dr Bhavanari Satyanarayana, Academic Secretary, for inviting him to present this talk.

.

References CHATTERS A.W. & HAJARNIVAS C.R [1]. . "Rings with Chain Conditions Research Notes in Mathematics", Pitman Advanced publishing program, Boston- London -Melbourne, 1980.



GOLDIE A.W. [1]. "The Structure of Noetherian Rings Lectures on Rings and Modules", Springer -

Verlag, New York, 1972. LAMBEK J. [1]. "Lectures on Rings and Modules", Blaisdell Publishing Company, (1966). REDDY Y.V. & SATYANARAYANA BH. [1]. "A Note on Modules", Proc. of the Japan Academy, 63-A (1987) 208-211. SATYANARAYANA BHAVANARI. [1] Lecture on "Modules with Finite Goldie dimension and Finite Spanning dimension",

International Conference on General Algebra, Krems, Vienna, Austria, August, 21-27, 1988.

[2] "A Note on E-direct and S-inverse systems", Proc. of the Japan Academy, 64-A (1988) 292-295.

[3] "The Injective Hull of a Module with FGD", Indian J. Pure & Appl. Math. 20 (1989) 874-883. [4] "On Modules with Finite Goldie Dimension" J. Ramanujan Math. Society. 5 (1990)

61-75. [5] Lecture on "Modules with Finite Spanning Dimension", Asian Mathematical Society

Conference, University of Hong Kong, Hong Kong, August 14-18, 1990. [6] "On Essential E-irreducible submodules", Proc., 4

th Ramanujan symposium on

Algebra and its Applications, University of Madras Feb 1-3 (1995), pp 127-129.

[7]. Lecture on "Finite Goldie Dimension in Modules", Pure Mathematics Seminar

2001, Calcutta University, Kolkata, Feb. 20-21,2001. [8] “Contributions to Near-ring Theory”, Doctoral Dissertation, Acharya Nagarjuna University, 1984. Published by VDM Verlag Dr Mullar, Germany, 2009 (ISBN: 978-3-639-22417-7). SATYANARAYANA BHAVANARI AND MOHIDDIN SHAW Sk [1] “Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978-3-639-23197-7). SATYANARAYANA BHAVANARI, AND NAGARAJU D. [1] “Dimension and Graph Theoretic Aspects of Rings (Monograph)” VDM Verlag Dr Muller, Germany, 2011. (ISBN: 978-3-639-30558-6) SATYANARAYANA BHAVANARI, AND RAMA PRASAD J.L [1] “Prime Fuzzy Submodules”, VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978-3-639-24355-0). SATYANARAYANA BHAVANARI AND SYAM PRASAD K [1] “Discrete Mathematics and Graph Theory”, Printice Hall of Inida, New Delhi, 2009. (ISBN:978-81-203-3842-5). SATYANARAYANA BHAVANARI, SYAM PRASAD K & NAGARAJU D. [1]. "A Theorem on Modules with Finite Goldie Dimension", Soochow Journal of

Mathematics, 32 (2006) 311 – 315.

LIVE AND LET LIVE - HAVE FEW WANTS TO HARM OTHERS IS A SIN – DO NOT THINK TO HARM OTHERS

LOVE THE HUMANITY - EACH SOUL IS POTENTIALLY DEVINE



“IDEAL MAPPING IN

GAMMA RINGS”

----------------------------------------------------------------------------------------------------------- Abstract: In this lecture, we collect some results related to the concepts: Γ-ring, ideal

of a Γ-ring, Γ-homomorphisms, m-system, prime ideal, g-system, prime radical. Finally,

we prove a theorem: If M, M1 are two Γ-rings, f: M → M1 a Γ-epimorphism and S ⊆ M,

then S is an m-system in M ⇔ f(S) is an m-system in f(M).

Introduction: Historically, the first step towards Γ-rings was taken by Nobusawa

1964 and the next step was taken by Barnes 1966. Γ-rings of Barnes were much studied.

Many authors studied the system Γ-ring in different aspects. The radical theory in

Γ-rings was studied by several authors like Booth, Godloza, Satyanarayana, Pradeep Kumar and Srinivasa Rao. 1.1 Definition : Let M be an additive group whose elements are denoted by a, b, c, ...

and Γ another additive group whose elements are α, β, γ, ... . Suppose that aαb is defined

to be an element of M and that αaβ is defined to be an element of Γ for every a, b, α and

β. If the products satisfy the following three conditions for every a, b, c ∈ M, α, β ∈ Γ:

(i) (a + b)αc = aαc + bαc; a(α + β)b = aαb + aβb; aα(b + c) = aαb + aαc;

(ii) (aαb)βc = aα(bβc) = a(αbβ)c; and

(iii) If aαb = 0 for all a and b in M, then α = 0,

then M is called a ΓΓΓΓ-ring.

The definition of Γ-ring in the sense of Barnes [1966] is as follows: 1.2 Definition: Let M and Γ be additive Abelian groups. M is said to be a ΓΓΓΓ-ring if

there exists a mapping M x Γ x M → M (the image of (a, α, b) is denoted by aαb) satisfying the following conditions (i) and (ii):

(i) (a + b)αc = aαc + bαc; a(α + β)b = aαb + aβb; aα(b + c) = aαb + aαc; and

(ii) (aαb)βc = aα(bβc) for all a, b, c ∈ M and α, β ∈ Γ.

The concept Γ-ring in the sense of Barnes is a generalization of the concept Γ-ring in the

sense of Nobusawa. Several authors preferred to study the Γ-ring in the sense of Barnes.

Henceforth, all Γ-rings considered (in this paper) are Γ-rings in the sense of Barnes.

A natural example of a Γ-ring can be constructed in the following way: 1.3 Example (Barnes [1]): Let (X, +), (Y, +) be two Abelian groups. Write

M = Hom(X, Y), Γ = Hom(Y, X). M and Γ are additive Abelian groups with respect to

the usual addition of mappings. (That is (f + g)(x) = f(x) + g(x) for all x ∈ X).

Let a, b ∈ M and α ∈ Γ. Then b: X → Y, α: Y → X and a: X → Y. Suppose aαb is the

usual composition of mappings. Since a, α, b are homomorphisms, we have that

15 Min. Talk _____________________________________________________

Mrs. Shakeera Sk.



aαb: X → Y is a homomorphism. Therefore aαb ∈ M. With these operations, it is easy

to verify that M is a Γ-ring.

1.4 Example: Let R be a ring. Write M = R, Γ = R. Take a, b, c ∈ M, α, β, γ ∈ Γ. aαb

is the product of a, α, b in R. So aαb ∈ R = M. Then M is a Γ-ring.

1.5 Example: Let M be any additive non Abelian group. Take Γ = {0}. Define

aαb = a0b = 0M and αbβ = 0b0 = 0Γ for all a, b ∈ M, α, β ∈ Γ. Then it is easy to verify

that M is a Γ-ring. 1.6 Example: Suppose M be a right R-module and suppose there exists m ∈ M such that

(0:m) = 0. Take Γ= HomR(M, R). Define aγb = a.γ(b) for all a, b ∈ M, γ ∈Γ. Then it is

easy to verify that M is a Γ-ring. 1.7 Example: Let U, V be vector spaces over the same field F. Write M = Hom(U, V),

Γ = Hom(V, U). Then M is a Γ-ring with respect to point wise addition and composition of mappings. 1.8 Notation: Let M be a Γ-ring. For A ⊆ M, B ⊆ M, ∆ ⊆ Γ we denote the set

{aαb / a ∈ A, α ∈ ∆, b ∈ B} by A∆B.

The set AΓB will be denoted by AB.

1.9 Definitions: (i) A subset A of a Γ-ring M is said to be a right ideal of M if A is an

additive subgroup of M and AΓM ⊆ A. (ii) A subset A of a Γ-ring M is said to be a left ideal of M if A is an additive subgroup of M and MΓA ⊆ A. (iii) If A is both left and right

ideal of M then A is said to be an ideal of M. The smallest left ideal containing a ∈ M is

denoted by ⟨a⟩l. This is the intersection of all left ideals of the Γ-ring M containing the element a. We may also call this left ideal as the left ideal generated by the element a ∈ M. The smallest left ideal containing a subset X of M is denoted by ⟨X⟩l. We may also call this left ideal as the left ideal generated by the subset X of M. The smallest ideal

containing a subset X of M is denoted by ⟨X⟩. We may also call this ideal as the ideal generated by the subset X of M. The ideal ⟨{a}⟩ is denoted by ⟨a⟩. 1.10 Definition: Let M be a Γ-ring and I an ideal of M. Consider M/I = {x + I / x ∈ M}, the quotient group of M with respect to the addition subgroup I. Define

(x + I)γ(y + I) = xγy + I for all x, y ∈ M and γ ∈ Γ. Then M/I becomes a Γ-ring. It is

called as quotient ΓΓΓΓ- ring of M with respect to the ideal I. 1.11 Definition (Barnes [ 1 ]): Let M, M1 be two Γ-rings. A mapping h: M → M1 is said

to be a Γ-ring homomorphism (or Γ-homomorphism) if it satisfies the following two

conditions: (i) h(a + b) = h(a) + h(b) for all a, b ∈ M; and (ii) h(aγb) = h(a)γh(b) for all a,

b ∈ M and γ ∈ Γ.

1.12 Theorem (Barnes [1]): (i) If M, M1 are Γ-rings and f: M → M1 is a

Γ-homomorphism, then ker f = {x ∈ M / f(x) = 0} is an ideal of M.



(ii) If f is onto Γ-homomorphism, then f(A) is an ideal of M1 if A is an ideal of M; and

B = {x / f(x) ∈ B1} is an ideal of M for all ideals B1 of M.

1.13 Definition: A subset S of a Γ-ring M is said to be an m-system if S = φ or if a, b ∈ S

implies ⟨a⟩ ⟨b⟩ ∩ S ≠ φ.

1.14 Examples: Consider M = Γ = ℤ, the ring of integers. Then M is a Γ-ring.

(i) If n > 0 and n is a prime number, then nℤ is a prime ideal of the Γ-ring M.

(ii) Let n ∈ ℤ such that n > 0. Write S = {n, n2, n3, ...}. Now we verify that S is an

m-system. Let x, y ∈ S ⇒ x = nk, y = ns for some k, s. Now nk+s+1 = nknns = xny ∈ ⟨x⟩ ⟨y⟩ and so nk+s+1 ∈ ⟨x⟩ ⟨y⟩ ∩ S. Therefore ⟨x⟩ ⟨y⟩ ∩ S ≠ φ. Hence S is an m-system. 1.15 Definition: An ideal P of M is said to be a prime ideal if for any two ideals A, B of

M, AB ⊆ P ⇒ A ⊆ P or B ⊆ P. 1.16 Theorem (Barnes [1]): An ideal P of a Γ-ring M is prime if and only if C(P) = M\P is an m-system. 1.17 Theorem (Barnes [1]): If I and P are ideals of a Γ-ring M, I ⊆ P and P is prime, then

P/A is prime in M/A. Conversely, if P1 is a prime ideal of M/A and f: M→ M/A is the

canonical Γ-epimorphism, then P = f –1(P1) is a prime ideal of M.

1.18 Definition: Let A be an ideal of M. The prime radical of A (denoted by r(A)) is defined as the set of all elements x of M such that every m-system containing x contains an element of A. The prime radical of M is defined as the prime radical of the zero ideal.

1.19 Result (Barnes [1]): If A is any ideal of the Γ-ring M, then the prime radical r(A) is equal to the intersection of all prime ideals containing A.

1.20 Definition (Hsu [5]): Let M be a Γ-ring. We define g as a function of M into the

family of all ideals of M satisfying the following two conditions: (i) a ∈ g(a); and

(ii) x ∈ g(a) + A ⇒ g(x) ⊆ g(a) + A for any element a ∈ M and for any ideal A of M. We fix such an ideal mapping g on M.

1.21 Lemma (Satyanarayana [9]): g(x) = g(0) + (x) for all x ∈ M. 1.22 Definitions: A subset S of the Γ-ring M is said to be a g-system if either S = φ or S

contains an m-system S1 such that g(a) ∩ S1 ≠ φ for every element a of S where S1 is called a kernel of S. Any m-system is a g-system with kernel itself. (Hsu [5]) An ideal P of the

Γ-ring M is said to be a g-prime if either M\P is a g-system in M or P = M.

(Satyanarayana [ 9 ]) A Γ-ring M is said to be prime ΓΓΓΓ-ring if (0) is a prime ideal of M.

A Γ-ring M is said to be a g-prime ΓΓΓΓ-ring if the ideal (0) is a g-prime ideal.



1.23 Note: (i) Let P be a prime ideal ⇒ M\P is an m-system (by Theorem 1.16) ⇒ M\P is a

g-system (by Definitions 1.22) ⇒ P is a g-prime ideal; (ii) The converse of (i) is not true. (For this observe the following Example 1.24). The following example provides an example of a g-prime ideal which is not prime.

1.24 Example: Let M = ℤ, the set of all integers and Γ = { 0, ± 2, ± 4, ... }. Clearly M is a

Γ-ring. Let P = ⟨32⟩. We define g(a) = ⟨{a, 2n}⟩ for all a ∈ M where n is a fixed positive integer. Let S1 = { 2, 22, 23, ... }. Now as in Example 2.3.2 (ii), we can verify that S1 is an

m-system. Let x ∈ M\P. Then 2 n ∈ g(x) ∩ S1. Hence M\P is a g-system with kernel S1.

Therefore P is a g-prime ideal. Clearly ⟨3⟩ ⟨3⟩ ⊆ ⟨32⟩ = P. But ⟨3⟩ ⊆ P. Therefore P is not a prime ideal. Thus we conclude that P is a g-prime ideal but not a prime ideal.

1.25 Lemma (Hsu [5]): For any g-prime ideal P, a, b ∈ M we have that

g(a)g(b) ⊆ P ⇒ a ∈ P or b ∈ P.

1.26 Definition: The set {x ∈ M / every g-system containing x contains 0} is called the g-prime radical of M. It is denoted by rg(M). 1.27 Theorem (Hsu [5]): rg(M) = the intersection of all g-prime ideals of M. 1.28 Lemma (Satyanarayana [9]): If A is an ideal of M containing g(0), then the following two conditions are equivalent: (i) A is prime; and (ii) A is g-prime. 1.29 Theorem (Satyanarayana [9]): If A is an ideal of M, then either rg(A) = A or rg(A) = r(A). Moreover, rg(M) = (0) or rg(M) = r(M).

1.30 Theorem (Satyanarayana [ 9 ]): If r(M) ≠ 0, then the following two conditions are equivalent: (i) rg(M) = r(M); and (ii) every g-prime ideal is a prime ideal.

1.31 Lemma: Suppose f: M → M1 is a Γ-epimorphism from the Γ-ring M onto the Γ-ring

M1. If a ∈ M, then f(⟨a⟩) = ⟨f(a)⟩ . 1.32 Lemma: Let M, M1 be two Γ-rings and f: M → M1 is a Γ-ring epimorphism. If S is an m-system in M, then f(S) is an m-system in f(M) = M1. 1.33 Lemma: Let M, M1be two Γ-rings. If S* is an m-system in f(M) = M1, then f -1(S*) is an m-system in M. 1.34 Theorem: Let M, M1 be two Γ-rings, f: M → M1 a Γ-epimorphism and S ⊆ M. Then

S is m-system in M ⇔ f(S) is an m-system in f(M). Conclusion: In the first part of the paper, we collected some results related to the concepts:

Γ-ring, ideal of a Γ-ring, Γ-homomorphisms, m-system, prime ideal, g-system, prime radical. Finally, in the last section, we presented the proof of a new theorem (1.34): If M,

M1 are two Γ-rings, f: M → M1 a Γ-epimorphism and S ⊆ M, then S is an m-system in M

⇔ f(S) is an m-system in f(M). We presented necessary examples.



Acknowledgements: The author acknowledge Dr A V Vijaya Kumari, Organizing Secretary, and Prof. Dr Bhavanari Satyanarayana for inviting her to present this talk.

REFERENCES

[1] BARNES W.E "On the Γ-rings of Nobusawa" Pacific J. Math 18 (1966) 411-422.

[2] BOOTH G.L "A Contribution to the Radical Theory of Gamma Rings". Ph.D,

Thesis, University of Stellenbosch, South Africa,1985. [3] BOOTH G.L. & GODLOZA L "On Primeness and Special Radicals of

Γ-rings", Rings and radicals, Pitman Research notes in Math series (contains

selected lectures presented at the international conference on Rings and Radicals,

held at Hebei, Teachers University, Shijazhuang, Chaina, August 1994) pp 123-130. [4] FACCHINI ALBERTO "Module Theory", Progress in Mathematics, Vol.167,

Birkhäuser Verlag, Switzerland, 1998.

[5] HUS D.F "On Prime Ideals and Primary Decompositions in Γ-rings", Math.

Japonicae, 2 (1976) 455-460. [6] NOBUSAWA "On a Generalization of the Ring theory" Osaka J. Math. 1(1964) 81-

89.

[7] PRADEEP KUMAR T.V. "On g1-γ-Prime Left Ideals and related Prime Radical in

Γ-rings", M. Phil., dissertation, Acharya Nagarjuna University, 1998.

[8] SATYANARAYANA BHAVANARI "A Note on Γ-rings" Proc. Japan Acad. 59-A (1983) 382-33.

[8 A] SATYANARAYANA BHAVANARI, “Contributions to Near-ring Theory”, VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-639-22417-7)

[9] SATYANARAYANA BHAVANARI "A Note on g-prime Radical in Gamma rings", Quaestiones Mathematicae, 12(4) (1989) 415-423.

[9A] SATYANARAYANA BHAVANRI AND MOHIDDIN SHAW Sk, “Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978-3-639-23197-7). [9B] SATYANARAYANA BHAVANRI AND NAGARAJU.D, “Dimension and Graph Theoretic Aspects of Rings (Monograph)” VDM Verlag Dr Muller, Germany, 2011. (ISBN: 978-3-639-30558-6) [10] SATYANARAYANA BH., PRADEEP KUMAR T.V. & SRINIVASA RAO M.

"On Prime Left Ideals in Gamma rings", Indian J. Pure and Appl. Math. 31(6) (2000) 687-693.

[10A] SATYANARAYANA BHAVANRI AND RAMA PRASAD J.L., “Prime Fuzzy Submodules”, VDM Verlag Dr Mullar, Germany, 2010. (ISBN: 978-3-639-24355-0). [11] SATYANARAYANA BHAVANRI AND SHAKIRA SHAIK “Gamma rings

and m-systems”, Proceedings of the National Seminar on Present Trends in Mathematics and its Applications (Editors: Dr Eswaraiah Setty, Dr Bhavanari Satyanarayana, and Dr Kunchum Syam Prasad) November 2010, PP 141-147.

[12] WIEGANDT RICHARD "Radical Theory of Rings", The Mathematics Student 51 (1983)145-185.



Uniform and Essential Ideals

in Associative Rings

_________________________________________________________________________________________________________________

Abstract

We present a study of uniform ideals and essential ideals in Associative Rings.

Introduction The dimension of a vector space is defined as the number of elements in the basis. One can define a basis of a vector space as a maximal set of linearly independent vectors or a minimal set of vectors which span the space. The former when generalized to modules over rings become the concept of Goldie dimension. Goldie proved a structure theorem for modules which states that “a module with finite Goldie dimension (FGD, in short) contains a finite number of uniform submodules U1, U2, …, Un whose sum is direct and essential in M”. The number n obtained here is independent of the choice of U1, U2, …, Un and it is called as Goldie dimension of M. The concept Goldie dimension in Modules was studied by several authors like Reddy, Satyanarayana, Syam Prasad, Nagaraju (refer [1, 1, 1]).

If we consider ring as a module over itself, then the existing literature tells about dimension theory for ideals (i.e., two sided ideals) in case of commutative rings; and left (or right) ideals in case of associative (but not commutative) rings. So at present we can understand the structure theorem for associative rings in terms of one sided ideals only (that is, if R has FGD with respect to left (right) ideals, then there exist n uniform left (or right) ideals of R whose sum is direct and essential in R). This result cannot say about the structure theorem for associative rings in terms of two sided ideals. So to fill the gap, we have to prove the structure theorem for associative rings with respect to two sided ideals. Before proving structure theorem, the concepts Uniform Ideal and Essential Ideals are to be studied.

Throughout the paper R denotes an associative ring (need not be commutative).

The paper is divided into three sections. In the beginning, we introduce and study the concepts: complement, essential with respect to two sided ideals of R. Later, we introduce the concept: uniform ideal and study few fundamental results. Let R be a fixed

(not necessarily commutative) ring. We write K ⊴ R to denote ‘K is an ideal of R’.

We use the term “ideal” for “two sided ideal”. The ideal generated by an element a ∈ R is denoted by <a>. We do not include the proofs of some results when they are easy or straight forward verification.

Essential Ideals

1.1 Definitions: Let I, J be two ideals of R such that I ⊆ J.

15 Min. Talk ________________________________________

Mr. K.S. Balamurugan



(i). We say that I is essential (or ideal essential) in J if it satisfies the following

condition: K ⊴ R, K ⊆ J, I ∩ K = (0) imply K = (0).

(ii). If I is essential in J and I ≠ J, then we say that J is a proper essential extension of I.

If I is essential in J, then we denote this fact by I ≤e J.

1.2 Definition: If K ⊴ R, A ⊴ R and K is a maximal element in {I / I ⊴ R,

I ∩ A = (0)}, then we say that K is a complement of A (or a complement in R).

1.3 Note: Let I and J be ideals of R.

(i) I ≤e J ⇔ I ∩ K = (0), K ⊴ R ⇒ J ∩ K = (0).

(ii) B is a complement in R ⇔ there exists an ideal A of R such that B ∩ A = (0)

and K1 ∩ A ≠ (0) for any ideal K1 of R with B ⊊ K1. In this case B + A ≤e R.

1.4 Result: (i) The intersection of finite number of essential ideals is essential;

(ii) If I, J, K are ideals of R such that I ≤e J, and J ≤e K, then I ≤e K;

(iii) I ≤e J ⇒ I ∩ K ≤e J ∩ K;

(iv) If I ⊆ J ⊆ K, then I ≤e K if and only if I ≤e J, and J ≤e K; and

(v) If R1, R2 are two rings, f: R1 → R2 is a ring isomorphism, and A is an ideal of R1,

then A ≤e R1 ⇔ f(A) ≤e R2.

1.5 Note: (Refer page 158 of [1]) If R is a ring, a ∈ R, then

<a> = 1

/ , , 0, , , ,k

i i i i

i

ra as na ras k n k r s s r R=

+ + + ∈ ∈ ≥ ∈

∑ � � .

1.6 Remark: If a, b ∈ R and x ∈ <a>, then there exists y ∈ <b> such that

x + y ∈ <a + b>.

1.7 Lemma: (i) L1, L2, K1, K2 are ideals of R such that Li ⊆ Ki for i = 1, 2 and

K1 ∩ K2 = (0). Then L1 ≤e K1 and L2 ≤e K2 ⇔ L1 + L2 ≤e K1 + K2; and (ii) Let K1, K2, … Kt, L1, L2, … Lt are ideals of R such that the sum

K1 + K2 + … + Kt is direct and Li ⊆ Ki for 1 ≤ i ≤ t. Then

L1 + L2 + … + Lt ≤e K1 + K2 + … + Kt ⇔ Li ≤e Ki for 1 ≤ i ≤ t.

1.8 Note: Consider ideals A, B, C of R as in Note 1.3 (ii) & (iii). Here A ⊕ B ≤e R

and A ⊕ B ⊆ C ⊕ B ⊆ R. Using Result 1.4(iv), we get that A ⊕ B ≤e C ⊕ B. By

Lemma 1.7, it follows that A ≤e C. Note that C is a complement ideal which is also an essential extension of A.

Uniform Ideals 2.1 Definition: A non-zero ideal I of R is said to be uniform if (0) ≠ J ⊴ R, and

J ⊆ I ⇒ J ≤e I.

2.2 Note: Let R1, R2 be two rings and f : R1 → R2 is a ring isomorphism. I, J ⊴ R1.

Then f -1(I) ∩ f -1(J) = f -1(I ∩ J).



2.3 Theorem: (i) I is an uniform ideal ⇔ L ⊴ R, K ⊴ R, L ⊆ I, K ⊆ I, L ∩ K = (0)

⇒ L = (0) or K = (0).

(ii) Let R1 and R2 be two rings and f: R1 → R2 be ring isomorphism. If U is ideal of

R1, then U is uniform in R1 ⇔ f(U) is uniform in R2.

(iii) Let H and K be two ideals of R such that H ∩ K = (0). For an ideal U of R

contained in H, we have that U is uniform ⇔ (U + K)/K is uniform in R/K.

(iv) If U and K are two ideals of R such that U ∩ K = (0), then U is uniform in R ⇔ (U + K)/K is uniform in R/K.

2.4 Remark: Let K be an uniform ideal of R and L ⊴ R such that L ⊆ K. Then either L = (0) or L is uniform.

Acknowledgements: The author acknowledge Dr A V Vijaya Kumari, Organizing Secretary, and Prof. Dr Bhavanari Satyanarayana for inviting him to present this talk.

References ADHIKARI M. R [1]. "Groups, Rings & Modules with Applications", University Press (India) Ltd., Hyderabad, India, (1999). GOLDIE A. W [1]. "The Structure of Noetherian Rings", Lectures on Rings and Modules, Springer-

Verlag, New York (1972). HERSTEIN I. N [1]. "Topics in Algebra" (Second Edition), Wiley Eastern Limited, New Delhi (1998). REDDY Y. V & SATYANARAYANA BH [1]. "A Note on Modules", Proc. Of the Japan Academy, 63-A (1987) 208-211. SATYANARAYANA BHAVANARI [1]. "A note on E-direct and S-inverse Systems", Proc. of the Japan Academy, 64-A

(1988) 292-295. SATYANARAYANA BHAVANARI [1] “Contributions to Near-ring Theory”, Doctoral Dissertation, Acharya Nagarjuna University, 1984. Published by VDM Verlag Dr Mullar, Germany, 2009 (ISBN: 978-3-639-22417-7). SATYANARAYANA BHAVANARI AND MOHIDDIN SHAW Sk. [1] “Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978-3-639-23197-7). SATYANARAYANA BHAVANARI AND NAGARAJU D [1] “Dimension and Graph Theoretic Aspects of Rings (Monograph)” VDM Verlag Dr Muller, Germany, 2011. (ISBN: 978-3-639-30558-6) SATYANARAYANA BHAVANARI AND RAM PRASAD J.L [1] “Prime Fuzzy Submodules”, VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978-3-639-24355-0). SATYANARAYANA BHAVANARI, SYAM PRASAD K & NAGARAJU D [1]. "A Theorem on Modules with Finite Goldie Dimension", Soochow J. Mathematics,

32(2) (2006) 311-315.

Lecture on

Interval valued C-Prime Fuzzy Ideals of Nearrings

ByJAGADEESHA B,Research Scholar,

Department of Mathematics,Manipal Institute of Technology,

Manipal University, Manipal-576104

Abstract

In this lecture we introduce interval valued C-prime ideals of a nearring.An interval valued C-prime ideal of a nearring is an extended notionof c-prime ideal of nearring. Some characterization and properties arediscussed.Key Words

Nearring, Interval, C-prime, Interval Valued C-prime, Level subset,Characteristic functionIntroduction

Zadeh introduced the concept of a fuzzy subset of a nonempty set X asa function from X to [0,1]. Rosenfield introduced the concept of fuzzysubgroups following Zadeh, fuzzy algebra theory has been developedby many researchers. In 1975 Zadeh introduced the concept of intervalvalued fuzzy subsets where the values of the membership functions areintervals of numbers instead of the numbers. In this lecture we defineinterval valued c-prime fuzzy ideals of nearrings.

1. Basic definitions and preliminaries

Definition 1.1. A nonempty set N together with two binary oper-ations + and . denoted by (N,+, .) is called as nearring if for alla, b, c ∈ N

(i) (N,+) is a group (Not necessarily abelian).(ii) (a.b).c = a.(b.c)(iii) (a+ b).c = a.c+ b.c

More precisely we have defined a right nearring.

Throughout this paper N and M denote right nearrings.

Definition 1.2. Let (N,+,.) is a near-ring. Let (I,+) is a normalsubgroup of (N,+)I is called as right ideal if for all i ∈ I, n ∈ N then i.n ∈ I

I is called left ideal for all n,m ∈ N , i ∈ I if n(m+ i)− nm ∈ I

I is called an ideal if I is a left ideal as well as right ideal1

2

Definition 1.3. An ideal I of N is called c-prime if a, b ∈ N withab ∈ I for all implies a ∈ I or b ∈ I

Definition 1.4. Let X be a nonempty set, the mapping µ : X → [0, 1]is called fuzzy subset of X

Definition 1.5. For t ∈ [0, 1], the set µt = {x ∈ N | µ(x) ≥ t} iscalled level subset of µ

Definition 1.6. An interval number in [0,1], say a is a closed subin-terval of [0,1]. That is a = [a−, a+] where 0 ≤ a− ≤ a+ ≤ 1.If a+ = a− = c then a = [c, c]. For the sake of convenience we considerit as a closed interval. Let D[0,1] denotes the set of all closed intervalsof [0,1]

Definition 1.7. If if I1 = [a1, b1], I2 = [a2, b2] Ii = [ai, bi] are elementsof D[0,1], thenI1 ∧ I2 = [a1 ∧ a2, b1 ∧ b2].I1 ∨ I2 = [a1 ∨ a2, b1 ∨ b2].∧i{Ii} = [∧i{ai},∧i{bi}].∨i{Ii} = [∨i{ai},∨i{bi}].I1 ≤ I2 if and only if a1 ≤ a2 and b1 ≤ b2

Definition 1.8. Let X be a nonempty set, the mappingµ : X → D[0, 1] is called an interval valued(i-v) fuzzy subset of X.

Definition 1.9. Let µ be an i-v fuzzy subset of a nearring N . Then µis called interval valued fuzzy ideal(IVFI) if(i) µ(x− y) ≥ µ(x) ∧ µ(y)(ii) µ(y + x− y) = µ(x)(iii) µ(xy) ≥ µ(x)(iv) µ(x(y + i)− xy) ≥ µ(i)

Definition 1.10. A i-v fuzzy ideal I of N is called as an i-v c-primefuzzy ideal if for all a, b ∈ Nµ(a) ∨ µ(b) ≥ µ(ab)

Definition 1.11. Let µ be a i-v fuzzy subset of N then for t ∈ D[0, 1]the set µt = {x ∈ N | µ(x) ≥ t} is called as level subset of N withrespect to µ. We denote µ∗ = {x ∈ N | µ(x) ≥ µ(0)}

Definition 1.12. Let f : M → N be a mapping. Then any i-v fuzzysubset µ is called f invariant if f(x) = f(y) implies µ(x) = µ(y) for allx, y ∈ N

Definition 1.13. A i-v fuzzy ideal µ is said to have Insertion FactorsProperty(IFP) if for a, b ∈ N , µ(anb) ≥ µ(ab) for all n ∈ N

3

2. Interval Valued C-prime fuzzy Ideals Of Nearring

Definition 2.1. A L-fuzzy ideal µ of N is called i-v c-prime fuzzy idealif for all a, b ∈ N, {µ(a) ∨ µ(b)} ≥ µ(ab).

We can is easily verify that if µ is an i-v fuzzy ideal of N thenµ(0) ≥ µ(x) for all x ∈ NExample 2.2. Let N = {0, a, b, c} be a set with two binary operations+ and · defined as follows

+ 0 a b c0 0 a b ca a 0 c bb b c 0 ac c b a 0

. 0 a b c0 0 0 0 0a 0 a 0 ab b b b bc b c b c

Define a i-v fuzzy subset µ by

µ(x) =

{β if x ∈ {0, a}α if x ∈ {b, c}

Where α < β . Then µ is an i-v c-prime fuzzy ideal of N .

Proposition 2.3. Let µ be an i-v fuzzy ideal of N . Then µt = µt−∩µt+

Proposition 2.4. Let µ be a i-v fuzzy ideal of N. Then µ is an i-vc-prime fuzzy ideal of N if and only if and only if the level subset µt isan i-v fuzzy ideal of N for all t ∈ D[0, 1]

Corollary 2.5. Let µ be a i-v c-prime fuzzy ideal of N. Then µ∗ is ai-v c-prime L fuzzy ideal of N

Proposition 2.6. Let µ be an i-v c-prime fuzzy ideal of N if and onlyif µ− and µ+ are c-prime fuzzy ideals of N .

Theorem 2.7. Let µ be an i-v fuzzy ideal of N . Then the set Nµ ={x ∈ N | µ(x) = µ(0)} is an i-v c-prime fuzzy ideal of N

4

Theorem 2.8. Let f : M → N be a homomorphism. If µ is an i-vc-prime fuzzy ideal of M then f−1(µ) is an i-v c-prime fuzzy ideal ofN .

Acknowledgments

I thank Manipal University and St. Joseph Engineering college fortheir encouragement. I thank Dr Bhavanari Satyanarayana academicsecretary and organizers of this National Seminar on ”Present Trends inAlgebra and its Applications” . I thank my research guide Dr KunchamSyam Prasad, and co-guide Dr Kedukodi Babushri Srinivas for theirsuggestions.

References

[1] S. Abou-Zaid, On fuzzy subnear-rings and ideals, Fuzzy Sets and Syst. 44(1991), 139-146.

[2] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, seconded., Springer-Verlag, USA, 1992.

[3] B. Davvaz, (ε, ε∨q)-fuzzy subnear-rings and ideals, Soft Computing 10(2006),206-211.

[4] B. S. Kedukodi, Fuzzy and Graph Theoritical aspects of Nearrings, DoctoralThesis, Manipal University, 2009.

[5] B. S. Kedukodi, S. P. Kucham, and S. Bhavanari, C-prime fuzzy ideals ofnearrings, Soochow Journal of Mathematics (2007), 891-901.

[6] S. Bhavanari, S. P. Kuncham, and B. S. Kedukodi, Graph of a nerring withrespect to an ideal, Commn. Algebra (2010), 1957-1962.

[7] G. Pilz, Near-Rings, Revised edition, North Hollond, 1983.[8] S. Bhavanari, S. P. Kuncham and P. K. Tumurukota, On IFP N-groups and

fuzzy IFP ideals, Indian Journal of Mathematics 46 (1) (2004), 11-19.[9] L. A. Zadeh, Fuzzy sets, Information and control 8, 3(1965), 338-353 .



67

A graceful numbering of

a new class of graphs Author: S.A. Mariadoss St.Aloysius College, Mangalore, Karnataka

E-mail: [email protected]

------------------------------------------------------------------------------------------------------------

Abstract: In [4], it is proved that for every even perfect number n, there exists a gracefulnumbering of a graph G = (V, E), with |E| = 2k = D(n),where D(n) is the set of all divisors of n. Since there are only 46 perfect numbers known (as on today), there are precisely 46. such graceful graphs. The question arises whether it is possible to have infinitely many such graceful graphs. Since every perfect number is known to be even, a graceful graph is constructed for an even integer n. This leads to the question whether it is possible to extend the result for every even integer and possibly for an odd integer too.

In this paper, we are able to obtain a class of graceful graphs for every positive

integer n ≥ 4. It is interesting to observe that every such graph has always have two spanning trees such that one is (k, 1) graceful, or graceful or, and the other is almost graceful or (k, 1) –graceful. Thus there exists infinitely many (k,1)-graceful trees.

Equivalently, there is an arbitrarily graceful tree for every k≥ 2. Further, for each n ≥ 4 there exist a tree that is graceful or, almost graceful according as n is even or odd.

It is interesting to note that this graceful graph G always contains a graceful minimal spanning tree Tg and also contain a (k, 1) – graceful maximal spanning tree.

Key Words: (k, d) – graceful, graceful and almost graceful graphs. And G is a lattice of k – spanning trees.

Graceful Graphs Definition 1.1: [1] An injective map f:V → S = {0, 1, 2,.....,q} of a graph G

induces the map f+ : E → {1, 2,....,q}, defined by f+(uv) = |f(u) – f(v)|, for every e = uv in E. If f+ is also injective, f is geodetic labeling. An optimal geodetic labeling is called (k, d) – graceful, if f+(E) = {k, k+d,..., k+(q – 1)d} for some positive integers k and d. If k = d = 1, (k, d) graceful is same as ‘graceful’. Any (k, 1) graceful-graph, need not be graceful.

Definition 1.2: A graph G = (V, E) is called almost graceful, if the vertex

labeling f resulting in an edge-labelling f+ such that f+(E) ≠ {1, 2, ....., q}, but f+(E)

contains a large sequential subset A of S′ = S – {0}, with 2 ≤ |A| < |S′|.

Example 1.3: let G = (V, E) be a graph with |V| = 5. S = {0, 1,....., 11}.

For K = {0, 2, 7, 8, 11}, the vertex-labeling is bijective. The induced edge-labels are 1, 2,.....9; 11. The largest sequential subset with atleast 2 elements, is

{1, 2,...,9}. So G is almost graceful, but not graceful.

Theorem 1.4: For every positive integer n ≥ 4, there exist a graceful graph

G = (V, E). If n = 2k, k≥ 2. |V| = k + 2, |E| = 2k and if n = 2k + 1, k ≥ 2,

|V| = k + 3, |E| = 2k + 1.

Research Paper (Oral Presentation)

------------------------------------------------------

Presenter: S.A. Mariadoss



68

Proof: (i) let n = 2k (k ≥ 2). G = (V, E), |E| = 2k

Define f : V → S = {0, 1, ...., 2k}

f(Vi) = 0, i = 1,....., k – 1

f(Vj) = j, j = k, k + 1,...., 2k

let V = {V1, V2...., V2k} then |V| = k + 2

let Ej = {(0, j) : j = k, k + 1, ...., 2k}

Ei = {(k+1, i) : i = k + 2,...., 2k}

If E = Ej ∪ Ei, for ej ∈ Ej, f+(ej) = |j|, for j = k,..., 2k

for ei ∈ Ei, f+(ei) = |(k+1 + i) – (k + 1)| = i, for i = 1,...., k – 1

Since E = Ei ∪ Ej,

f+(E) = f+(Ei) ∪ f+(Ej)

= {1,....k – 1, k, k+1,..., 2k}

∴ G = (V, E) is graceful

case (ii): n = 2k+1, k ≥ 2, G = (V, E)

f:V → {0, 1, ....., 2k, 2k+1}

Vi → f(Vi) = 0, for i = 1, 2,...., k – 1

Vj → f(Vj) = j, j = k, k+1,....., 2k+1

V = {V1, Vk, Vk+1,...., V2k+1}, |V| = k + 3

Let E = Ei ∪ Ej, where Ei = {(k+1, i): I = k + 2, ..., 2k}

Ej = {(0, j) : j = k, k+1, ...., 2k}

For ei ∈ Ei, f+(ei) = |(k+1 + i) – (k+1)| = i, for i = 1,...., k - 1

ej ∈ Ej, f+(ej) = |j – k – 1|, for j = k + 2,...., 2k, 2k + 1

Then f+(E) = f+(Ei) ∪f+(Ej) = {1, 2,..., 2k+1}

∴ G = (V, E) is graceful, with |E| = 2k + 1, |V| = k + 3

Combining case (i) and case (ii), we conclude that for every positive integer

n ≥ 4, we can construct a graceful graph for G. If n is even, graph of G is a triangular-book with a book-mark. And if n is odd, G is a triangular book-graph with two book-marks.

•

•

•

•

• k+1

k

2k

k+3 •

O

k+2

•

•

•

• k+1

k

2k+1

2k

k+3 •

O

k+2



69

The following results concerns the existence of two special spanning trees of the graceful graph G. These spanning trees are either (k, 1)-graceful, graceful or almost-graceful.

Theorem 1.5: let n be a positive integer ≥ 4. Then there exists a graceful graph

G = (V, E), with |E| = n. If n = 2k (k ≥ 2), then there exists two spanning trees

Tk,1, Tg each with (k + 1) edges such that Tk,1, is (k, 1) graceful and Tg is graceful.

If n = 2k+1 (k≥ 2), then there exists spanning trees Tk,1, and Tag with (k + 2) edges so that Tk,1 is (k, 1) – graceful and Tag is almost graceful.

Proof: Case (i) n = 2k, k≥ 2.

By Theorem 1.4. there exists a graceful graph G = (V, E), with |V| = k +2,

|E| = 2k.

Let Tk,1 = (V, Ek), |Ek| = k + 1 be a spanning tree of G,

For Ek = E – E1, where E1 = {(k + 1, j) : j = k + 2, ...., 2k}

Then edge-labels are k, k+1,...,2k

∴f+(Ek) = {k, k+1,...., 2k} ∴Tk,1 is (k, 1) – graceful.

If E2 = {(0, j) ; j = k + 2,..., 2k}

Then Eg = E – E2, the edge labels are 1, 2,..., k-1, k, k+1.

∴ the spanning tree Tg = (V, Eq) of G is a graceful graph with (k+1) edges.

Case (ii) : n = 2k + 1, k ≥ 2.

Evidently G = (V, E) is graceful with |V| = k + 3, |E| = 2k + 1.

For E1 = {(k+1, j): j = k+2,...., 2k+1}

The Ek = E – E1, the induced edge-labels = f+(Ek) = {k,....,2k} and |Ek| = k + 2 and Tk,1 = (V, Ek) is the spanning tree of G and it is (k, 1)-graceful.

Also, if Eag = E – E2, E2= {(0, j) ; j = k+2,....., 2k}

The edge-labels are 1, 2,...., k-1, k, k+1; 2k + 1; and k+2,...., 2k ∉f+(Eag) .

Therefore, f+(Eag) ≠ {1, 2,...., 2k+1}

But the edge-label set contains a sequencial subset A, with 2 ≤ |A| < |f+(Eag)|.

∴ Tag is an almost graceful tree. But Tag = (V, Eag) is a spanning tree of

G = (V, E)

•

•

•

• •

• 2k

O

k

k+1

k+2 k+3



70

∴ Tag = (V, Eag) is a spanning tree and is almost graceful. Hence the theorem.

Remark: In case i) of Theorm 1.5 for k = 4, n = 8 and G = (V, E) has a graceful – labeling; and G has two spanning trees one is T4,1; (4,1) - graceful and Tg is graceful and Tg = (V, Eg);

| Eg| = k+1 = 5.

Tg is tree-complete in the sense it has all subtrees with the edges 1, 2, 3, 4, 5. Further each of these subtrees are also graceful.

Corollary 1.6: For all positive integers n≥4, there exists a graceful graph

G = (V, E). And for every such G we have

(i) A spanning tree that is (k, 1) – graceful for all k ≥ 2, where k = 2

n or,

k = 2

1n −according as n is even or, odd.

(ii) A spanning tree T such that, T is graceful or, almost – graceful according as n is even or, odd.

Definition 1.7: A minimum spanning tree of a weighted graph is a set of

(n-1) – edges of minimum total weight.

If T is a tree, w(T) = total weights of edges in T.

If T1, T2,....,Ti are spanning trees, T is minimal, only if w(T) ≤ w(Tj) for all

j = 1, 2,..., i.

Lemma 1.8: For all positive integers n ≥ 4, n = 2k, k ≥ 2, the graceful graph

G = (V, E), with

|E| = 2k, |V| = k + 1 contains a spanning tree Tg that is graceful with

|E(Tg)| = k + 1.

One can modify Tg into Tag for all j > i ≥ 1, wj(Tag) ≥ wi(Tag)

Proof: For n = 2k, k ≥ 2, theorem 1.4 guarantees the existence of a graceful graph G = (V, E).

By theorem 1.5, there is a graceful spanning tree Tg of G with (k + 1) edges with

edge-label set equal to {1, 2,..., k + 1}. ∴w(Tg) = ( ) ( )

2

2k1k ++

Tg can be reduced to Tag by deleting a selected number of j edges and adding j other edges.

• •

•

•

•

k

0

2k

k+4

k+3 • k+2

k+1

• 2k+1



71

For j = 1, we delete (k+1, 2k) and add (0, 2k) to obtain E(Tag) = {1, 2,...., k – 2; k, k + 1, 2k}, provided min{k} = 3, For j = 2, we delete (k + 1, 2k), (k + 1, 2k – 1) and include edges (0, 2k) and (0, 2k – 1), only if min(k) = 4.

Proceeding this way, we get the following formula:

Wj (Tag) = ( ) ( )

2

)kmin(jk1k +++. Also for all j > i ≥ 1, wj(Tag) ≥ wi(Tag) and

w0(Tag) = w(Tg). Hence the lemma is true.

Theorem 1.9: For every positive integer n ≥ 4, there exists a graceful graph

G = (V, E). Then the spanning trees Tk, 1, Tag or Tg satisfy the following inequality:

w(Tk, 1) ≥ w(Tag) ≥ w(Tg). Thus G contains a minimal spanning tree Tg

Proof: Case (i) : n = 2k, k ≥ 2. Theorem 1.5 gives spanning trees Tk, 1 and Tg each having (k + 1) – edges such that Tk, 1 is (k, 1) – graceful and Tg is graceful.

For Tk, 1, edge label set E(Tk, 1) = {k, k + 1,..., 2k}

∴ w(Tk,1) = =−−+

2

k)1k()1k2(k2 ( )2

1kk3 + ...................(1)

For Tg, w(Tg) = ( ) ( )

2

2k1k ++, W(Tg) =

( )2

2k3k2 ++ ......................................... (2)

By Lemma 1.8, the graceful tree Tg can be modified to Tag such that

wj(Tag) ≥ wi(Tag) for j >i ≥ 1 ....................................... (3)

Case (ii) : let n = 2k + 1, k ≥ 2. By theorem 1.5, G contains spanning trees Tk, 1 and Tg with

(k + 2) – edges and E(Tk, 1) = {k, k + 1, ...., 2k + 1}

∴w(Tk , 1) = ( ) ( )

2

2k7k3

2

1k32k 2 ++=

++ for Tag, w(Tag) =

2

4k7k2 ++ ................ (4)

for all k ≥ 2, w(Tk,1) ≥ w(Tag) ∴ we get w(Tk,1) ≥ w(Tag) ≥ w(Tg) and for all j > i ≥ 1,

wj(Tag) and for all j > i ≥ 1, wj(Tag) ≥ wi(Tag).

∴ G contains a minimal spanning tree, namely Tg.

Remark: Theorem 1.9 gives a minimal spanning tree. The question arises whether one can find all spanning trees of G. The following result settles this question.

Theorem 1.10: For every positive integer n ≥ 4, there exists a graceful graph

G = (V, E) with |E| = 2k, (n = 2k or 2k + 1, k ≥ 2) Such that,

(1) There are k spanning trees of G: T0, T1,..., Tk-1.

(2) There exists a maximal spanning tree Tk, 1 and a minimal spanning tree Tg.

For any spanning tree Ti, we have w(Tg) ≤ w(Ti) ≤ w(Tk, 1)

(3) L = {T0, T1,..., Tk -1) is a lattice w.r.t. partial order Ti ≤ Tj iff w(Ti) ≤ w(Tj)

Case (i) : n = 2k, k ≥ 2, the k-spanning trees are T0, T1,.., Tk-1.

Trees (T) edges: E(T)



72

T0 (0, k), (0, k + 1), (k + 1, j) for j = k + 2, ..., 2k

T1 (0, k), (0, k + 1), (0, 2k) and (k + 1, j) for j = k + 2,..., 2k – 1

T2 (0, k), (0, k + 1), (0, 2k), (0, 2k – 1); and (k + 1, j), for j = k + 2,...2k – 2

.

Tk –1 (0, k), (0, k + 1), (0, 2k),.....(0, k + 2)

Every Ti, i = 0, 1, .., (k – 1), has (k + 1) – edges and w(Ti) = ( ) ( )

2

i22k1k +++

and w(T0) ≤ w(T1) ≤ ..... ≤ w(Tk-1) –– (*)

and T0 is a graceful tree and Tk-1 is a (k, 1) – graceful tree.

Case (ii) : let n = 2k + 1, k ≥ 2. The k-spanning trees are T0, T1, ..., Tk-1 with the additional edge (0, 2k+1). Along with edges listed in case (i)

So the formula becomes, w(Ti) = ( ) ( )

2

i21k4k7k2 ++++ –– (* *)

The new formula (* *) again gives the inequality (*).

Combining both cases, we conclude their exists exactly k-spanning trees for the

graceful graph G = (V, E) satisfying. w(Tg) = w(T0) ≤ w(T1) ≤...... w(Tk-1)

= w(Tk, 1)

∴ T0 = Tg, the unique minimal spanning tree and Tk,1 is unique maximal.Hence the theorem.

Conclusion: In view of theorem 1.5, we conclude that for all positive integers

n≥7, there exists a graceful numbering of a graph G = (V, E), with |E| = n.

When n = 2k (k ≥ 2), there always exists a (k, 1) – graceful spanning tree, with

(k+1) –edges, of G. When n = 2k + 1 (k ≥ 2), there exists a (k, 1) – graceful spanning tree, with (k + 2)-edges, of G. Thus there exists infinitely many

(k, 1) – graceful trees. So there are infinitely many arbitrary graceful trees. It is interesting to note that the graceful numbering of a graph G gives rise to a lattice of k-spanning trees of G.

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting him. Reference 1. B.D. Acharya and S.M. Hegde, “Arithmetic graphs”, J.Graph theory, 14, (1990),

275 – 299

2. B.D. Acharya and S.M. Hegde, “Technical proceedings of the graph discussion on

“Graph Labeling problem”, DST / MS / GD 105 / 99 (1999)

3. J.A. Gallian, “Dynamic survey of graph labeling problems” Electronic Journal of

Combinatorics, DS # 6, 2005, 1 – 158.

4. S.A Mariadoss, “Graceful graphs using perfect numbers”, International J. Mathematics

and Computer Science, Vol. 2, No. 3, 2007, 199 - 208 (ICMCS 2007), 225 – 227.



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ENUMERATION OF HAMILTON CYCLES AND TRIANGLES IN AN ARITHMETICAL GRAPH ASSOCIATED WITH EULER TOTIENT FUNCTION ΦΦΦΦ

Author: L.Madhavi Yogi Vemana University, Kadapa, A.P., INDIA. E-mail: [email protected]

----------------------------------------------------------------------------------------------------------------------------------

Abstract: Hamilton cycles are cycles of largest length and triangles are cycles of smallest length in a graph. Berrizbeitia, P. and Giudici, R.E., [2, 3] and Dejter, I., and Giudici, R.E., [6] have studied the cycle structure of Cayley graphs associated with certain arithmetical functions. In [4] the Enumeration of Triangles and Hamilton Cycles in Quadratic Residue Cayley Graphs have been studied by the authors. In this paper the number of Hamilton cycles and triangles in a class of

Cayley graphs associated with Euler totient function Φ(n), where n ≥ 1 is an integer, are determined.

1.INTRODUCTION

Let ( X, .) be a group. A subset S of X is called a symmetric subset if s-1 ∈ S, ∀ s ∈ S. The graph

G whose vertex set V = X and the edge set E = { ( g,h) / g-1 h ∈ S or hg-1 ∈ S } is called the Cayley graph of X corresponding to the symmetric set S and it is denoted by G(X,S) . Clearly G(X,S) is an undirected graph and it does not contain loops if the identity element e of X is deleted from S. It is easy to see that the Cayley graph G(X,S) is |S| - regular and the number of

edges in G(X,S) is 2

|||| SX.

2. EULER TOTIENT CAYLEY GRAPH AND ITS PROPERTIES

For any positive integer n let Zn = { 0,1,2,.. . . . n−1}. Then (Zn ,⊕), where ⊕ is addition modulo n is an abelian group of order n . For any positive integer n let S denote the set of all positive

integers less than n and relatively prime to n. Then | S | = ϕ(n), the Euler totient function. It is

easy to see that S is a symmetric subset of the group (Zn ,⊕) and S is a multiplicative subgroup of

order ϕ(n) of the semigroup (Zn*,Ȩ) where Zn

* = Zn – { 0 } and Ȩ is multiplication modulo n .

Definition 2.1 : For each positive integer n let ( Zn ,⊕ ) be the additive group of integers modulo n and let S be the set of all numbers less than n and relatively prime to n. The Euler Totient Cayley Graph G( Zn , ΦΦΦΦ ) is defined as the graph whose vertex set V is Zn =

{ 0 , 1, 2, …., n – 1 } and the edge set is given by E = { ( x , y ) | x – y ∈ S or y – x ∈ S} .

Since the graph G(Zn ,Φ) is the Cayley graph of the group (Zn , ⊕) associated with the symmetric set S, the following Lemma is immediate.

Lemma 2.2 : The graph G(Zn ,Φ) is ϕ(n) – regular . Moreover the number of edges in G( Zn , Φ )

is 2

)(nnϕ .

Lemma 2.3 : The graph G(Zn ,Φ) is hamiltonian and hence it is connected.


------------------------------------- Presenter: L. Madhavi



74

Proof : Let s be an element of S. Then s < n and s is relatively prime to n . Hence s is a

generator of ( Zn ,⊕ ). So s , 2s , …., (n -1)s are all distinct and { s, 2s ,…., ns } = Zn. For

1 < r < n we have ( r + 1 ) s – rs = s ∈ S . So for each r , 1 ≤ r ≤ n there is an edge between (r +

1)s and rs. This shows that Cs : ( 0 s 2s … ns = 0 ) , is a cycle and this is clearly a Hamilton

cycle in G (Zn ,Φ ) . Therefore G(Zn ,Φ ) is hamiltonian and hence it is connected.

Definition 2.4 : For s ∈ S the cycle Cs = ( 0 s 2s … ns = 0 ) is called the Hamilton cycle corresponding to the element s in S.

Lemma 2.5: For n ≥ 3 the graph G(Zn ,Φ) is Eulerian.

Proof : Clearly the graph G(Zn, Φ) is ϕ(n) - regular. But by the Theorem 2.5 (e) of [1], ϕ(n) is

even for n ≥ 3. That is, the degree of each vertex in G (Zn ,Φ) is even so that it is Eulerian.

Theorem 2.6 : If n is an even number then the graph G (Zn ,Φ) is a bipartite graph.

Proof : First we shall show that G(Zn ,Φ) has no odd cycles. To see this let (i1 i2 … ir i1) be a

cycle in G(Zn ,Φ). Then (i1 , i2) , ( i2 , i3 ), ..….., ( ir , i1 ) are edges in G(Zn ,Φ) so that is – is+1 ∈ S

for 1 ≤ s ≤ r – 1 and ir – i1 ∈ S . That is is – is+1 and ir – i1 are relatively prime to n for i ≤ s ≤ r

– 1. Since n is even is – is+1 and ir – i1 must be odd for 1 ≤ s ≤ r – 1. That is, one of is and is+1 is

even and the other is odd for 1 ≤ s ≤ r – 1 and the same is true for i1 and ir . Thus i1 , i2 ,…., ir , il are even and odd or odd and even alternately . This shows that half of i1 , i2 ,…, ir are even and the other half are odd so that their number is even. This shows that the cycle ( i1 i2 … ir i1 ) is an

even cycle and G(Zn ,Φ) has no odd cycles so that [5, pp.14,15 ] the graph G(Zn ,Φ) is bipartite.

Corollary 2.7 : If n is even then G(Zn ,Φ) has no triangles .

Proof : Let n be an even integer. Then by the Theorem 2.6 G(Zn ,Φ) has no odd cycles. Since a

triangle is a 3 – cycle, which is an odd cycle, G ( Zn ,Φ ) can not have triangles.

3. ENUMERATION OF DISJOINT HAMILTON CYCLES

Lemma 3.1 : For any s ∈ S the Hamilton cycles associated with s and n – s are one and the same.

Proof : Let s be an element of S. Then by the Lemma 2.3 the graph G(Zn ,Φ) has a Hamilton cycle Cs = ( 0 s 2s … (n – 2)s (n – 1)s ns = 0 )

In the abelian group (Zn ,⊕), nt = 0 for 1 ≤ t ≤ n . So for any r, 0 ≤ r ≤ n, we have

( n – r )s = ns – rs = 0 – rs = rn – rs = r(n – s ). Hence the cycle

C(n – s ) : ( 0 n – s 2 (n – s) … (n – 1) (n – s) 0 ) is same as Cs .

Lemma 3.2 : For s, t ∈ S, t ≠ s and t ≠ n – s the Hamilton cycles Cs and Ct are edge disjoint.

Proof : Let s, t ∈ S such that t ≠ s and t ≠ n – s . Then the Hamilton cycles Cs and Ct are given

by

Cs = ( 0 s 2s … (n – 1)s ns = 0 )

= Cn – s ( by the Theorem 3.1 )



75

= ( 0 (n – s) 2 (n – s) … (n – 1)(n – s) n (n – s) = 0 ) and

Ct = ( 0 t 2t … (n – 1)t nt = 0 ) .

We claim that the Hamilton cycles Cs and Ct are edge disjoint .If possible assume that Cs and Ct

are not edge disjoint. Then there exists an edge ( it, (i+1)t ) in Ct such that either

( it, (i+1)t ) = ( js, (j+1) s ) , or, (it , (i+1) t ) = ( k (n−s) , (k+1) (n−s))

for some 0 ≤ j ≤ n−1 and 0 ≤ k ≤ n−1.

But (it, (i +1)t) = ( js, (j+1) s ) implies that it = js and (i +1)t = ( j +1)s and this gives t = s

which is a contradiction.

Also (it, (i+1)t) = (k(n−s), (k+1)(n−s)) implies that it = k(n−s) and (i +1)t = (k+1) (n−s) and this

gives t = n – s which is again a contradiction. Therefore the two Hamilton cycles Cs and Ct are

edge disjoint.

Theorem 3.3 : For n ≥ 3 the Euler totient graph G(Zn ,Φ) can be decomposed into 2

)(nϕ

edge disjoint Hamilton cycles.

Proof : Let n ≥ 3 be any integer. First we shall show that s ≠ n – s for all s ∈ S. If s = 1 then n – s

= n – 1 ≥ 2 and n ≥ 3 so that n – 1 ≠ 1 . On the other hand if s ≠ 1 and s = n – s then n = 2s

so that the gcd of s and n is the same as the gcd of s and 2s which is s. Since s ≠ 1, this is a

contradiction to the fact that the gcd of s and n is 1 as s ∈ S . So s ≠ n – s for all s ∈ S . Hence S is

partitioned into 2

)(nϕ disjoint pairs ( s, n – s ) of distinct numbers. By the Lemma 3.1 the

Hamilton cycles corresponding to this pair are one and the same. Thus by Lemma 3.2 these

2

)(nϕ distinct pairs produce

2

)(nϕ edge disjoint Hamilton cycles.

Since each Hamilton cycle contains | Zn | = n edges, the total number of edges contributed by

these 2

)(nϕ edge disjoint Hamilton cycles is | Zn |

2

)(nϕ and by the Lemma 2.2 this is clearly

equal to the total number of edges in the graph G ( Zn ,Φ) . Hence the graph G(Zn ,Φ) is

decomposed into 2

)(nϕ edge disjoint Hamilton cycles.

4. ENUMERATION OF TRIANGLES

In this section we give a formula for the number of triangles in G(Zn ,Φ) in terms of a well known

arithmetic function, namely, Schemmel totient function ϕ(2)(n) which denotes the number of

pairs of consecutive positive integers less than n and relatively prime to n[7,p.147] .



76

G(Zn ,Φ) being a Cayley graph, it is vertex transitive. So to count the number of triangles we

concentrate our attention on the triangles of the form ( 0 , a, b) where a, b ∈ S and (a – b)∈ S .

Trivially 1 ∈ S. Hence for any b ∈ S the trio (0, 1, b ) will be a triangle if ( b – 1) ∈ S . A

triangle of this form is called a fundamental triangle. The set of all fundamental triangles is

denoted by ∆01 . That is,

∆01 = { ( 0 , 1, b ) | b ∈ S and ( b – 1 ) ∈ S } .

Theorem 4.1: If n ≥ 3 is an odd integer then | ∆01 | = ϕ(2) (n).

Proof : Let n be an odd positive integer ≥ 3 . Then the triplet ( 0 , 1, b ) is a fundamental triangle

if and only if b ∈ S and b – 1 ∈ S ⇔ b and b – 1 are a pair of consecutive numbers less than n and relatively prime to n . Thus there are as many fundamental triangles in G

(Zn ,Φ) as there are pairs of consecutive positive integers less than n and relatively prime to n. So,

| ∆01 | = ϕ(2) ( n ) .

Definition 4.2 : For each µ ∈ S, we define

∆µ = { (0 , µ , k ) | k , ( k − µ ) ∈ S } .

That is , ∆µ is the set of all triangles of the form ( 0 , µ , k ) for a fixed µ ∈ S .

Theorem 4.3. : For any µ ∈ S , | ∆µ | = | ∆01 | = ϕ(2) ( n ) .

Proof : We claim that the mapping f : ∆01 � ∆µ given by f ( 0 , 1, b ) = ( 0 , µ , µ b ),

is a bijection. To see this, let ( 0 , µ , µ b1 ) = ( 0 , µ , µ b2 ) for some b1 , b2 ∈ S . Then

µ b1 = µ b2 . Since ( S,Ȩ ) is a group this gives b1 = b2 so that ( 0 , 1, b1 ) = ( 0 , 1, b2 ) and f is

one – to – one .

Let ( 0 , µ , k ) be any element of ∆µ . Then µ , k and ( k − µ ) are in S . For k, µ ∈ S, there is a

unique element b in S such that k = µ b so that ( k − µ ) ∈ S implies ( µ b − µ ) ∈ S or

µ (b – 1) ∈ S . This gives b – 1 ∈ S and (0, 1, b) ∈ ∆01 . But (0,µ ,k ) =(0 , µ , µb)= f( 0 , 1 ,b ) .

This shows that the function f is onto and so f is a bijection . Hence | ∆µ | = | ∆01 | = ϕ(2) ( n ) .

Theorem 4.4 : Let ∆(0) denote the set of all triangles with 0 as one vertex . Then for any

odd integer n ≥ 3

| ∆( 0 ) | = 2

1 ϕ(n) ϕ(2)

( n ) .

Proof : It is evident that ∆(0) = { ( 0 , µ, k ) | µ , k ∈ S and ( k − µ ) ∈ S },

and for a fixed µ ∈ S

∆µ = { ( 0 , µ , k ) | k ∈ S and ( k - µ ) ∈ S } .



77

So , ∆( 0 ) = µ∈µ

∆∪s

. But the triangle ( 0 , µ , k ) appears twice in the union, once in ∆µ and

once in ∆k . Hence by the Theorem 4.3

| ∆ ( 0 ) | = 2

1 ∑

∈µµ

∆S

|| = 2

1 ∑

∈µ S

ϕ(2) ( n )= 2

1 ϕ(2) ( n ) ∑

∈µ S

1 = 2

1 ϕ(2) ( n ) | S | =

2

)()2(nϕ

ϕ (n) .

Theorem 4.5 : For any integer n ≥ 3 the total number of triangles T(Φ) of G ( Zn , Φ) is given by

0 if n is even and 6

n ϕ (n) ϕ(2)

( n ) if n is odd .

Proof : If n is even then by Corollary 2.7, T (Φ) = 0 . So let n be an odd positive integer. Since

G( Zn , Φ) is vertex transitive, and ϕ(n) - regular, the number of triangles through each vertex is

the same and their number is 2

)()2( nϕ ϕ( n) . So the total number of triangles in

G( Zn , Φ) is n ϕ ( n ) 2

)()2(nϕ

, since the number of vertices in G(Zn Φ) is n. However each

triangle in G(Zn , Φ) is counted once by each of its three vertices . So the number T(Φ) of distinct

triangles in G(Zn , Φ ) is given by

0 if n is even

)()( 6

(2)nn

nϕϕ if n is odd

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting her.

REFERENCES: [1] Apostol, Tom M ., Introduction to Analytic Number Theory, Springer International Student Edition , (1989). [2] Berrizbeitia, P. and Giudici, R.E., Counting pure k-cycles in sequences of Cayley graphs, Discrete Math., 149(1996), 11-18. [3] Berrizbeitia, P. and Giudici, R.E., On cycles in the sequence of unitary Cayley graphs. (Reporte Techico No. 01-95, Universidad Simon Bolivar, Dpto. de Mathematicas, Caracas, Venezuela , 1995). [4] Bommireddy Maheswari and Madhavi Levaku., Enumeration of Triangles and Hamilton Cycles in Quadratic Residue Cayley Graphs. Chamchuri Journal of Mathematics. Vol.1 (2009) No.1, 95-103. [5] Bondy, J.A., and Murty , U.S.R., Graph Theory with Applications, Macmillan , London

(1976). [6] Dejter, I., and Giudici, R.E., On unitary Cayley graphs, JCMCC, 18 (1995), 121-124. [7] Dickson, E., History of Theory of Numbers,Vol.1, Chelsea Publishing Company ( 1952 ).

T(Φ) =

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 78-81 78

PRIME RIGHT ALTERNATIVE RINGS Author: D. Bharathi, Department of Mathematics, Sri Venkateswara University, Titupathi, A.P, India. Email: [email protected]

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract: In this paper we show that a 2-divisible prime right alternative ring R with

( )( )x,y , x 0= is either alternative or strongly (-1,1). If R has an idempotent 0,1,e ≠

then we show that it must be alternative. Key words: 2-divisible, prime ring, right alternative ring, idempotent, strongly (-1,1) ring, alternative ring Introduction: E.Kleinfeld and H.F. Smith [1] studied the prime rings in the join of alternative and (-1,1) rings. Thee rings are the subvariety of right alternative rings which satisfy the identities (i) S (xy,z,x)=S(y,z,x)x, (ii) S(x,y,zx) =x S(x,y,z) and (iii)

( )( ) ( )( )S x,y,z , , t,S x,y,z , . .t u u R E= − Roomeldi [2] proved that if a 2-divisible

simple right alternative ring satisfies (i) and (ii), then it is alternative. In [1] it is proved that if R is a 2-and 3-alternative or strongly (-1,1). Without assuming (i), (ii), and (iii), we derive some interesting identities of right alternative rings satisfying

( )( )x,y , x 0= and their consequences in this paper. We show that a 2-divisible prime

right alternative ring R with ( )( )x,y , x 0= is either alternative or strongly (-1,1). If R

has an idempotent 0,1,e ≠ then we show that it must be alternative.

Through this section we shall assume R to be a 2-divisible right alternative

ring with ( )( )x,y ,x 0= and A is the subset of R generated by all alternators. R is said

to be right alternative if ( ), , 0x y y = for all x,y R.∈ it is left alternative if ( )y,y,x 0=

and if R satisfies ( )( ), , 0x y z = then it is strongly (-1,1). R is prime if whenever S and

T are ideals of R such that ST=0 then either S=0 or T=0. Also ann

( ) { }A = x R/xA=Ax=0∈ and ( ) ( ) ( ){ }ANN A = x ann A / A,R,x =0∈ . We define

( ) ( ) ( ) ( )S x,y,z = x,y,z + y,z,x + z,x,y .

First we list the well known identities in right alternative rings.

( ) ( ) ( ) ( ) ( )wx,y,z - w,xy,z + w,x,yz =w x,y,z + w,x,y z (1)

( ) ( )( ) ( ) ( )wx,y,z + w,x, y,z =w x,y,z + w,y,z x (2)

( )( ) ( )( ) ( )( ) ( )x,y ,z + y,z ,x + z,x ,y =2S x,y,z (3)

( ) ( ) ( ) ( ) ( )xy,z =x y,z + x,z y+2 x,y,z + z,x,y (4)

( ) ( )2x,y ,z = x,y,yz+zy (5)

( ) ( )x,yz,y = x,z,y y. (6)

The linearization of ( )( )x,y , 0x = gives ( )( ) ( )( ), , , ,x y z z y x= − (7)

Also the following identities are satisfied in a right alternative ring with

( )( ), , 0x y x = [3].

Research Paper (oral Presentation)

-----------------------------------------------------------------------------------------------------

Presenter: Dr. D. Bharathi.


( ) ( )( ), , , 0w x y z = (8)

(( , ), ), ) 0,w x y z = (9)

3(( , ), 2 ( , , ),x y z S x y z= (10)

S(x,y,(a,b))=0 (11) We derive some more identities of R in the following lemma. Lemma 1:- In a 2-divisible right alternative ring R with ((x,y),x)=0, the following identities hold:

(a) ( )( ) ( )( ) ( )( ) ( )( ), , , , , , , , , , , ,w x y z y z w x w x y z x w y z− = − ,

(b) ( )( ), , , 0x x x y =

(c) ( )( ) ( )( ) ( )( ) ( )( ), , , , , , 2 , , , 2 , , , ,x y x y x y y x x y x y y x x y= − = − =

(d) ( )( ), , 0xy x y =

Proof : By interchanging w and x in (2) and subtracting the result equation from (2) we get

( )( ) ( )( ) ( )( ) ( )( ) ( )( ), , , , , , , , , , , , , , ,w x y z w x y z x w y z w x y z x w y z+ − = − (12)

However, it follows from the identity (11) that

( )( ) ( )( ) ( )( ), , , , , , , , , 0w x y z x y z w y z w x+ + = . This implies

( )( ) ( )( ) ( )( ), , , , , , , , , ,w x y z x w y z y z w x− = − since R is alternative.

By substituting (12) we get ( )( ) ( )( ) ( )( ) ( )( ), , , , , , , , , , , , .w x y z y z w x w x y z x w y z− = −

This establishes (a). In a right alternative ring, we know that (x,x,x)=0. But since

( ) ( )2x,x,x = x ,x , we get ( )2x ,x 0= . From (7) implies

( )( ) ( )( )2 2, , , , 0.x x y y x x= − = By substituting ( ), ,y x z x y= = in (4) we get

( )( ) ( )( ) ( )( )2 , , , , ,x xy x x x y x x y x= + + ( )( )2 , ,x x x y + ( )( ), , , 0.x y x x = We have

( )( ) ( )( )2 , , , , , , 0x x x y x y x x+ = , since ( )( ), , 0.x x y = This implies ( )( ), ; , 0x x x y = ,

Since R is right alternative and 2-divisible.Thus (b) is proved.

We linearize (b) by replacing one x by y. Thus ( )( ) ( )( ), , , , ,y x x y x y x y+ +

( )( ), , , 0.x x y y = This implies ( )( ) ( )( )x,y, x,y =- y,x, x,y . But ( )( )x,y, x,y =

( )( )- x, x,y ,y and ( )( ) ( )( )- y,x, x,y , , , .y x y x= Also from the identity (11). We have

( )( ), , 0S x y x y = ( )( ) ( )( ) ( )( ), , , , , , , , , .x y x y y x y x x y x y= + + This establishes (c).

From (4) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ), , , , , , 2 , , , , , , .xy x y x y x y x x y y x y x y x y x y= + + +

However, the last two terms cancel each other because of (c).

Two more are zero because of ( )( ), , 0.x y x = Thus ( )( ), , 0xy x y = . Thus (d) is

proved.

Lemma 2:- If R is a 2-divisible right alternative ring with ( )( ), , 0x y x = , then

( ), , 0S xy x y = and ( )( ), , , 0y x x y = .

Proof : - By substituting z xy= in the identity (10) and using the identity (d) of

lemma (1) we get ( )2 , , 0S x y xy = . Since R is 2-divisible, this implies ( ), , 0S x y xy =

or ( ), , 0S xy x y = . Thus linearization in y of the identity gives


( ) ( ) ( )2 20 , , , , , , ,S x x y S xy x x S x x y= + = that is, ( )2 , , 0.S x x y = Hence

( ) ( ) ( ) ( )2 2 2 2, , , , , , , ,x x y x y x y x x x x y= − − = by linearizing the right alternative

identity and (6) from the identity (1) we then have

( ) ( ) ( ) ( )2 2, , , , , , , ,x x x y x x y x x y x x xy y= − + − ( ) ( ), , , ,x x x y x x y x= using (6) and

right alternative identity. Thus ( )( ), , , 0.x x x y = Next from linearization of this

identity and right alternative identity we see.

( )( ) ( ) ( )( ) ( )( ), , , , , , , , , , , ,y x x y x y x y x x y y x y y x= − − = (13)

By substituting w y= and z x= in the identity (a). we get

( )( ) ( )( ) ( )( ) ( )( ), , , , , , , , , , , .y x y x y x y x y x y x x y y x− = − That is

( )( ) ( )( ), , , , , , .x y y x y x x y= − By substituting this in (13) we obtain

( )( )2 , , , 0.y x x y = Thus ( )( ), , , 0y x x y = , since R is 2-divisible • .

Lemma 3: In a 2-divisible right alternative ring ,R A RA+ is a left ideal and A AR+

is an ideal of R containing A RA+ .

Proof : From the identity (1) we have ( ) ( ) ( )2, , , , , ,a b b x ab b x a b x= − ( ), ,a b bx+ =

( ) ( ), , , , ,ab b x a b xb− using (5). Modulo A thus gives with (6) that

( ) ( ) ( ) ( ) ( ), , , , , , , , , , .a b b x b ab x b xb a b b x a b b a x= − − = − − Hence for , ,a b x R∈ , we

have ( ) ( ) ( ), , , , , ,a b b x b b x a b b a x≡ − − modulo A. (14)

The right side of (14) hence the left one, too, is symmetric in a and x. Also in any

right alternative ring we have the identity ( )( ) ( )( ), , , , , , , ,a b c y z a b c y z=

( )( ), , ,a b y z c+ + ( )( )( ) ( ) ( ) ( )( ), , , , , , , , , ,a y z b c a b c y z a b c y z− + ( )( ), , , .a b y z c−

If we put b=a in this identity, we get

( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ), , , , , , , , , , , , , , , , , , , , , ,a a c y z a a c y z a a y z c a y z a c a a c y z a a c y z= + + − +

( )( ), , , .a a y z c− (15)

By applying (14) to (15) we obtain for

, , ,a c y z R∈ ( )( ) ( )( ) ( )( ), , , , , , , , ,a a c y z y z a a c c a a y z≡ ≡ modulo A.

From (14) and (16) we see that ( ), , .A R R A RA A AR⊂ + ⊂ +

Cleary ( ) ( ), , , , .A A R R A R R A+ = + This proves that A RA+ is a left ideal and

A RA+ is an ideal.

From Theorem 4 in [2] elements of the form ( )( ), , ,x x y z are in the

commutative center U. Since A is the ideal generated by alternators of R, we have

( )( ), , , .x x y z U A∈ ∩ From the Proof of theorem 6 in [4] we have 0.U A∩ = Thus

we get ( )( ), , , 0x x y z = .

Theorem 1: A 2-divisible prime right alternative ring R with ( )( ), , 0x y x = is either

an alternative ring or a strongly (-1,1) ring.

Proof : From lemma 2 ( ), , 0S xy x y = and ( )( ), , , 0y x x y = . By linearizing

( )( ), , , 0y x x y = and using the identity ( )( ), , , 0.x x y z = we get


( ) ( )( ) ( )( )( ), , , , , , , , 0a b x x y y x x a b= − = (18)

From 4 we have (a,b,c) – a(b,c)-(a,c)b-2(a,b,c)+(c,a,b).

By interchanging a and b we also have ( ) ( ) ( ) ( ) ( ), , , 2 , , , , .ba c b a c b c a b a c c b a− − = +

By adding these last two equations and using (x,y,z)=-(x,z,y), we arrive at (aob,c)-ao(b,c) –bo(a,c)=2{(a,b,c)+(b,a,c)}, where aob=ab+ba. In this equation we let c=(s,t) be a commutator. Then using linearized (x,x,(y,z)=0 we see (aob,c)-ao(b,c)-bo(a,c)=0. In this last equation we next let a =(x,x,y) be an alternator. From Lemma 3 the linear span A of the alternators is an ideal, and also (A,C)=0 by (18) . Thus we obtain 2a(b,c)=ao(b,c)=0. This implies a(b,c)=0, since R is 2-divisible. That is (x,x,y) (b,(s,t)=0=(b,(s,t) (x,x,y). Thus we have established (b,(s,t) ∈ ANN(A)={x∈ann(A)/(A,R,x)=0}. But by theorem 2 in [4] ANN(A) is an idealof R. Since A (ANN(A))=0 and R is prime, this means either A=0 and R is alternative, or each (b,(s,t))=0 and R is strongly (-1,1). This completes the proof of the theorem. • .

Theorem 2: If R is a 2-divisible right alternative ring satisfying ( )( ), , 0x y x = with

an idempotent 0,1,e ≠ then R is alternative.

Proof: From the identity (b)in lemma (1) we have (x,x,(x,y))=0. In particular, for any idempotent e we have (e,e,(e,y))=0. Thus using (5) (e,e,(e,y))=0 and (6) we see (e,e,x)=(e,e2,x)=(e,e,ex+xe) =2(e,e,ex)= 2(e,e,x)e. Iteration and the right alternative

identity then give 2(e,e,x)e= ( ) ( )4 , , 4 , , ,e e x e e e e x e= so ( )2 , , 0e e x e = . This in

turn means ( ) ( ), , 2 , , 0e e x e e x e= = for any idempotent e. At this point the argument

given in section of [5] shows that R is alternative, which completes the proof of the theorem. • .

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting her. References 1. Kleinfeld,E and Smith, H,F : Prime rings in the joining of alternative and (-1,1) rings.

Contemporary mathematics, vol. 131 (1992), 613-623. 2. Roomeldi, R.E : Nilpotency of ideals in a(-1,1) ring with minimum

condition, algebra i Logika 12(1973), 333-348. 3. Kleinfeld, E : A generalization of strongly (-1,1) rings, J. of Algebra,

119 (1988), 218-225. 4. Hentzel, I.R. and Smith H.F : Semiprime locally (-1,1) rings with minimal condition,

Algs, Gps, and Geom., 2(1985), 26-52. 5. Kleinfeld, E and Smith H.F : Prime (-1,1) rings with chain condition, comm..

Algebra, 7 (1979), 163-176 .

“Virtue is the knowledge of goodness” “Sin is the ignorance of goodness”

Secret of Success is the concentration CONCENTRATION IS ESSENTIAL FOR EVERY PERSON. CONCENTRATION IS ESSENTIAL FOR EVERY PERSON. CONCENTRATION IS ESSENTIAL FOR EVERY PERSON. CONCENTRATION IS ESSENTIAL FOR EVERY PERSON.

Concentration is achieved by continuous striving just as Yogi.

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July 11-12, 2011)

(Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 82-90 82

A NOTE ON ANTI FUZZY

IDEALS IN NEAR

SUBTRACTION SEMIGROUPS

Authors: T. Nagaiah, P. Narasimha swamy Department of Mathematics, Kakatiya University, Warangal – 506009, A.P, India. E-mail: [email protected], [email protected]. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract: In this paper, the notion of anti fuzzy ideals in near subtraction semigroups is introduced and investigated some related properties. Keywords: Subtraction semigroup, near – subtraction semi group, anti fuzzy ideal, f-invariant, and normal fuzzy ideals.

2000 Mathematical Subject Classification: 04A72, 03E72.

1 Introduction The notion of fuzzy set was first introduced by L.A.Zadeh [13]. Fuzzy set theory has been developed in many directions by many scholars and has evoked great interest among mathematicians working in different fields. A. Rosenfeld [11] introduced the fuzzy sets in the realm of group theory. Since then many mathematicians have been involving in extending the concepts and results of abstract algebra to the broader frame work of the fuzzy setting. A fuzzy ideal in near subtraction semigroup has been studied in [3, 4, 5, 6], and [14]. Akram [1] introduce the notion of fuzzy sub quasigroups with respect s-norm and studied some of its properties.

In this paper the notion of relationship between f-invariant anti fuzzy ideals and anti fuzzy ideals in near subtraction semigroups is introduced. Further some properties of an anti fuzzy ideals in a near - subtraction semigroups are also discussed. The proofs are almost similar to that of Fuzzy ideals in near subtraction semigroups [10].

2 Preliminaries

We briefly recall few definitions and examples from [4] which are used here.

Definition 2.1: A non empty set X together with a binary operation “ - ” is said to be subtraction algebra if it satisfies the following:

1) x - ( y - x ) = x 2) x - ( x - y ) = y - (y - x) 3) ( x - y ) - z = ( x - z ) - y, for every x, y, z ∈X

In subtraction algebra the following holds:

1) x - 0 = x and 0 - x = 0. 2) (x - y) - x = 0 3) (x - y) - y = x - y 4) (x - y) - (y - x) = x - y, where 0 = x – x is an element that does not depend on the choice of x∈X.

Definition 2.2: [9] A nonempty set X together with two binary operations “ - ” and “ ⋅ ” is said to be subtraction semigroup if it satisfies the following :

Paper (Oral Presentation) ----------------------------------------------------

Presenter: T. Nagaiah



1) (X , -) is a subtraction algebra 2) (X, ⋅ ) is a semigroup. 3) x (y - z) = x y – x z and (x – y) z = x z – y z for every x, y, z ∈ X.

3 Near – subtraction semigroup

Definition 3.1: [4] A nonempty set X together with two binary operations “ - ” and “ ⋅ ” is said to be a near subtraction semigroup (right) if

1) (X , -) is a subtraction algebra. 2) (X , ⋅ ) is a semigroup and 3) (x - y ) z = x z - y z, for every x, y, z ∈ X.

Note that it is clear 0 ⋅ x = 0, for every x ∈X. Similarly we can define a near – subtraction semigroup (left) . Hereafter a near – subtraction semigroup means it is a near – subtraction semigroup (right) only.

Example 3.2: Let X = {0, a, b, 1} in which “- ” and “ ⋅ ” are defined by

01011

00

10

00000

10

bbb

baa

ba−

101

10

00000

10.

ba

bab

aaaaa

ba

∴ (X, -, ⋅ ) is a near – subtraction semigroup.

Definition 3.3: A near - subtraction semigroup X is said to be zero-symmetric if x ⋅ 0 = 0 for every x ∈X.

Definition 3.4: A near – subtraction semi group X is said to have an identity if there exists an element 1∈X such that 1 ⋅ x = x ⋅ 1 = x, for every x ∈X.

Definition 3.5: [4] A non empty subset S of a subtraction algebra X is said to be a subalgebra of X, if x – y ∈S, whenever x, y ∈ S.

Definition 3.6:[4] Let (X, -, ⋅ ) be a near – subtraction semigroup. A nonempty subset I of X is called

i) a left ideal if I is a sub algebra of (X, - ) and x i - x (y – i) ∈ I for all x, y ∈ X and i∈ I.

ii) a right ideal if I is a subalgebra of (X, -) and IX ⊆ I

iii) an ideal if I is both a left and right ideal and IX ⊆ I.

Note: 1) Suppose if X is a subtraction semigroup and I is a left ideal of X, then for i∈I and x, y ∈X, we

have x i – x (y - i) = x i – (x y – x i) = x i∈I by Property 1 of subtraction algebra. Thus we have XI ⊆ I.

2) If X is a zero symmetric near-subtraction semigroup, then for i∈I and x ∈X, we have x i – x (0 - i) = x i - 0 = x i∈I.

Definition 3.7: [7] Let X be a non empty set. A map µ : [ ]1,0→X is called a fuzzy set in X , and the

complement of fuzzy set denoted by cµ , is the fuzzy set in X given by )(xcµ = 1 - )(xµ for all x∈X.

Definition 3.8: [10] A fuzzy set µ in a near subtraction X, is called a fuzzy ideal of X if it satisfies the

following conditions:

i) µ (x - y) ≥ min { µ (x), µ (y)} for all x, y∈X,



ii) µ (a x – a (b - x)) ≥ µ (x) for all a, b, x∈X and

iii) µ (x y) ≥ µ (x) for all x, y∈X.

4 Anti Fuzzy: In this section we introduce the notion of anti fuzzy ideals in near- subtraction semigroups. In what follows, let X denote a near – subtraction semigroup, unless otherwise specified.

Definition 4.1: A fuzzy set µ in a near subtraction semigroup X , is called an anti fuzzy ideal of X, if it

satisfies the following conditions:

i) µ (x - y) ≤ max {µ (x), µ (y)} for all x, y∈ X,

ii) µ (a x – a (b - x)) ≤ µ (x) for all a, b, x∈ X and

iii) µ (x y) ≤ µ (x) for all x, y∈ X.

Note that µ is an anti fuzzy left ideal of X if it satisfies (i) and (ii), and µ is anti fuzzy right ideal of X if it

satisfies (i) and (iii).

Example 4.2: Let X = {0, a, b, 1} in which “ - ” and “ ⋅ ” are defined by

01011

00

10

00000

10

bbb

baa

ba−

101

10

00000

10.

ba

bab

aaaaa

ba

Then (X, - , ⋅ ) is a near – subtraction semigroup. We define the fuzzy set µ : [ ]1,0→X

{ }

=

∈

∈

=

18.0

,04.0

6.0

)(

xif

axif

bxif

xµ

The routine calculation shows µ is an anti fuzzy ideal of X.

Theorem 4.3: Let µ be an anti fuzzy ideal of X then the set µX = {x ∈ X / µ (x) = µ (0)} is an ideal of X.

Proof: Suppose µ is an anti fuzzy ideal of X and let x, y∈ µX then

µ (x – y) ≤ max { µ (x), µ (y)} = µ (0) (since x, y µX∈ , µ (x) = )0(µ and µ (y) = )0(µ ).

.)0()( µµµ XyxyxThus ∈−⇒=− For every a, b, x∈ µX , a x- a (b – x) X∈ an d

µ (a x- a (b - x)) ≤ µ (x) = µ (0) ⇒ a x – a (b - x) µX∈

µand (x y) ≤ µ (x) = µ (0).

Therefore uX is an ideal of X.

Theorem 4.4: Let A be a non – empty subset of X and Aµ be a fuzzy set in X defined by

Aµ (x) = ∈

otherwise t,

A x if s,

for all x ∈ X and s, t∈[0, 1] with s > t. Then Aµ is an anti fuzzy ideal of X if

and only if A is an ideal of X. Moreover Aµ

X = A



Proof: Suppose Aµ is an anti fuzz ideal of X and let x, y ∈A. Then Aµ (x - y) ≤ max { Aµ (x), Aµ (y)}=s.

Then x – y ∈A. For every a, b∈ X and x∈A, we have Aµ (a x - a (b - x)) ≤ Aµ (x) = s ⇒ a x - a (b - x)∈A.

For all x, y ∈ A, then Aµ (x y) ≤ µ (x) = s, Thus x y ∈ A. Hence A is an ideal of X.

Conversely suppose that A is an ideal of X, let x, y ∈ X, if atleast one of x and y does not belongs to A, then

Aµ (x - y) ≤ t = max { }.)(µ),(µ AA yx If x, y ∈ A then x - y ∈A, we have

Aµ (x - y) ≤ s = max{ }.)(µ),(µ AA yx Let a, b, x∈ X and if x ∈A such that a x – a (b - x)∈A, we have

Aµ (a x – a (b - x)) ≤ s = Aµ (x). If x A∉ such that a x- a (b - x) A∉ , we have Aµ (a x – a (b - x)) ≤ t = Aµ (x).

For all x, y A∈ , then x y ∈A. We have Aµ (x y) ≤ s = Aµ (x). Hence Aµ is an anti fuzzy ideal of x.

Moreover )}0()(/{XAµ AA xXx µµ =∈=

= })(/{ sxXx A =∈ µ = }/{ AxXx ∈∈ = A.

Definition 4.5: Let f : X → 'X be a mapping where X and

'X are non-empty sets and µ is a fuzzy sub set of 'X .

The pre-image of µ under f written as fµ , is a fuzzy subset of X defined by ( )f(x)µ)(µ

f =x for all x ∈ X.

Definition 4.6: Let X and 'X are near – subtraction semigroups. A map f : X → 'X is called homomorphism of near

subtraction semigroup, if f (x - y) = f (x) – f (y) and f (x y) = f (x) f (y) for any x, y ∈ X.

Definition 4.7: Let X and 'X be two near – subtraction semigroups and f be a function of X into

'X . If ν is

fuzzy set of'

X , then the image of µ under f is the fuzzy set in X defined by

≠

=

−

∈ −

otherwiseo

yfifxSup

xfyfx

φµ

µ

)()(

))((

1

)(1

for each y ∈ 'X

A fuzzy subset µ in X is said to have an sup property if for every subset N ⊆ X, there exists Nn ∈0 such that

.)()( 0 nSupnNn

µµ∈

=

Theorem 4.8: Let f : X → 'X be an onto homomorphism of near – subtraction semigroups. If µ is an anti fuzzy

ideal 'X then

fµ is an anti fuzzy ideal of X.

Proof: (i) suppose µ is an anti fuzzy ideal of'X , then for all x, y X∈ , we have

fµ (x - y) = µ ( f (x - y)) ( ))()( yfxf −= µ ≤ max { ))((,))(( yfxf µµ } = max { µ f

(x), µ f (y)}

ii) For all a, b, x ∈X, we have µ f (a x – a ( b - x)) = µ (f(a x – a (b - x))) = µ (f(a x) – f (a (b - x)))

= µ [f(a) f(x) – f(a) (f(b) – f(x))] ≤ µ (f(x) ) =

)(µ fx

iii) For all x, y ∈ X we have µ f

(x y) = µ (f (x y)) = µ (f (x) f (y)) ≤ µ (f (x) ) = µ f (x)

Therefore µ f is an anti fuzzy ideal of X.

Theorem 4 .9: Let f : X → 'X be a homomorphism of near – subtraction semgroup. If µ f is an anti fuzzy ideal of X,

then µ is an anti fuzzy ideal of 'X .

Proof: Suppose µ f is an anti fuzzy ideal of X.

(i) Let ',' yx ∈ 'X there exists x, y ∈ X such that f(x) = x′ , f(y) = 'y then

µ ( '' yx − ) = µ ( f(x) – f (y) )= µ ( f(x–y) )= µ f (x – y) ≤ max { µ f

(x), µ f (y) }

= max { ))((,))(( yfxf µµ } = max { µ ( x′ ), µ ( 'y )}

(ii) Let',' ba , 'x ∈ 'X there exist a, b, x ∈ X such that f(a) = 'a , f(b) =

'b and f(x) = x′ we have

µ ( 'a x′ – '

b ( 'a – x′ )) = µ [f(a) f(x) –f(b) (f(a) – f(x) )]



= µ (f (a x) - f (a (b - x) )= µ ( f[a x - a (b - x)])= µ f (a x - a (b - x) ) ≤ µ f

(x) = µ (f (x) )= µ ( x′ )

(iii) Let x′ , 'y ∈ 'X there exists x, y ∈ X such that f(x) =

'x and f(y) =

'y then

µ ( '' yx ) = µ (f(x) f(y)) = µ (f(x y)) = )(xyfµ ≤ µ f (x) = µ (f(x)) = µ ( x′ )

Therefore µ is an anti fuzzy ideal in'X .

Theorem 4.10: An onto homomorphic image of an anti fuzzy ideal with sup property is anti fuzzy ideal.

Proof: Let f : X →'X be an onto homomorphism of near subtraction semigroup and µ be an anti fuzzy ideal of X

with sup property

Given ',' yx ∈ 'X , let x0 ∈

1−f ( 'x ), y0 ∈

1−f ( 'y ) be such that µ (x0 ) =

)'(1

)(supxfn

n−∈

µ ,

)( 0yµ = )'(1

)(supyfn

n−∈

µ respectively then we have

µf ( '' yx − ) =

)''(1

)(supyxfn

n−∈ −

µ ≤ max { µ (x0), µ (y0) }

= max

∈∈

(n) Sup ,(n) Sup)(y'fn)(x'fn 1-1-

µµ = max { })'(),'( yxff µµ

Given ',' ba , x′ ∈ 'X we let

suchbexfxbfbafa )'(),'(,)'( 1

01

01

0−−− ∈∈∈

that

)()(sup))'('''( 0))''('''(

'

1

xzxbaxaxbaxafz

f µµµ ≤=−−−−∈ −

= )'(1

)(supxfn

n−∈

µ )'(xfµ=

Given ,',' 'Xyx ∈ we let )'()'( 1

01

yfyandxfxo−− ∈∈

be such that )'(

0)'(

011

)(sup)(,)(sup)(yfnxfn

nyandnx−− ∈∈

== µµµµ respectively. Then we have

Theorem 4.11: Let µ and σ be are fuzzy subsets of X. If µ and σ are anti fuzzy ideal of X, then so σµ ∪ where

σµ ∪ is defined by max))(( =∪ xσµ { } .)(),( Xxallforxx ∈σµ

Proof: (1) For all x, y X∈ then { })(),(max))(( yxyxyx −−=−∪ σµσµ

{{ })(),(maxmax yx µµ≤ { } =)(),(max, yx σσ max{ max ( )(),( xx σµ ), max ))(),(( yy σµ }

= { }.))((),)((max yx σµσµ ∪∪

(ii) For all x, y X∈ , we have

( ) max)()( =−−∪ xbaaxσµ { }))((),(( xabaxxbaax −−−− σµ

{ } max)(),(max =≤ xx σµ { } ))(()(),( xyx σµσµ ∪=

(iii) For all x, y X∈ , then have =∪ ))(( xyσµ { })(),(max xyxy σµ

{ })(),(max xx σµ≤ ))(( xσµ ∪= . Hence σµ ∪ is an anti fuzzy ideal of X.

Theorem 4.12: If { }∧∈ii /µ is a family of anti fuzzy ideals of a near- subtraction semigroup of X then so is

∧∈

∨i

iµ .

Proof: Let { }∧∈ii /µ be a family of anti fuzzy ideal of X and x,y∈X, then we have

'

)'(0

)(

)'()(sup)()(sup)''(11

XofidealfuzzyantianisHence

xnxzyx

f

f

xfnyxfz

f

µ

µµµµµ ==≤=−− ∈′′∈



{ } { }

{ } ( )

∨

∨=∧∈∧∈=

∧∈≤∧∈−=−

∨

∧∈∧∈

∧∈

)(,max}/)({},/)({max

/)(),(max{/)()(

yxiySupixSup

iyxSupiyxSupyx

ii

iiii

iiii

i

µµµµ

µµµµ

{ }

{ } ( )xixSup

ixbaaxSupxbaaxhaveweXxbaallFor

iii

ii

i

∨=∧∈≤

∧∈−−=−−

∨∈

∧∈

∧∈

µµ

µµ

/)(

/))(())((,,,

Theorem 4.13: Let µ be an anti fuzzy ideal in X and let [ ] [ ]1,0)0(,0: →µφ be an increasing function. Let φµ be

a fuzzy set in X, defined by φφ µµφµ ThenXxallforxx .))(()( ∈= is an anti fuzzy ideal of X.

Proof: (i) Let x, y ∈ X then

{ }

{ } { }

( ) ( ) ( ) )()()(()(

,,)(

)(),(max))(()),((max

)(),(max())(()(

xxxbaaxxbaax

thenXxbaLetii

yxyx

yxyxyx

φφ

φφ

φ

µµφµφµ

µµµφµφ

µµφµφµ

=≤−−=−−

∈

==

≤−=−

(iii) Let x, y ∈ X then

( ) ( ) ).()()()( xxxyxy φφ µµφµφµ =≤=

Therefore φµ is an anti fuzzy ideal of X.

Definition 4.14: A fuzzy ideal µ of X is said to be normal if there exists a∈ X such that µ (a) = 1. We note that µ is

normal fuzzy ideal of X if and only if µ (1) =1.

Let FN(X) denote the set of all normal fuzzy ideal of X.

Definition 4.15 : A fuzzy ideal µ of X is said to be complete , if it is normal and there exists Xz ∈ such

that )(zµ =0.

Theorem 4.16 : Let µ be an anti fuzzy ideal of X and w be a fixed element of X such that 0)()1( ≠= wµµ .

Define a fuzzy set *µ in X by )()1(

)()()(*w

wxx µµ

µµµ−

−= for all x∈X.

Then *µ is a complete an anti fuzzy ideal of X.

Proof: (i) For any x, y, ∈ X, we have

( ))()1(

)()(*

w

wyxyx

µµ

µµµ

−

−−=−

{ }

{ }

( )x

idealfuzzyantianisceixSup

ixySupxyhavewethenXyxallFor

ii

i

ii

i

∨=

∧∈≤

∧∈=

∨∈

∧∈

∧∈

µ

µµ

µµ

)sin(/)(

/))()(,,



{ }

{ })(*),(*max

)()1(

)()(,

)()1(

)()(max

)()1(

)()(),(max

yx

w

wy

w

wx

w

wyx

µµ

µµ

µµ

µµ

µµ

µµ

µµµ

=

−

−

−

−=

−

−≤

(ii) For any x, y ∈ X then

( ) ( ))(*

)()1(

)()(

)()1(

)()()(* x

w

wx

w

wxbaaxxbaax µ

µµ

µµ

µµ

µµµ =

−

−≤

−

−−−=−−

(iii) For any x, y ∈X then

)(*)()1(

)()(

)()1(

)()()(* x

w

wx

w

wxyxy µ

µµ

µµ

µµ

µµµ =

−

−≤

−

−=

Therefore ∗µ is an anti fuzzy ideal of X .

Hence )(* Xf N∈µ since )(* wµ = 0 thus *µ is a complete anti fuzzy ideal of X.

Theorem 4.17: Let µ be an anti fuzzy ideal of X and let +µ be a fuzzy set in X given by

+µ (x) = µ (x) + 1 - µ (1) for

all x∈X then µ+ is a normal anti fuzzy ideal.

Proof: i) For all x, y ∈ X, then

µ+ (x – y) = µ (x - y) + 1 - µ (1) ≤ max {µ (x), µ (y)} + 1 – µ (1)

= max {µ (x) + 1 - µ (1), µ (y) + 1 - µ (1)} = max { µ +(x), µ +

(y)}.

ii) For all x, a, b ∈ X, then µ

+ (a x – a (b – x)) = µ (a x – (b - x)) + 1 - µ (1)

≤ µ (x) + 1 - µ (1) = µ+ (x).

iii) For all Xyx ∈, then µ+

(x y) = µ (x y) + 1 - µ

(1) ≤ µ (x) + 1 - µ (1) ≤ µ

+(x).

Since +µ (1) = 1, then

+µ is normal anti fuzzy ideal.

Lemma 4.18: If µ is an anti fuzzy ideal of X, then .)()0( Xxallforx ∈≤ µµ

Theorem 4.19: Let X be a near subtraction semigroup and µ be an anti fuzzy ideal of X. If x ≤ y then ).()( yx µµ ≤

Proof: Let µ be an anti fuzzy ideal of X, and let x, y X∈ . If x ≤ y this implies x-y=0

Consider µ (x) = µ (( x - y) + y) ≤ { } { } )()(),0(max)(),(max yyyyx µµµµµ ==−

Therefore ).()( yx µµ ≤

Theorem 4.20: Let X be a near subtraction semigroup. A fuzzy set µ of X is an anti fuzzy ideal if and only if cµ is

fuzzy ideal.

Proof: Let µ be an anti fuzzy ideal in X.

i) For each x, y∈X, we have cµ (x - y) = 1 - µ (x-y) ≥ 1 - { })(),( yxMax µµ

= { })(1),(1 yxMin µµ −− = { })(),( yxMin

cc µµ

For all x , y∈X, )()(1)(1)( xxxyxycc µµµµ =−≥−=

For all x, a, b X∈ then cµ ( )( xbaax −− ) = 1 - µ ( )( xbaax −− ) )(1 xµ−≥ =

cµ (x)

Hence cµ is a fuzzy ideal in X.

Definition 4.21: [2] Let X and X ′ be any sets and let f : X → X ′ be any function. A fuzzy set µ is called f-

invariant if and only if for all x, y ∈X, f(x) = f(y) implies µ (x) = µ (y).

Theorem 4.22: Let XXf ′→: be an epimarphism of near subtraction semigroup then µ is f- invariant anti fuzzy

ideal of X if and only if f ( µ ) is an anti fuzzy ideal of X ′ .

Proof: '.',') XyxLeti ∈ Then there exists x, y∈X such that f(x) = 'x , f(y) = 'y .



Suppose µ is f- invariant anti fuzzy ideal of X.

{ } { })')(()')(()(),()(

))()(())()()(()'')((

yfxfMaxyxMaxyx

yxffyfxffyxf

µµµµµ

µµµ

=≤−=

−=−=−

ii) '.',' XyxLet ∈ Then there exists x, y∈X such that f(x) = 'x , f(y) = 'y .Then

)')(()()(

))()(())()()(()'')((

xfxxy

xyffyfxffyxf

µµµ

µµµ

=≤=

==

iii) thatsuchXxbaexiststhereThenXxbaLet ∈∈ ,,.',',' ' ')(')(,')( xxfandbbfaaf ===

)')(()())(())(()((

]))()()[()()()(())''('')(( '

xfxxbaaxxbaaxff

xfbfafxfaffxbaxafhaveWe

µµµµ

µµ

=≤−−=−−=

−−=−−

idealfuzzyantianisfHence )(µ

Conversely suppose that )(µf is an anti fuzzy ideal of 'X , then for any x∈X

( ) ( ) ( ) ( ) ( ){ }( ) ( ) ( ){ } ( )xxtXtt

xftfXttxffxff

µµµµ

µµµ

==∈=

=∈==−

,/inf

,/inf)()())((1

Theorem 4.23: [11] Let I be an ideal of near- subtraction semigroup X. If µ is a an anti fuzzy ideal of X, then the

fuzzy set µ of I

X defined by ( ) ( )Ix

xaIa∈

+=+ µµ inf is an anti fuzzy ideal of quotient near subtraction

semigroup I

X .

Proof: Clearly µ is well defined. Let I

XIyIx ∈++ , then

)]()[()]()[(

])[(])[()}()({)

,

vyuxInfvuyxInf

zyxInfIyxIyIxi

IvuIvuz

Iz

+−+=−+−=

+−=+−=+−+

∈∈−=

∈

µµ

µµµ

{ })(),(max

)(),(max)}(),({max,

IyIx

vyInfuxInfvyuxInfIvIuIvu

++=

++=

++≤∈∈∈

µµ

µµµµ

havewethenIXIxandIbIaiii ∈+++ ,)

( ) )])(()[(})()()[()()({ IxbIaIaxIxIbIaIxIa +−+−+=+−++−++ µµ

)]))([()])(()[( IxbaaxIxbaIax +−−=+−−+= µµ

)]))([( zxbaaxInfIz

+−−=∈

µ

)][( zxInfIz

+≤∈

µ ][ Ix += µ

Hence µ is an anti fuzzy ideal in near - subtraction semigroup of X.

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari

Satyanarayana (Academic Secretary) of the National Seminar, for inviting her.

REFERENCES

][(][(][

])[()}()({)

IxzxInfzyxInf

IyxIyIxii

IzIz

+=+≤+=

+=++

∈∈

µµµ

µµ



[1] M. Akram : Fuzzy subquasigroups with respect to a s-norm ,buletinul academiei destinte, No 2 (57), 2008, pp 3 -

13.

[2] M. Akram: Anti fuzzy ideals of lie algebras, Quasigroups and systems 14 (2006), 123-132.

[3] P. Dheena and G. Mohanraaj: On prime and fuzzy prime ideals of subtraction algebra international

Mathematical forum, 4, 2009, no. 47, 2345 - 2353.

[4] P. Dheena and G. Satheesh Kumar: On strongly regular Near – Subtraction Semigroups, Commun.

Korean.Math.Soc. 22(2007) NO.3, pp. 323-330.

[5] Y. B. Jun, H.S. Kim and E.H. Roh: ideal theory of subtraction algebras, scientiae mathematics Japonicae, 61, No. 3

(2005), 459-464.

[6] Y.B. Jun and K.H. Kim: prime and irreducible ideals in subtraction algebra, International

Mathematical forum, 3 No.10 (2008) 457-462.

[7] K.H. Kim, Y.B.JUN And Y.H. YON: on anti fuzzy ideals in near – rings Vol.2, No.2, (2005) pp .71 -80.

[8] K.H. Kim and Y. B. Jun: A note on fuzzy R – subgroups of near rings, Soochow Journal mathematics Vo.28, No. 4

(2002), pp. 339 - 346.

[9] K. H. Kim: On subtraction semigroups, scientiae Mathematicae Japonicae 62 (2005) No .2, 273- 280.

[10] Prince Williams: Fuzzy ideals in near subtraction semigroups , international journal of computational and

mathematical sciences 2.1.2008.

[11] A. Rosenfeld: Fuzzy groups, J. Math. Anal. Appl.35 (1971) 512-517.

[12] S. E. Yehia: Fuzzy ideals and fuzzy subalgebras of Lie algebras, Fuzzy Sets and Systems 80 (1996), PP. 23-244.

[13] L. A. Zadeh: Fuzzy sets, Information Control 8 (1965), 338 − 353.

[14] B. Zelinka: Subtraction semigroups, Math. Bohemica 120, (1995), 445-447.

John Napier invented Logarithms.

Trigonometry is invented by Hipparchus. The first calculating machine to do multiplications was invented by Leibnitz in 1671.

The writer of “Siddhantha Siromani” is Bhaskara Charya.

Evariste Galois first used the term “Group”.

Noether is known as “Mother of Modern Algebra”.

“VIRTUE IS THE KNOWLEDGE OF GOODNESS”

“SIN IS THE IGNORANCE OF GOODNESS”.

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya

Kumari) PP: 91-96 91

Rings with [x, y

n] – [xn, y]

in the center. Authors: K.SUVARNA AND K.MADHUSUDHAN REDDY, SRI KRISHNADEVARAYA UNIVERSTIY. ANANTAPUR – 515055. [email protected] and [email protected] ----------------------------------------------------------------------------------------------------------------------- ABSTRACT: If R is a semisimple nonassociative ring with unity satisfying [x,

yn] – [xn

, y] in the center, then R is commutative. INTRODUCTION: RamAwtar [1] studied nonassociative rings with unity satisfying the identity (xy)n

= xny

n. In this paper we prove that if R is a semisimple nonassociative ring with unity satisfying [x, y

n] – [xn, y] in the center, then R is

commutative. PRELIMINARIES: Throughout this section R denotes a nonassociative ring

with unity. The center U of R is defined as U = {u∈R / [u, R] = 0}. It is also called as commutative center. A ring R is of characteristic ≠ n if nx = 0 implies

x = 0 for all x in R and n a natural number. MAIN RESULTS First we prove the following Lemma: Lemma 1: For any positive integers n and m the following relations hold:

0

0 if ( 1) ( 1 )

! if

mk n

k

m m nm k

k n m n=

> − + − =

= ∑

Proof : We prove the Lemma by using induction on m.

Case 1: We consider m > n. If m = 1 then

∑=

−

−=

1

0

)2(1

)1(..k

nk kk

SHL .

Since m > n, we have n = 0 because m = 1

1 10

0 0

0 1

1 1. . ( 1) (2 ) ( 1)

1 1 ( 1) ( 1) 1 1 0.

0 1

k k

k k

L H S kk k= =

= − − = −

= − + − = − =

∑ ∑

Therefore result is true for m = 1.

Now we assume that result is true for m.

Research Paper

(Oral Presentation) ------------------------------------------------------ --------------------------

Presenter: MADHUSUDHAN REDDY



i.e., ∑=

=

>=−+

−

m

k

nk

nmn

nmkm

k

m

0 if !

if0)1()1( 1

Now consider ∑+

=

−+

+−

1

0

)2(1

)1(m

k

nk kmk

m

nmm

k

nk mmm

mkm

k

m))1(2(

1

1)1()2(

1)1( 1

0

+−+

+

+−+−+

+−= +

=

∑

)1()1()11()!1(!

)!1()1( 1

0

+

=

−++−+−+

+−=∑ m

m

k

nkkm

kmk

m

.)1(1....)1(1

)1()!)(1(!

!)1()1( 1

0

1 +

=

− −+

++−+

+−+

−−+

+−=∑ m

m

k

nnk kmn

kmkmkmk

mm

.)1(

)1(

1.............................

)1(1

)1(

)!(!

!)1()1( 1

0

21

+

=

−−

−+

−++

+−+

+−+

−−+= ∑ m

m

k

nn

k

km

kmn

km

kmk

mm

.)1()1(

1....)1()1()1( 1

0

1 +

=

− −+

−+++−+

−+= ∑ m

m

k

nk

kmkm

k

mm 2

Now by 1

{ } 01....)1()1(0

1 =++−+

−∑

=

−m

k

nk kmk

m. Since n – 1, n – 2,………< m,

2 is equal to

1

0

)1()1(

1)1()1( +

=

−+−+

−+= ∑ m

m

k

k

kmk

mm

1

0

)1()1(

)1()1( +

=

−+

−+

+−=∑ m

m

k

k

k

m

km

m

1

0

)1()!(!

!

)1(

)1()1( +

=

−+−−+

+−=∑ m

m

k

k

kmk

m

km

m

1

0

)1()!1(!

)!1()1( +

=

−+−+

+−=∑ m

m

k

k

kmk

m



1

0

)1(1

)1( +

=

−+

+−=∑ m

m

k

k

k

m

110 )1(

1)1(........

1

1)1(

0

1)1( +−+

+−++

+−+

+−= mm

m

mmm

+

+−+

+−++−

++

+−= +

1

1)1(

1)1(........

2

2

1

11 1

m

m

m

mmmmm

+

+−+

+

−−+−

+

+

+

−= +

1

1)1(

1)1(.....

21101 1

m

m

m

m

m

mmmmmmm

= 1 – 1 = 0. Thus the induction completes.

Case 2: For m = n, we adopt the same technique of induction. �

Theorem 1: Let R be a semisimple nonassociative ring of char. ≠ n! with unity

satisfying [x, yn] – [xn

, y] ∈ U for all x, y in R. Then R is commutative.

Proof : By hypothesis [x, yn] – [xn

, y] ∈ U

i.e., xyn – y

nx – x

ny + yx

n ∈ U. 3 Now we replace y with y + 1 in 3.3.3. Then

x(y + 1)n – (y + 1)nx – xn(y + 1) + (y + 1)xn

∈ U

i.e., x(y + 1)n – (y + 1)nx – xn

y + yxn ∈ U. 4

From 3 and 4, we get

x(y + 1)n – (y + 1)nx – xy

n + y

nx ∈ U. 5

Now we apply Bionomial expansion for 5. Then

+

−++

+

+

−−

n

ny

n

ny

ny

ny

nx

nnn

1.......

210

21

xn

ny

n

ny

ny

ny

nnnn

+

−++

+

+

− −−

1.......

210

21Uxyxy nn ∈+−

xyn

ny

ny

nxxy

nnn +

−++

+

+ −−

1...............

21

21



xxyn

ny

ny

nxy

nnn −

−++

+

−− −−

1...............

21

21Uxyxy nn ∈+− .

On simplification, we get

.

1.......

21

1.......

21

21

21 Ux

yn

n

yn

yn

yn

ny

ny

nx

nn

nn ∈

−++

+

−

−++

+

−−

−−

By continuing the same process as above for (t – 1) times, we obtain

UEEEEEE nttntt ∈′−′−′−+++ ++ ........................ 11 . 6

Where for r = t, t + 1, t + 2,………….,n, we have

and

−−

−−

−−=′

−−

−−

−−=

+−−−

=

+−−−

=

∑

∑

)()1(2

)1(1

)1(

)()1(2

)1(1

)1(

122

0

122

0

xyktk

t

r

ntE

xyktk

t

r

ntE

rnrt

k

k

r

rnrt

k

k

r

7

Now again by replacing y with y + 1 in 6 and substracting 6 from resulting expression, we obtain

UGGGGGG nttntt ∈′−′−′−+++ ++++ ....................... 2121 . 8

Where for r = t + 1, t + 2,……………., n, we have

)()(1

)1(1

121

0

+−−−

=

−

−−

−= ∑ rnr

t

k

k

r xyktk

t

r

ntG

and )()(1

)1(1

121

0

xyktk

t

r

ntG rnr

t

k

k

r

+−−−

=

−

−−

−=′ ∑ .

The coefficients of xyn-r+1 and yn-r+1

x (t + 1 ≤ m ≤ n) are

21

0

)(1

)1(1

−−

=

−

−−

−∑ mt

k

k ktk

t

m

nt .

We replace t with n in 6. Then UEE nn ∈′− .

Therefore )()1(2

)1(1

)1( 122

0

+−−−

=

−−

−−

−− ∑ nnn

n

k

k xyknk

n

n

nn



Uxyknk

n

n

nn nnn

n

k

k ∈−−

−−

−−− +−−

−

=

∑ )()1(2

)1(1

)1( 122

0

.

i.e., −−−

−−

−− −

−

=

∑ )()1(2

)1(1

)1( 22

0

xyknk

n

n

nn n

n

k

k

Uyxknk

n

n

nn n

n

k

k ∈−−

−−

−− −

−

=

∑ )()1(2

)1(1

)1( 22

0

or Uyxxyknk

n

n

nn n

n

k

k ∈−−−

−−

−− −

−

=

∑ )()1(2

)1(1

)1( 22

0

. 9

Now by using Lemma 1, we have

=

>=−+

−∑

= nmn

nmkm

k

mn

m

k

k

if !

if 0)1()1(

0

So here, we have

2

2

0

22

0

)1)2((2

)1()1(2

)1( −−

=

−−

=

−+−

−−=−−

−− ∑∑ n

n

k

knn

k

k knk

nkn

k

n

= (n – 2)!

and we know that nn

n

n=

=

− 11. Using these 9 can be written as.

(n – 1)n (n – 2)! (xy – yx) ∈ U.

i.e., n(n – 1) (n – 2)! (xy – yx) ∈ U, or n! (xy – yx) ∈ U. 10

Since R is of char. ≠ n!, we have xy – yx ∈ U.

Now if U = 0 or U = R then R is essentially a commutating ring. Therefore

suppose 0 ≠ U ≠ R. We take R to be a simple ring. Consider the principle ideal (xy

– yx)R. Since U ≠ R and R is simple, (xy – yx)R = 0. Now if R is a division ring then xy – yx = 0. Hence R is commutative. If R is a simple ring which is not a

division ring, then R is homomorphic to D2, the complete matrix ring of 2×2

matrices over a division ring D which must satisfy the condition [x, yn] – [xn

, y] ∈

U. Infact, if we choose

=

00

01x and

=

00

10y then this condition fails.

Hence R is commutative.



Now semisimple ring is a subdirect sum of simple rings, each of which is shown to be commutative. Hence semisimple ring R is also commutative under given hypothesis, because a subdirect sum of simple rings will also satisfy the identity satisfied by the simple rings. �

We present two examples to show that the existence of a unity in R is necessary in the hypothesis of the Theorem 1.

Example 1: Let R be a the subring generated by the matrices

000

100

000

,

000

000

100

,

000

000

010

in the ring of all 3×3 matrices over Z2. The ring of integers mod 2. For each

integer n ≥ 1 and all x, y ∈ R, [x, yn] = [xn

, y] holds. However, R is not commutative.

Example 2: Let R be the subring generated by the matrices

000

100

000

,

001

000

000

,

010

000

000

in the ring of all 3×3 matrices over Z2, the ring of integers mod 2. For each integer

n ≥ 1 and x, y ∈ R, [x, yn] = [xn

, y] is satisfied in R, but R is not commutative. � Acknowledgements: The authors thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting them. REFERENCE [1] Ram Awtar - “On the commutativity of nonassociative rings”, Publicationes Mathematicae, 22 (1975) 177–185

What ever you think that you will beWhat ever you think that you will beWhat ever you think that you will beWhat ever you think that you will be If you think your self weak, weakIf you think your self weak, weakIf you think your self weak, weakIf you think your self weak, weak you will beyou will beyou will beyou will be

If you think your self strong, strong you will beIf you think your self strong, strong you will beIf you think your self strong, strong you will beIf you think your self strong, strong you will be ---------------------------------------------------------------------------------------------

GOD IS PRESENT IN EVERY JIVA,

THERE IS NO OTHER GOD BESIDES THAT

“WHO SERVES JIVA, SERVES GOD IN DEED”


(July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP:97-100

97

CANCELLATIVE LEFT(RIGHT) REGULAR SEMIGROUPS Authors: G. Shobhalatha, P. Sreenivasulu Reddy and K. Hari Babu Gates institute of technology Gooty, Anantapur(dist) – 515 416, A.P, India. ------------------------------------------------------------------------------------------------------- Abstract: A semigroup S is called regular semigroup if for every a∈S there exists x in S such that axa = a introduced by J.A.Green. In this paper some preliminaries and basic concept of regular Semigroup are present and proved that a cancellative semigroup S is left(right) regular semigroup if and only if it is a :

(i) completely regular semigroup (ii) Clifford semigroup (iii) an E-inversive semigroup (iv) g-regular semigroup.

Introduction: Regular semigroups were introduced by J.A.Green in his influential paper (1951) “On the structure of semigroups”. This was also the paper in which Green’s relations were introduced. The concept of regularity in a semigroup was adopted from an analogous for rings already considered by J.Von Neumann. The suggestion that the notion of regularity be applied to semigroups was first made by David Ree’s. Left (right) regularity in semigroups has long been studied. In 1954 Clifford proved in his paper that semigroup is a band of groups if and only if it is both left and right regular. Kiss generalized left(right) regular elements of semigroups in 1972. It was shown by Anjaneyulu in 1981 that in a duo semigroup, the set of all left regular elements and the set of all right regular elements coinside. John Howie’s latest book is a substantial updating of his 1976 book “An introduction to semigroup theory (Academic press). Like its predecessor the new book doesnot attempt to cover the whole field, but concentrates instead on the algebraic theory with a particular emphasis on the class of regular semigroups. Regular semigroups are easier to handle than arbitrary semigroups, but, more importantly they play a paradigmatic role in semigroup as a whole. 1.1 Definition: An element a of semigroup (S, .) is left (right) regular if there exists an element x in S such that xa2 = a (a2x = a) . 1.2 Definition: A semigroup (S, .) is called left(right) regular if every element of S is left (right) regular. 1.3 Definition: An element a of a semigroup (S, .) is said to be regular if there exist x in S such that axa = a 1.4 Definition: A semigroup (S, .) is called regular if every element of S is regular. Examples of regular semigroups: i) Every group is regular. ii) Every inverse semigroup is regular. iii) Every band is regular in the sence of this article, through this is not what is

meant by regular band. iv) The bicyclic semigroup is regular. v) Any full transformation semigroup is regular. vi) A Rees matrix semigroup is regular.

Research Paper

(Oral Presentation) -----------------------------------------------------------------------------

Presenter: HARI BABU



98

vii) The set of integers(Z) with respect to addition. The set of real numbers (R) with respect to multiplication 1.5 Definition: A regular semigroup S is called Clifford semigroup if all its idempotents form centre. 1.6 Definition: A semigroup S is said to be g-regular semigroup if every element of S is g-regular. Examples: (i) Every regular semigroup is g-regular (ii) Every inverse semigroup is g-regular (iii) Every simple semigroup with idempotent element is g-regular semigroup. Let {0. e1, e2, ….} with the operation eiej = 0 if i ≠ j, eiej = ei if i = j. Then S is a g-regular semigroup. 1.7 Definition: An element a is said to be g-regular if there exist an element x such that x = xax.

1.8 Definition: Let S be a semigroup. The center Z(S) of S is the set Z(S) = {a∈S:

ax = xa for every x∈S}. 1.9 Definition: An element a of a semigroup S is called an E-inversive if there is an

element x in S such that (ax)2 = ax i.e,ax∈ E(S). Where E(S) is set of all idempotent elements of S. 1.10 Definition: A semigroup S is called an E –inversive semigroup if every element of S is an E-inversive.

Examples: (i) Regular semigroups (a = axa implies that ax ∈ E(S)).

(ii) Eventually regular semigroups (an regular for some n >1 implies that an xan = an and

a(an-1x) ∈E(s) for some x ∈S). (iii) Semigroup S which contain idempotents and are totally ordered with respect to the

natural partial order a ≤ b iff a = xb = by, xa = a = ay for some x,y ∈ S1.(See Mitsch

(1986)) are E-inversive. Infact if a∈ S and a ≥ e for some e∈ E(S) then e = xa = ay and

ay∈ E(S), if a ≤ e ; then a = xe = ey ,xa = a implies that a2 = (xe)(ey) = x(ey) = xa = a.

Hence a.a = a ∈ E(S). 1.11 Theorem: A cancellative left regular semigroup is commutative

Proof: Let S be a cancellative left regular semigroup. Let a,b∈S ⇒ (ab)2 = abab ⇒ a2b2

= abab ⇒ a.ab2 = abab ⇒ ab2 = bab. Since S is cancellative, abb = bab ⇒ ab = ba. Hence S is commutative. Therefore, a cancellative left regular semigroup is commutative. Similarly we can prove that a cancellative right regular semigroup is commutative. 1.12 Theorem: A cancellative semigroup is left (right) regular semigroup if and only if it is completely regular. Proof: Let S be a cancellative semigroup. Assume that S is left regular semigroup.Then

for any a∈S there exist x∈S such that xa2 = a ⇒ xxa2 = xa ⇒ x2a2 = xa ⇒ (xa)2 = xa

⇒ xaxa = xa. Since S is cancellative, xaxa = xa ⇒ axa = a ⇒ axa = a. Therefore a is a

regular for every a∈S. Hence S is a regular semigroup. From Theorem 2.2.1, S is commutative. Thus, ax = xa. Therefore, S is completely regular semigroup.

Conversely, Let S be a completely regular semigroup.Then for any a∈S there exist x∈S such that axa = a and xa = ax. To prove that S is left regular, consider axa = a

⇒ xaxa = xa ⇒ (xa)2 = xa ⇒ x2a2 = xa ⇒ xxa2 = xa ⇒ xa2 = a. ⇒ a is left regular. Hence S is a left regular semigroup. Similarly we see that a cancellative semigroup is a regular if and only if it is completely regular.



99

1.13 Theorem: A cancellative semigroup is left (right) regular if and only if it is Clifford semigroup. Proof: Let S be a cancellative semigroup. Suppose S is left regular semigroup then from

Theorem. 2.2.2, S is a regular semigroup.Since S is a regular semigroup ⇒ S is an E-

inversive semigroup. For any a∈ S there exist x∈ S such that ax, xa∈E(S).Again from Theorem(2.2.1) . S is commutative. Therefore, all the idempotent elements commutes. Hence S is a Clifford semigroup. Conversely, Assume that S is a Clifford semigroup. By the definition 1.1.22, S is regular and idempotent elements commutes. Since S is regular, then from Theorem [2.2.2.], S is a left regular semigroup. 1.14 Theorem: A calcellative semigroup is regular if and only if it is g-regular. Proof: Let S be a cancellative semigroup. Assume that S is a regular semigroup. For any

a∈S there exist an element x ∈S such that a = axa ⇒ ax = axax. Since S is cancellative,

a(x) = a (xax) ⇒ x = xax ⇒ a is g-regular for every a in S. Therefore S is g-regular

semigroup. Conversely, Assume that S is g-regular semigroup. For any a∈S there exist

an element x ∈S such that x = xax ⇒ xa = xaxa ⇒ a = axa ⇒ a is regular, for all a in S. Therefore, S is a regular semigroup. 1.15 Theorem: A cancellative semigroup S is left(right) regular if and only if it is g-regular. Proof: Proof is similar to Theorem 2.2.4. 1.16 Note: Every regular semigroup is g-regular semigroup but the converse is need not be true. Example: (i) Let {1, e, 0} be a semigroup with identity 1 and 0 and ee = 0 and let S be the N*{1, 0, e} *N.

Define an operation on S by (m,a,n). (p,b,q)=

=

<+−

>+−

pifnqabm

pifnqbqnm

pifnnpqam

),,(

),,(

),,(

Then S becomes a simple semigroup. Since S has idempotent elements, it is a g-regular semigroup. But any element of the D-class N*{e}*N is not regular. Hence S is g-regular semigroup which is not a regular semigroup. (ii) Let S be the subsemigroup of &({1,2,3,…7}) generated by X

=

5765432

7654321 We can prove that easily x has index 4 and period 3. The

Kernal Kx is {x4, x5, x6 }. And x6 is the identity element of Kx. Since S is periodic, it is regular semigroup. Also x6 is unique idempotent of S. 1.17 Theorem: A cancellative left regular semigroup is an E-inversive semigroup. Proof: Let S be a cancellative left regular semigroup then by Theorem 2.2.1, S is

commutative. Let a∈S. Then there exist x∈S such that xa2 = a ⇒ xxa2 = xa ⇒ x2a2 = xa

⇒ (xa)2 = xa and we have (ax)2 = ax and (xa)2 = xa ⇒ ax and xa are elements of E(S)

⇒ a is an E-inversive element of S. Therefore S is an E-inversive semigroup.

Conversely, Assume that S is an E-inversive semigroup. For any a∈ S there exist x∈S such that (ax)2 = ax , (xa)2 = xa. To prove that S is a left regular, consider ax = (ax)2

⇒ ax = axax. Since S is a cancellative, ax = axax ⇒ a = axa ⇒ a = (ax)a ⇒ a = (xa)a

⇒ a = xa2 ⇒ a is left regular. Hence S is a left regular semigroup. Similarly we can



100

prove that a cancellative semigroup is right regular semigroup if and only if it is an E-inversive semigroup. Acknowledgements: The authors thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting them. References: [1] Clifford, A.H. and Preston, G.B. “The algebraic theory of semigroups”, Math.Surveys No.7,Amer.math.soc., Vol. I, (1967). [2] Grillet, P.A. “The Structure of regular Semigroups-.I”, Semigroup Forum 8 (1974), 177-183. [3] Grillet, P.A. “The Structure of regular Semigroups-.II”, Semigroup Forum 8 (1974), 254-259. [4] Howie, J.M. “Introduction to semigroup theory”, Academic Press, London, 1976. [5] Mitsch, H. “ Subdirect products of E-inversive semigroups” J.Austral.math.soc. (series A) 48 (1990);66-78. [6] Mun Gu Sohn and Ju Pil Kim “G- Regular semigroups”, Bull. Korean Math.Soc.25, No.2, (1988) 203-209

The mind is demoralized by contact with the worth less. It becomes like those with whom it associates.

So with excellent it attains to excellent. -----------------------------------------------------------------------------------------------

Even an insect, if it be on a flower, may ascend the head of the great. ------------------------------------------------------------------------------------------

A stone consecrated by a man of mighty power, becomes divine --------------------------------------------------------------------------------------------

The eastern mountain is lighten up by the nearness of the sun

-----------------------------------------------------------------------------------

SO EVEN THE MAN DEVOID OF

EXCELLENCE IS LIGHTED UP BY THE

NEARNESS OF THE WISE.


12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 101-104 101

Some Results on Weakly Periodic Rings Authors: P. Prathapa Reddy* and K. Suvarna**

* Department of Humanities, GATES Institute of Technology, Gooty, Anantapur, A.P, India, [email protected]. ** Department of Mathematics, Sri Krishnadevaraya University, Anantapur, A.P, India, [email protected]. ---------------------------------------------------------------------------------------------------------------------

Abstract: A ring R is called periodic if for each x ∈ R, there exit distinct positive integers m and n such that xn = xm. An element x of R is potent if xk = x for some integer k > 1. A ring R is called weakly periodic if every x in R can be written in the form x = a + b for some nilpotent element a and some potent element b in R. let w = w(x, y) be a word or monomial in the noncommuting indeterminates x and y. In this paper using some properties of weakly periodic rings proved by Abu-Khuzam et. al., we prove that if R is a weakly periodic ring satisfying

[xa – xa2x, x] = 0 for all x ∈ R, a ∈ N, then R is commutative. Also it is proved that in a weakly

periodic ring R with [a, b] potent for all a ∈ N, b ∈ N, if there exists a word w = w(x, y) such that

w[[x, y], xy] = 0 = [[x, y], xy]w′, then R is commutative.

1. Introduction

Bell and Klein [2] established sufficient conditions for finiteness, commutativity or periodicity of weakly periodic rings. In [1] Abu-Khuzam et. al., using a word w = w(x, y), studied weakly periodic rings with conditions on commutators. Yaqub [4] shown that a weakly periodic ring R in which certain extended commutators are potent must have a nil commutator ideal and the set N of nilpotents form an ideal, which coincides with the Jacobson radical J of R. Recently Rosin and Yaqub [3] studied the structure of weakly periodic rings with a particular emphasis on conditions which imply that such rings are commutative or have a nil commutator ideal.

In this paper, using some properties of weakly periodic rings proved by Abu-Khuzam et.

al., we prove that if R is a weakly periodic ring satisfying [xa – xa2x, x] = 0 for all x ∈ R, a ∈ N,

then R is commutative. Also it is proved that in a weakly periodic ring R with [a, b] potent for all

a ∈ N, b ∈ N, if there exists a word w = w(x, y) such that w[[x, y], xy] = 0 = [[x, y], xy]w′, then R is commutative.

2. Preliminaries A ring R is called weakly periodic if every element of R is expressible as a sum of a

nilpotent element and a potent element of R. It is well known that if R is periodic then it is

weakly periodic. For x, y ∈ R, [x, y]1 = xy – yx is the usual commutator, and for every positive integer k > 1, we define inductively [x, y]k = [[x, y]k–1, y]. A word w(x, y) is a product in which each factor is x or y. The empty word is defined to be 1. We begin with the following results for an arbitrary ring R, which are useful to prove main results.

Lemma 1: Suppose that R is a ring with identity 1. If xn[x, y] = 0 and (x+1)n[x, y] = 0 for some x, y in R and some integer n > 0 , then [x, y] = 0. A similar statement holds if we assume [x, y] xn

= 0 and [x, y] (x+1)n = 0.

Research Paper

(Oral Presentation) ------------------------------------------------------

Presenter: P. Prathap Reddy



Lemma 2: If P is the set of potent elements, N*= {x ∈ N/x

2 = 0} is commutative and N

multiplicatively closed, then PN ⊆ N. In particular, if N is commutative and P ∪ N generates R, then N forms an ideal. Lemma 3: Suppose that R satisfies the following condition: for each x, y ∈ R there exist f(x), g(x) in x

2Z[x] such that [x – f(x), y – g(y)] = 0. If for each a ∈ N

∗ and x ∈ R, there exists a positive integer k such that [a, x]k (= [[a, x]k-1, x]) = 0, then R is commutative. Lemma 4: Let R be a subdirectly irreducible ring. Then the only central idempotent elements of R are 0 and 1. Lemma 5: Let R be a weakly periodic ring. Then the Jacobson radical J of R is nil. If, furthermore, xR ⊆ N for all x ∈ N, then N = J and R is periodic.

3. Main Results Lemma 6: Let R be an arbitrary ring (not necessarily weakly periodic) which satisfies

w[[x, y], xy] = 0 =[[x, y], xy] w′, then the idempotents of R are central.

Proof: Suppose e2 = e ∈ R, x ∈ R, f = e + ex – exe. Then f 2 = f, ef = f, fe = e and hence [e, f] =

(ex – exe). Now, by hypothesis, there exists a word w = w(e, f) such that w[[e, f], ef] = 0. Since ef=f and fe =e we have w= w(e, f) = e or w = f. Hence, we have 0 = w [[e, f], ef] =

e[[e, f], ef] or f[[e,f], ef] = – (ex – exe). Thus ex = exe for all x in R.

Now, by taking f ′ = e + xe – exe, and by hypothesis, an argument similar to the above shows

that [[e, f ′], e f ′] w′= 0; w′= e or w′ = f ′. It is easily seen that [[e, f ′], e f ′] = –[f ′, e] = exe – xe. Thus xe = exe for all x in R. Hence the idempotents of R are central. � Now we are able to prove the main results of this section.

Theorem 7: Let R be a weakly periodic ring. Suppose that (i) for each x∈ R there exists

f(x) ∈ x2Z[x] such that x – f(x) ∈ C(N) and (ii) for each x ∈ N + P and a ∈ N, [xa – xa

2x, x] = 0.

Then R is commutative.

Proof: By (i), we can easily see that N is commutative. Hence, by Theorem 4.1.2, N is a commuatative ideal.

Now, let x ∈ N + P and a ∈ N. Then x – xn ∈ N and x2

– xn+1∈ N.

In particular, R satisfies the condition [x – f(x), y – g(y)] = 0, for each x, y ∈ R and f(x), g(x) in

x2Z[x]. Further, since N2

⊆ C, we have [xa – xa2x, x] = 0.

Then [xa – a2x

2, x] = 0.

ie., (xa – a2x

2)x – x(xa – a2x

2) = 0,

ie., xax – x2a = 0.

So x[a, x] = 0.

We substitute x+1 for x. Then (x + 1)[a, (x + 1)] = 0.

By Lemma 1 [a, x] = 0, for all x ∈ R and a∈ N. Hence by Lemma 3, R is commutative. �

Theorem 8: Let R be a weakly periodic ring such that for all x, y ∈ R, there exists a word w = w(x, y) and a positive integer k = k(x, y) for which w[[x, y], xy] = 0. Then the commutator ideal of R is nil, the set N of nilpotents is an ideal of R and R is periodic. Proof: First, we observe that all the hypotheses are inherited by any homomorphic image of the ground ring R. Moreover, any division ring D which is weakly periodic must satisfy the condition “x

n(x) = x, n(x) >1, for all x in D” and hence by a well known theorem of Jacobson, D is

commutative. Thus the theorem is true for division rings.



Next, suppose that R is a primitive ring which satisfies the hypothesis of the theorem and suppose R is not a division ring. Since the hypothesis involving the existence of the word w=

w(x, y) is inherited by all subrings and all homomorphic images of R, therefore some complete matrix ring Dm over some division ring D, with m > 1, satisfies the word hypothesis. This, how ever, is false, as can be seen by taking

x = E11, y=E11 + E12; x, y ∈ Dm. In this case,[[x, y], xy] = ± E12 for all positive integer k and, moreover any word w=w(x,y) must be equal to x or y (since x2

= x, y2 = y, xy = y, yx = x).

Hence w[[x, y], xy] = ± xE12 = ± E12 or w[[x, y], xy] = ± yE12 = ± E12. In either case we obtain w[[x, y], xy] = ± E12 ≠ 0. In other words, the “word” hypothesis is not satisfied, a contradiction. This contradiction shows that any primitive ring which satisfies the hypotheses must be a division ring, and hence must be commutative as remarked above. Therefore, theorem is also true for all semi simple rings, which implies R/J is commutative, and thus

C(R) ⊆ J. Combining this with lemma 5, we see that

C(R) ⊆ J ⊆ N.

Hence the commutator ideal C(R) of R is nil. This, as is readily verified, implies that N is an ideal of R.

To prove R is periodic, let x ∈ R.

Then by the definition of weakly periodic, x = a + b, for some a ∈ N, b potent (bn = b, n > 1).

Thus, x – a = b = bn = (x – a)n

(a ∈ N). Since N is an ideal, this implies that x – xn ∈ N, and

thus, xα =x

α +1f(x) for some integer α ≥ 1 and some polynomial f(λ) with integer coefficients.

Hence R is periodic, by Chacron’s Theorem. This completes the proof of the theorem. � Theorem 9: Let R be a weakly periodic ring such that (i) [a, b] is potent for all nilpotent elements a, b in R,

(ii) for all x, y in R, there exist words w = w(x, y), w′ = w′(x, y) such that

w[[x, y], xy] = 0 = [[x, y], xy] w′, then R is commutative. Proof: By theorem 8 we have R is periodic and the set N of nilpotents is an ideal of R. 3.1

Now, by Hypothesis (i), [a, b] is potent for all a ∈ N, b ∈ N .

Thus [a, b]n = [a, b] for some n > 1 (a ∈ N, b ∈ N). 3.2

By re-iterating in 3.2, we obtain

[a, b]λ(n – 1 )+1 =[a, b] for all positive integers λ. 3.3

Since N is an ideal and a ∈ N, b ∈ N, we have [a, b] ∈ N. Hence by 3.3, we conclude that

[a, b]=0 for all a ∈ N, b ∈ N 3.4 i.e, N is commutative. Moreover by lemma 6, the set E of idempotents of R is contained in the center C. 3.5 As is well known, we have

R ≅ a subdirect sum of subdirectly irreducible rings Ri. 3.6

Let σ: R → Ri be the natural homomorphism of R onto Ri. In view of 3.1 Ri must be periodic also. Hence by lemma 2 [1], we have

the nilpotents of Ri= σ(N). 3.7



But N is commutative. Hence by 3.7, the set Ni of nilpotents of Ri is commutative. 3.8 We now distinguish two cases

Case 1: 1∉ Ri .

Let xi ∈ Ri and let σ: x → xi. Since R is periodic, Let xr= x

s, r > s ≥ 1.

Let e =xs(r-s). Then e2

= e and hence e ∈ C. Thus, ei =xi

s(r-s) is a central idempotent of Ri. 3.9 Since we are assuming that Ri does not have an identity, the central idempotent element ei of the subdirectly irreducible ring Ri must be equal to zero.

Hence xis(r-s) = 0 for all xi ∈ Ri.

Thus Ri = Ni is commutative, by 3.8.

Case 2: 1i ∈ Ri. The above argument in Case 1 shows that xis(r –s) is a central idempotent in the

subdirectly irreducible ring Ri, and hence

xis(r–s) = 0, or xi

s(r –s) = 1i, for all xi ∈ Ri . Thus, every element of Ri is nilpotent or is a unit in Ri. 3.10 Moreover, Ri (as a homomorphic image of R) satisfies all the hypotheses of theorem 3.8. Hence, the set Ni of nilpotents of Ri forms an ideal of Ri. This ideal Ni is also commutative, by 3.8. We have thus shown that the set Ni of nilpotents of Ri is a commutative ideal of Ri. 3.11 We claim that Ni is contained in the center Ci of Ri. 3.12

Suppose not. Let ai ∈ Ni, xi ∈ Ri, be such that

[ai, xi] ≠ 0, ai ∈ Ni, xi ∈ Ri. 3.13

By 3.11, xi ∉ Ni , and hence by 3.10, xi is a unit in Ri.

Let ui = 1i + ai. Then ui is a unit in Ri (since ai ∈ Ni). By hypothesis (ii), there exists a word w = w(ui, xi) and a positive integer ki = ki(ui, xi) such that 0 = w[[ui, xi], (ui – 1i) xi] = w(ui – 1i)[[ui, xi], xi]. Therefore wai[[ui, xi], xi] = 0. Hence w[ui, xi]2 = 0. By theorem 2 [1] it follows that R is commutative. � Acknowledgements: The authors thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting them. References [1] Ab-Khuzam, H., Hasanali, M., and Yaqub, A., “Weakly periodic rings with

conditions on commutators”, Acta Math. Hungar 71 (1996), 145-153. [2]. Bell, H.E. and Klein, A., “On finiteness, commutativity and periodicity in rings”, Math. J. Okayama Univ., 35 (1993), 181-188. [3]. Rosin, A. and Yaqub, A., “Weakly periodic and subweakly periodic rings”, Intl. J. Math. Math. Sci, 33 (2003), 2097-2107. [4]. Yaqub, A., “Structure of weakly periodic rings with potent extended commutators”, Intl. J. Math. Math. Sci. 25-5 (2001), 299-304.


Regular semigroups satisfying the identity abc = cb

Authors: G. Shobhalatha & P. Sreenivasulu Reddy Sri Krishnadevaraya University, Anantapur – 515 005, Andhra Pradesh, India. Email; [email protected]. ------------------------------------------------------------------------------------------------------------ Absract: In this paper we proved that a regular semigroup S satisfying the identity abc = cb for every a,b,c in S is cancellative. It is also proved that a regular semigroup is left(right) regular, completely regular, commuting regular, quasi-seperative, weakly seperative, seperative, permutable, medial, normal, L-commutative, R-commutative, conditional commutative, external commutative, left(right) quasi-normal, left(right) semi-normal and left(right) semi-regular with the identity abc = cb which is defined by J.M.Howie[2]. Introduction: In this paper we discuss the structures of regular semigroups which satisfies the identity abc = cb for every a, b, c. The results obtained in this section based on the results of Yamada, M. and Kimura, N. [3] 1.1 Definition: A semigroup (S, .), all of whose elements are cancellable is a cancellative semigroup 1.2 Definition: An element a of semigroup (S, .) is left (right) regular if there exists an element x in S such that xa2 = a (a2x = a) . 1.3 Definition: A semigroup (S, .) is called left(right) regular if every element of S is left (right) regular. 1.4 Definition: An element a of a semigroup (S, .) is said to be regular if there exist x in S such that axa = a 1.5 Definition: A semigroup (S, .) is called regular if every element of S is regular. Examples of regular semigroups: i) Every group is regular. ii) Every inverse semigroup is regular. iii) Every band is regular in the sence of this article, through this is not what is meant

by regular band. iv) The bicyclic semigroup is regular. v) Any full transformation semigroup is regular. vi) A Rees matrix semigroup is regular. vii) The set of integers(Z) with respect to addition. viii) The set of real numbers (R) with respect to multiplication 1.6 Definition: A semigroup S is called left (right) permutable if for every a,b,c in S, abc = acb (abc = bac). 1.7 Definition: A semigroup S is called permutable if it is both left and right permutable.

1.8 Definition: A semigroup S is called quasi- seperative if for any x,y∈ S, x2 = xy = y2 implies x = y.

Research Paper

(Oral Presentation) ----------------------------------------------------------------------------------------

Presenter: P. Srinivasulu Reddy


1.9 Definition: A semigroup S is called weakly seperative if x2 = xy = yx = y2 implies x = y for all x,y in S 1.10 Definition: A semigroup S is called seperative x2 = xy x2 = yx

if ⇒ x = y and ⇒ x = y y2 = yx y2 = xy

1.11 Definition: A semigroup S is said to be commuting regular if for any a, b∈S there

exist an element z∈S such that abzab = ba 1.12 Definition: An element a of a semigroup S is called an E-inversive if there is an

element x in S such that (ax)2 = ax i.e, ax∈ E(S). Where E(S) is set of all idempotent elements of S. 1.13 Definition: A semigroup S is called an E –inversive semigroup if every element of S is an E-inversive.

Examples: (i) Regular semigroups (a = axa implies that ax ∈ E(S)).

(ii) Eventually regular semigroups (an regular for some n >1 implies that an xan = an and

a(an-1x) ∈E(s) for some x ∈S). (iii) Semigroup S which contain idempotents and are totally ordered with respect to the

natural partial order a ≤ b iff a = xb = by, xa = a = ay for some x,y ∈ S1.(See Mitsch

(1986)) are E-inversive. Infact if a∈ S and a ≥ e for some e∈ E(S) then e = xa = ay and

ay∈ E(S), if a ≤ e ; then a = xe = ey ,xa = a implies that a2 = (xe)(ey) = x(ey) = xa = a.

Hence a.a = a ∈ E(S). 1.14 Definition: A semigroup (S, .) is said to be left(right) normal if abc = acb (abc =bac) for all a,b,c in S. 1.15 Definition: A semigroup (S, .) is said to be normal if it satisfies the identity abca = acba for all a,b,c in S. 1.16 Definition: A semigroup (S, .) is said to be left(right) quasi-normal if it satisfies the identity abc = acbc (abc = abac) for all a,b,c in S. 1.17 Definition: A semigroup (S, .) is said to be left (right) semi-normal if it satisfies the identity abca = acbca (abca =abcba) for all a,b,c in S. 1. 18 Definition: A semigroup (S, .) is said to be left(right) semi-regular if it satisfies the identity abca =abacabca (abca = abcabaca) for all a,b,c in S. 1.19 Definition: A semigroup S is called a (i) L-commutative if xab = xba. (ii) R-commutative if abx = bax. (iii) External commutative if axb = bxa

(iv) Conditional commutative if axb = bxa ⇒ ab = ba, for all a,b,x∈S.

1.20 Theorem: A regular semigroup S satisfying the identity abc = cb where a,b,c∈S, is one of the following: (i) commutative. (ii) left (right) regular. (iii) completely regular. (iv) commuting regular.

Proof: Let S be a regular semigroup satisfying the identity abc = cb, where a,b,c∈S.

Since S is regular and a, b∈S ⇒ there exists elements x, y∈S such that a = axa, b = byb Since S is regular,S is an E-inversive. i.e., (ax)2 = ax, (xa)2 = xa and (by)2 = by,(yb)2 = yb

(i) To prove that S is commutative, consider ab = cba for any a,b,c ∈S. ab = cba = (cb)a = ab(ca) = ab(dac) = (abd) ac = d(bac) = (dca) = ac = (bc)a = d(cba)

= dab = ba ⇒ ab = ba ⇒ S is commutative.


(ii) To prove that S is left regular. Since S is regular and a∈S implies

a = (ax)a = (xa)a ( by (i)). ∴ a = xa2. Hence, S is a left regular Similarly we can prove that S is right regular

(iii) Since S is regular, for any a∈S there exist x∈S such that a = axa, and by (i) ax = xa. Therefore, S is completely regular.

(iv) Let a, b ∈S implies there exist x, y ∈S such that a = axa, b = byb. Consider abyxab = (a)byxa (b) = axabyxabyb = a(xaby)2 b = a(xa)2(by)2 b

⇒ abyxab = a(xa)(by)b (Theorem1.20 ) = (axa)(byb) = ab hence abyxab = ba (by (i)) Therefore S is commuting regular. 1.21 Theorem : A regular semigroup S satisfying the identity abc = cb for any

a,b,c∈S, is quasi –seperative.

Proof : Let S be a regular semigroup satisfying the identity abc = cb for any a,b,c∈S.

To prove that S is quasi-seperative, we have to prove that for any a,b ∈S, if a2 = ab = b2 implies a = b.

Consider, a2 = ab ⇒ aa = ab ⇒ a (ax)a = a(byb) ⇒ a(cxa)a = (yb)b

⇒ acxa2 = (yb)b ⇒ aca = yb2

Since c∈S is regular there exist z∈S such that c = czc

aca = yb2 ⇒ a(czc)a = b ⇒ zcca = b (acz = zc) ⇒ zc2a = b ⇒ ca = b ⇒ caxa = b

(cax = xa) ⇒ xaa = b ⇒ xa2 = b ⇒ a = b (by Theorem.)

Similarly, let b2 = ba ⇒ bb = ba⇒ b(byb) = b(axa) ⇒ b(cyb)b = (bax)a (bax = xa, by

= cyb) ⇒ bcyb2 = xaa ⇒ bcb = a ⇒ b(czc)b = a ⇒ zccb = a ⇒ zc2b = a ⇒

cb = a ⇒ c(byb) = a ⇒ ybb = a ⇒ yb2 = a ⇒ b = a (Since S is left regular)

So a2 = ab = b2 ⇒ a = b Hence S is quasi-separative.

1.22 Theorem : A regular semigroup S satisfying the identity abc = cb for all a,b,c∈S is weakly separative.

Proof : Let S be a regular semigroup satisfying the identity abc = cb for all a,b,c∈S.

To prove that S is weakly seperative, we have to prove a2 = ab = ba =b2 ⇒ a = b.

From Theorem1.21, a2 = ab = b2 ⇒ a = b By theorem 1.21, S is commutative. Therefore, a2 = ab = ba = b2

⇒ a = b. Hence S is weakly separative

1.23 Theorem: A regular semigroup S satisfying the identity abc = cb for all a,b,c∈S is seperative.

Proof : : Let S be a regular semigroup satisfying the identity abc = cb for all a,b,c∈S To prove that S is seperative, we have to prove that, if a2 = ab and b2 = ba implies a = b and a2 = ba and b2 = ab implies a =b Let a2 = ab and b2 = ba. But by Theorem.1.21. S is commutative. Therefore a2 = ab = ba =

b2. Then by Theorem.1.22, S is weakly seperative. i.e., a2 = ab = ba =b2⇒ a = b. Hence

a2 = ab and b2 = ba ⇒ a = b. Similarly we can prove that a2 = ba and b2 = ab ⇒ a =b. Therefore S is seperative.

1.24 Theorem: A regular semigroup S satisfying the identity abc = cb for any a,b,c∈S is cancellative.


Proof: Let S be a regular semigroup satisfying the identity abc = cb for any a,b,c∈S.

Since S is a regular there exist x,y∈S such that a = axa and b= byb. To prove that S is

cancellative, consider ac = bc for any c∈S. Then axac = bybc aca = bcb ( Since xac = ca, ybc = cb) a(czc)a = b(czc)b zc2a = zc2b (Since Theorem 3.2.1) ca = cb caxa = cbyb xaa = ybb xa2 = yb2 a = b

ac = bc ⇒ a = b Hence, S is right cancellative Similarly, ca = cb c(axa) = c(byb) (cax)a = (cby) xaa = ybb (cax = xa, cby = yb) xa2 = yb2 a = b

ca = cb ⇒ a = b. Hence S is left cancellative. Therefore, S is cancellative 1.25 Theorem : Let S be a regular semigroup satisfying the identity abc = cb for all

a,b,c∈S then S is permutable.

Proof : Let S be a regular semigroup satisfying the identity abc = cb for any a,b,c∈S.

Since S is regular, for any a, b∈S there exist x, y ∈S such that a = axa and b= byb

Consider abc = (axa)bc for some c∈S = a(xab)b = abac = a(ba)c = abybac = (aby) (bac) = (yb) bac = ybbac =

yb2ac = bac ⇒ abc = bac ⇒ S is left permutable Similarly, let abc = axabc = ax(abc) = ax(cb) = (ax)cb = (ax)2 cb = a(xax)cb = a(xa)cb =

axacb = acb ⇒ abc = acb ⇒ S is right permutable. Therefore, S is permutable. 1.26 Theorem : Let S be a regular semigroup satisfying the identity abc = cb for all

a,b,c∈S, then S is medial.

Proof : From Theorem 1.25, S is permutable. i.e., abc = acb for all a,b,c∈S ⇒

abcd = acbd for any d∈S. Hence, S is medial .

1.27 Theorem : A regular semigroup S satisfying the identity abc = cb for any a,b,c∈S is conditional commutative semigroup.


Since S is regular, there exist x,y ∈S such that a = axa and b= byb Let ab = ba

We have to prove that azb = bza for some z∈S. Consider, ab = axab = a(xa)b = a(zax)b = a(zxa)b = a(zxa)b (by theorem 1.25)

= a(xaz)b = axazb = azb ⇒ ab = azb And ba = byba = b(yb)a = b(zby)a = b(zyb)a (by Theoerem 1.25)

= b(zyb)a = b(ybz)a = bybza = bza ⇒ ba = bza


Hence ab = ba ⇒ azb = bza. Therefore, S is conditional commutative semigroup.

1.28 Theorem : A regular semigroup S satisfying the identity abc = cb for all a,b,c∈S is L- commutative and R-commutative.

Proof: Let S be a regular semigroup satisfying the identity abc = cb for any a,b,c∈S. Let

a, b are elements of S then there exists elements x, y∈S such that a = axa, b = byb

Let bac = b(axa)c for any c∈S

= b(ax)ac = b(bxa)ac = b(bxa2)c = b(ba)c = (bba)c = abc ⇒ bac = abc Hence, S is L- commutative. Similarly we can prove that xba = xab and hence S is R- commutative. 1.29 Theorem : Let S be a regular semigroup satisfying the identity abc = cb for all

a,b,c∈S then S is external commutative semigroup.

Proof: Let S be a regular semigroup satisfying the identity abc = cb for all a,b,c∈S.

Let azb = (az)b for z∈S (By Theorem 1.25)

= abz = baz ⇒ azb = bza. Hence, S is external commutative

1.30 Theorem : A regular semigroup S satisfying the identity abc = cb for all a,b,c∈S is one of the following: (i) left (right) quasi-normal. (ii) left (right) semi-normal. (iii) left (right) semi-regular.


Since S is regular, there exist x,y,z ∈S such that a = axa, b= byb and c = czc (i) left quasi-normal: abc = axabc (Since a = axa) = a(xab)c = a(ba)c abc = abac S is left quasi-normal right quasi-normal: abc = ab(czc) = a(bcz)c =a(zcb)c (by theorem 1.29) = (azc)bc = (acz)bc (by theorem 1.25) = a(xac)zbc (Since a = axa) = a(ca)zbc = aca(zbc) = ac(cb) = ac(acb) abc = acbc S is right quasi-normal. (ii) left semi-normal: Since S is right quasi-normal we have abc = acbc abca = acbca S is left semi-normal right semi-normal: abc = a(byb)c = ab(ybc) = ab(cb) = abcb ⇒ abca = abcba


Hence S is right semi-normal (iii) Left semi-regular: Since S is left quasi-normal, abc = abac abc = abxa2c (By Theorem 1.20) = abxaac = abcaxac (xa = cax) = abcac = abacac (bca = baca) = abacbca (ac = bca) abc = abacacb (By Theorem 1.29) abca = abacaxacba = abaca(xac)ba = abaca(cab)a (xac = ca) =abacabaca (By Theorem 1.29) = abacaba(czc)a (c = czc) = abacab(acz)ca = abacabzcca (acz = zc) = abacabzc2a abca = abacabca (Since S is left regular). Similarly we see that S is a right semi-regular.

1.31 Theorem: A regular semigroup S satisfying the identity abc = cb for all a,b,c∈S is permutable if and only if it is left semi-normal.

Proof: Let S be a regular semigroup satisfying the identity abc = cb a,b,c∈S. Since S is

regular, there exist x,y,z ∈S such that a = axa, b = byb and c = czc Assume that S is a permutable semigroup then abc = bac = (ba)c (ba = xab) = (xa)bc = ba(xb)c (xa = bax) = bacbxc (xb = cbx) = (ba)cbxc = x(abc)bxc (ba = xab) = xbac bxc (permutable) = xbac(bxc) (bxc = xbc) = xbacxbc = xbacx(bc) (bc = acb) = (xba)cxacb (xba = ab) = abcxacb = a(bcx)acb (bcx = cbx ) abc = acbxacb abca = acb(xac)ba = acbcaba (xac = ca) = acb(cab)a = acb(acb)a (permutable) = ac(bbc)a (acb = bc) = acbcba (permutable)


= ac(bcb)a (bcb = bc) = acbca abca = acbca Hence S is left semi-normal Conversely, let S be left semi-normal, then we have to show that S is permutable To Prove that S is permutable, Consider abca = acbca ab(ca) = acb(ca) (ca = xac) abxac = acbxac abxac = a(cbx)ac (cbx = xb) ab(xac) = axbac (xac = ca) abca = (axb)ac (axb = bx) abca = b(xac) (xac = ca) abca = bca abca = (bc)a (bc = acb) abca = (ac)ba (ac = bca) abca = bc(aba) (aba = ab) abca = b(ca)b (ca = aac) abca = ba(acb) (acb = bc) abca = (bab)c (bab = ba) ab(ca) = bac (ca = aac) (aba)ac = bac (aba = ab) (aba)c = bac abc = bac Similarlly, we can prove that abc = acb. Therefore, S is permutable. Acknowledgements: The authors thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting them. Reference: [1] A.H.Clifford and G.B.Preston :”The algebraic theory of semigroups” Math.surveys7;vol .I Amer.math. soc 1961. [2] J.M.Howie “ An introduction to semigroup theory” Academic Press (1976). [3] Miyuki Yamada and Naoki Kimura “ Note on idempotent semigroups.II” Proc.Japan Acad. 34;110 (1958).

[4] Naoki.Kimura “The structure of idempotent semigroups(1) Proc.Japan.Acad., 33,(1957) P.642.

[5] Naoki.Kimura “Note on idempotent semigroups I”.Proc. Japan Acad.,33,642 (1957). [6] Naoki Kimura “Note on idempotent semigroups III”. Proc.Japan Acad.;34;113 (1958). [7 ] P.A. Grillet. “The Structure of regular Semigroups-.I”, Semigroup Forum 8 (1974), 177-183. [8 ] P.A. Grillet. “The Structure of regular Semigroups-.II”, Semigroup Forum 8 (1974), 254-259. [9 ] P.A. Grillet. “The Structure of regular Semigroups-.III”, Semigroup Forum 8 (1974), 260-265.


THE IDEAL GENERATED BY SETS CONTAINED IN THE NUCLEUS Authors: K.SUVARNA AND D.S.IRFANA Department of Mathematics, Sri KrishnaDevaraya University Anantapur-515055, A.P.,India. -------------------------------------------------------------------------------------------------------

ABSTRACT: Let R be a nonassociative ring, and let S be an additive subgroup of R such that S+SR=S+RS.In this paper, we prove that if S is contained in any one of the three nuclei then the ideal of R generated by S is nilpotent if and only if the subring generated by S is nilpotent KEYWORDS: Nucleus, Nonassociative ring, associator ideal, Skew derivation, Derivation and Semiprime ring.

1. INTRODUCTION: Yen and Hentzel[2] proved that is if S is contained in two of the three nuclei, then the ideal of R generated by S is nilpotent if and only if the subring generated by S is nilpotent.

In this paper,we prove that if S is contained in any one of the three nuclei, then the ideal of R generated by S is nilpotent if and only if the subring generated by S is nilpotent.

2. PRELIMINARIES: The associator ( a,b,c ) and commutator [a,b] are defined by (a,b,c)=(ab)c-a(bc), [a,b] = ab-ba. The set of all elements n in R such that (n, R,R)=0 is called a left nucleus of a ring R. The set of all elements n in R such that (R,R,n)=0 is called a right nucleus of a ring R. The set of all elements n in R such that (R,n,R)=0 is called a middle nucleus of a ring R. By t he nucleus of a ring R,we mean the set of all elements n in R such that (n, R,R) =(R ,n ,R)=(R ,R, n)=0.The associator ideal I of R is the smallest ideal which contains all associators in R.An additive mapping d from R to R is called a derivation on R if d(xy)=d(x)y+xd(y)for all x,y in Rand d from a ring R to R is called a Skew derivation or a s - derivation [1] if d(xy) = d(x)y + s(x)d(y) holds for all x, y in R, where s is an automorphism of R. If R has a s - derivation d, then d(R)+d(R)R = d(R) + Rd(R) since s is invertible.We know that R is semiprime if the only ideal of R which squares to zero is the zero ideal. Throughout this paper R represents a nonassociative ring. Let S be an additive subgroup of R such that S +SR=S+RS.Examples of S are [R,R],(R,R,R) and d(R),where

is a skew derivation of R.Let I=S+SR=S+RS.By assumption SR⊂ I and RS⊂ I. We now prove the following Lemmas: 3. MAIN RESULTS: We now prove the following Lemmas: Lemma 1: S+SR=S+RS is a two-sided ideal of R.

Proof : If S is in the left nucleus then S+SR is a right ideal, i.e. IR=SR+S.R2⊂ I.

If S is in the right nucleus,then S+SR is a left ideal,i.e.RI=RS+R2.S⊂ I. If S is in the middle nucleus then I+RI=I+IR because of

RI=R(S+SR)=RS+RS.R⊂I+IR and

IR=(S+RS)R=SR+R.SR⊂I+RI. If S is in both the left and right nuclei,then I is a right and left ideal,so I is an ideal.

Research Paper


Presenter: D. S. Irfana


If S is in the middle nucleus,then I+IR=I+RI.So if I is either a right ideal or a left ideal, then I is an ideal.Since S is either in the left or right nucleus, I is a right or a left ideal

and infact I is an ideal. ♦ Lemma 2: Sn +Sn R = Sn +RSn for all positive integers n.

Proof: ∞∑i=n S

i is an associative subring contained in the same one of the three

nuclei.So (Si,Sj,R)= (Si,R,Sj )=(R,Si,Sj)=0 for all integers i,j ≥ 1.By induction,it

is true for n=1.Assume the result for n.Then we get Sn+1R=SnS.R=Sn.SR⊂Sn(S+RS)

=Sn+1 +SnR.S⊂Sn+1 +(Sn+RSn)S=Sn+1 +RSn+1 and

RSn+1=R.SnS=RSn.S⊂(Sn+SnR)S=Sn+1+Sn.RS⊂ Sn+1 +Sn(S+SR)=Sn+1 +Sn+1 R.♦ Lemma 3: Sn + Sn R=Sn +RSn is the ideal of R generated by Sn for all positive integers n.

Proof: By using Lemma 2 and we replace S by Sn in Lemma 1.♦

Lemma 4: Si R.Sj R ⊂ Si+j + Si+j R for all integers i,j≥1. Proof: Case 1: If S is in the left nucleus then by Lemma 3,

Si R.Sj R= Si.R(Sj R) ⊂ Si (Sj +Sj R) = Si+j + Si+j R. Case 2: If S is in the middle nucleus then by Lemma 3,

Si R.Sj R ⊂ Si R.(Sj +RSj)=(Si R+Si R.R)Sj⊂(Si +Si R)Sj =Si+j +Si.RSj =Si+j +Si+j R.

Case 3: If S is in the right nucleus then by Lemma 3,

Si (RSj).R⊂Si(Sj +RSj).R=Si+j +Si (RSj)R=Si+j +Si(Sj +Sj R)=Si+j +Si+j R.♦

Lemma 5: SiR.Sj⊂Si+j + Si+j R for all integers i,j≥1. Proof: Case1: If S is in the left nucleus then by Lemma 3,

Si R.Sj=Si. .RSj ⊂Si(Sj + Sj R)=Si+j + Si+j R. Case 2: If S is in the middle nucleus, then by Lemma 3,

Si R.Sj ⊂ (Si +Sj R)Sj =Si+j +Si R.Sj =Si+j +Si.RSj =Si+j + Si+j R. Case 3: If S is in the right nucleus, then by Lemma 3,

Si(RSj) ⊂ Si(Sj +RSj)=Si+j +Si (RSj)=Si+j + Si+j R. ♦ Lemma 6: (S+SR)n =Sn +Sn R for all positive integers n. Proof: We assume that the result for all positive integers m ≤ n. Then using this inductive hypothesis and Lemma 4 and 5, for all positive integers i and j, i,j ≤ n we get (S+SR)i (S+SR)j =(Si +Si R)(Sj +Sj R)=Si+j +Si R.Sj +Si.Sj R+

SiR.Sj R=Si+j +Si R.Sj +Si+j R+Si R.SjR=Si+j + Si+j R. ♦ Hence we have proved the following theorem. Theorem 1: Let R be a ring and let S be an additive subgroup of R such that S+SR=S+RS. If S is contained in one of the three nuclei, then the ideal of R generated by S is nilpotent if and only if the subring generated by S is nilpotent. Moreover, if R is

semiprime and the subring generated by S is nilpotent, then S =0. ♦ Acknowledgements: The authors thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting them. 4. REFERENCES: [1] Kharchenko,V.K., Automorphisms and derivations of associative rings, Kluwer academic publishers,Dordrecht/ Boston / London , 1991. [2] Yen, C.T., and Hentzel, I.R., The ideal generated by sets contained in the nuclei, Soochow Journal of Mathematics, Vol.22, No.3,435-438,1996.


Fuzzy Homomorphism, Flags and Cosets of Incline Algebra

Authors: AR. Meenakshi and N. Jeyabalan, Department of Mathematics, Karpagam University, Coimbatore – 641 021, India. [email protected], [email protected]

Abstract: In this paper we have proved if a mapping µ from incline R into [0, 1] is a fuzzy homomorphism then the incline order relation is preserved and the crisp sets are

ideals as a subincline of R. We have provided sufficient conditions for µ to be a fuzzy homomorphism. Here we have used the concepts of chains, keychains, flags, pinned flags for ideals as a subincline in incline R. Further we deal with C(a), the coset of a with respect to addition for some a in R and proved that every element of C(a) is the 1-inverse of a. ---------------------------------------------------------------------------------------------------

Reduction of the Region of Ambiguity in Rough Sets under Fuzziness Authors: B. Krishnaveni and G. Ganesan Department of Mathematics, Adikavi Nannaya University, Rajahmundry, A.P Email: [email protected], [email protected]

Abstract: In 1982, Z. Pawlak introduced the concept of rough sets. This theory found applications in approximating the given information with respect to the available knowledge. In this theory, each information set is approximated into two namely, lower and upper approximations. The lower and upper approximations respectively refer the certainty and possibility measures of the given data. Here, the datum which appears in between the upper and lower approximations is said to belong to the region of ambiguity or boundary. Our earlier work concentrated on reduction of the region of ambiguity using the algebraic approach. In 2004, G. Ganesan et.al., analyzed the notion of introducing the thresholds in fuzzy inputs. In this paper, we used the concept of thresholds in reduction of the region of ambiguity for fuzzy inputs. Keywords: Rough Sets, lower and upper approximations, thresholds, fuzzy inputs.

Paper (Oral Presentation) _____________________________

Presenter: N. Jeyabalan

Paper (Oral Presentation) _________________________________

Presenter:B. Krishnaveni

,

TRUTH AND LOVE ARE TWO SIDES OF THE SAME COIN.

SPEAK TRUTH – DONOT LIE – ONE LIE LEADS TO OTHER.


PURE FUZZY SUBGROUPS Author: N.V. Ramana Murty Andhra Loyola College, Vijayawada. E-mail: [email protected]. Abstract: Pure subgroups were introduced by Prufer and later on they were

developed by J.M.Maranda, L.Fuchs[1] etc. Pure subgroups played an eminent role in the theory of Abelian groups. In recent decades this notion was also developed in Fuzzy Algebra. F.I. Sidky, and M.A.Mishref [2] introduced the notion of Pure subgroups in Fuzzy theory. Later on they were developed by J.N.Mordeson [3]. This paper makes and attempt to study Fuzzy Pure subgroups. Throughout this paper all groups are additive Abelan groups.

By a Fuzzy subset of X, we mean a function µ from the set X into [0,1]. We

denote that the set of all Fuzzy subsets of X by I(X). In a similar fashion, it is defined that

a Fuzzy subset µ of a group G is called a Fuzzy subgroup of G if

i) ( ) ( ) ( ) , ,x y x y x y Gµ µ µ+ ≥ ∧ ∀ ∈ and ii) ( ) ( ) .x x x Gµ µ− ≥ ∀ ∈

The set of all fuzzy subgroups of G is denoted by I(G). A Fuzzy point is denoted by ay

and is defined as ( ) ,ya x a for x y= = otherwise 0 if ,x y≠ where [0,1]a ∈ . In crisp

Abelian group theory, A subgroup H of a group G is said to be a pure subgroup of G if

.nG nH G n Z= ∀ ∈I We denote that ( ) ( ){ }| .I I Gµ ν ν µ= ∈ ⊆ In Fuzzy algebra, it is

defined that the Fuzzy subgroup ( )Iν µ∈ of G is said to be pure in a Fuzzy subgroup µ

of G if ax ν∀ ⊆ with 0, , , ( )a a aa n N y n y xµ> ∀ ∈ ∀ ⊆ = implies az ν∃ ⊆ such that

( ) .a an z x= Then we say that the Fuzzy subgroup ν of G is a Fuzzy Pure subgroup of a

Fuzzy subgroup µ of G. Towards the properties of Fuzzy Pure subgroups we have the

following results:

Lemma 1: If ( ) ,Iν µ∈ then ν is pure in µ if and only if aν is pure in

( ){ }| 0 0 ,a

a a aµ ν∀ ∈ < ≤ where ( ){ }|a

x G x aµ µ= ∈ ≥ , a − level set of µ and

[ ]0,1 .a ∈

Lemma 2: If ( )I Gµ ∈ and ,n N∈ then i) ( )(0) 0 ;nµ µ= ii) ;nµ µ⊆

iii) nµ is a Fuzzy subgroup of G; iv) If µ has the supremum property, then

( ) ( );n G Gµ µ⊆

Similarly it has been discussed that under what conditions the Divisible Fuzzy subgroups become pure and the transitive property of Fuzzy Pure subgroups holds. References [1] Fuchs, L. “Infinite Abelian Groups”, Vol.I &II, Academic Press, New York 1970. [2] Sidky, F.I. and Mishref, M.A., “Divisible and Pure Fuzzy subgroups” Fuzzy Sets and Systems, Vol.34, 377-382, 1990. [3] Mordeson, J.N. “Fuzzy Commutative Algebra” World Scientific Company, Singapore, 1998.

Paper (Oral Presentation) ----------------------------------------------------------------------------

Presenter:

Dr N.V. Ramana Murty


Fuzzy ideals of Seminearrings Abstract: In this Paper, the algebraic system Seminearring is considered which is

generalization of both a semiring and a nearring. The algebraic systems with binary operations of addition and multiplication satisfying all the ring axioms except possibly one of the distributive laws and commutativity of addition are called “Nearrings”. A semiring is an algebraic system which is closed and associative under two operations, usual addition, multiplication, and satisfies both distributive laws. A seminearring S is an algebraic system with two binary operations: usual addition and usual multiplication such that S forms a semigroup with respect to both the operations, and satisfies the right distributive law. A natural example of a seminearring is obtained by considering the operations usual addition and composition of mappings on a set of all mappings of an additive semigroup S into itself.

We consider the s-ideal (left, right) of a seminearring defined by Javed Ahsan, Weinert and provide examples. If S is a nearring with 1 instead of a seminearring, then these s-ideals are just the S-subgroups of a nearring. We consider the fuzzy set as a seminearring S. The operations defined on a fuzzy set of S are similar to the fuzzy ideals defined in nearrings, rings and semirings. For a given s-ideal of a seminearring, the existence of its fuzzy s-ideal is obtained. These s-ideals play a vital role in developing new substructures in seminearrings and other similar generalized algebraic systems.

References

[1].Golan J. S. “The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science”, Longman Scientific and Technical Publishers, 1992. [2].Javed Ahsan. “Seminear-rings Characterized by their s-ideals I”, Proceedings of

Japan Academy, Series A, 101–103, 1995. [3]. Javed Ahsan. “Seminear-rings Characterized by their s-ideals II”, Proceedings of

Japan Academy, Series A, 111–113, 1995. [4].Kim S.D and Kim H.S. “On Fuzzy ideals of Nearrings”, Bulletin of Korean

Mathematical Society, vol.33, 593–601, 1996. [5].Pilz G. “Near-Rings: The theory and its Applications”, North-Holland Publishing

Company, 1983. [6].Sung Min Hong, Jun and Kim. “Fuzzy ideals in Nearrings”, Bulletin of Korean

Mathematical Society, 455–464, 1998. [7].Van Hoorn, Willy. G and Van Rootoselaar.B. “Fundamental notions in the theory

of seminear-rings”, Composition Math. , 18, 65–78, 1966. [8].Weinert H.J and Hebisch.U. “Semirings- Algebraic theory and applications in

Computer Science”, World Scientific Publishing Company Ltd., 1998. [9].Zadeh L.A. “Fuzzy Sets”, Information and Control, 338–353, 1965.

Paper (Oral Presentation)

P.Venu Gopala Rao, Andhra Loyola College (Autonomous),

Vijayawada- 520 008. E-mail: [email protected],

(ABOUT MAHATMA GANDHI): GENERATIONS TO COME, IT MAY BE, WILL

SCARCELY BELIEVE THAT SUCH A ONE AS THIS EVER IN FLESH AND

BLOOD WALKED UPON THIS EARTH …. ALBERT EINSTEIN



ON LOWER DIFFERENCE GRAPHS Authors: I.H.N. Rao1 and K.V.S. Sarma 2 1Director & Sr. Prof., G.V.P. College for P.G. Courses, Visakhapatnam. 2Asst. Prof. (Sr.Grade), Regency Institute of Technology, Yanam.

ABSTRACT: The concept of “Lower Difference Graphs” is introduced and some useful properties of these graphs are obtained. Further a study about the stability, nature of these graphs is made and observed a significant application of these graphs in the design of networks. The network designing with effectiveness mainly depends upon the stability of its nodes and links. In due course of time, the network begins to use its links or nodes or both. So it is desirable to construct the networks as stable as possible with regard to the destruction. The integrity of a graph is appropriate measure to deal this problem and this parameter is evaluated for the lower difference graphs.

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On Semi-Projective Modules Author: Manoj Kumar Patel (Research Scholar), Dep’t. Applied Mathematics, IT-BHU, Varanasi-221005, UP (India).

Abstract: In this paper, we have studied the properties of quasi-principally projective

modules related with generalized Hopfian and variants of supplemented modules. We have introduced the idea of generalized hollow module which is a generalization of hollow module and also studied some properties related to it. -----------------------------------------------------------------------------------------------------------

A note on finite injective modules Author: Varun kumar Department of Applied Mathematics Institute of Technology Banaras Hindu University Varanasi-221005.

Abstract: In this paper, we generalize the idea of finitely injective modules to small finitely injective modules and study some properties of finitely injective and small finitely injective modules. We prove that for a hollow module M every quasi-sf-injective module to be quasi-f-injective. Also study (FC2) and (FC3) conditions for quasi-f-injective modules.




Presenter: Manoj Kumar Patel


Presenter: Varun Kumar


11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) 118

1

Regular Delta Near Rings Authors: Dr. T.V. Pradeep Kumar and N.V. Nagendram Departnemnt of Mathematics, ANU College of Engineering Nagarjuna Nagar, A.P Lakireddy Balirddy Collee of Engineering Mylavaram. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

In recent decades interest has arisen in algebraic systems with binary operations addition and multiplication satisfying all the ring axioms except possibly one of the distributive laws and commutativity of addition. Such systems are called “Near-rings”. A natural example of a near-ring is given by the set M(G) of all mappings of an additive group G (not necessarily abelian) into itself with addition and multiplication defined by (f + g)(a) =

f(a) + g(a); and (fg)(a) = f(g(a)) for all f, g ∈ M(G) and a ∈ G. Modern Algebra presently, the basis for developing several new areas of technology like Digital Computing, Data Communication, Sequential Machines, Computer Systems and Radar Solar Systems and finite automata. In this paper I would like to explain about regular delta near rings. The concept of Noetherian near-rings and Noetherian δ–Near Rings was studied by by S. Ligh [ 8 ] , Y.V.Reddy , C.V.L.N.Murthy.[ 10 ] and some others. Later N V Nagendram,T V Pradeep Kumar and Y V Reddy [5,6] were studied regular delta near rings and obtained some results related to Noetherian regular delta near rings and their extensions, p-regular delta near rings and their extensions.

Acknowledgements: I am very much thankful to my college Management , Director and also Organizers (Dr A V Vijaya Kumari, Organizing Secretary, and Prof. Dr Bhavanari Satyanarayana, Academic Secretary) of the National Seminar on “Present trends in Algebra and its applications” 11 th and 12th of July 2011 at JMJ College for Women, TENALI.

References [ 1 ] A. Badawi, On pseudo-almost valuation rings, Comm. Algebra, 35 (2007), 1167-1181. [ 2 ] G.Pilz “ Near rings” North Holland, New York, 1983 [ 3 ] K. R. Goodearl and R. B. Warfield Jr, An introduction to non-commutative Noetherian rings, Cambridge University Press, 1989. [ 4 ] K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979. [ 5 ] N V Nagendram,T V Pradeep Kumar and Y V Reddy “On Noetherian Regular δ -Near Rings and Their Extensions”,IJCMS5-8-2011,Vol.6,No.6,PP.255-262. [ 6 ] N V Nagendram,T V Pradeep Kumar and Y V Reddy “On p-Regular δ -Near Rings and Their Extensions”, accepted by IJCM, @ Mindreader Publications. [ 7 ] S J Choi “P-Regularity of a Near-Ring “, M.Sc., thesis, university of Dong-A, 1991. [ 8 ] S.Ligh j. austral .math soc. 2 (1972) 141-146. [ 9 ] SU – Jeong Choi “Quasi ideals of a P-Regular Near-Rings” Intl. J. of Algebra Vol.4,2010 No.11 , 501 – 506. [10] Y.V. Reddy And C.V.L.N Murthy ‘On Strongly Regular Near-Rings’ Proc. Edinburgh

Math.Soc.27(1984),62-64.

Paper (oral Presentation) -----------------------------------------------------------------------------------------------------

Presenter: N V Nagendram

Secret of Success is concentration.



FLOW OF A JEFFREY FLUID THROUGH AN ARTERY WITH MULTIPLE STENOSES Authors: S.Sreenadh, A.RagaPallavi, T.Savitha and CH.Badari Narayana. Department of Mathematics, Sri Venkateswara University, Tirupati ---------------------------------------------------------------------------------------------------------------------

Abstract: A mathematical model is developed to study the steady flow of Jeffrey fluid through a tube of non-uniform cross section with multiple stenoses. Using appropriate boundary conditions, analytical expressions for the velocity and the volumetric flow rate have been derived. These expressions are computed numerically and the computational results are analyzed graphically. --------------------------------------------------------------------------------------------------------------------

PERISTALTIC TRANSPORT OF

A POWER-LAW FLUID IN

CONTACT WITH A NEWTONIAL

FLUID IN AN INCLINED

POROUS CHANNEL Authors: D.Venkateswarlu Naidu *, S.Sreenadh* and Vishwamohan** *Department of Mathematics, Sri Venkateswara University, Tirupati, ** Assistant Professor, M.L.I.E.T, Nellore.

Abstract: Peristaltic transport of a power-law fluid surrounded by a peripheral layer of a Newtonian fluid in an inclined porous channel is studied under long wavelength and low Reynolds number assumptions. The flow is examined in a wave frame of reference moving with the velocity of the wave. The expressions for stream function, the velocity and the pressure rise are obtained. The equation for the interface separating the two fluids is obtained. Numerical results are reported for various physical parameters of interest. --------------------------------------------------------------------------------------------------------------------------------

Peristaltic transport of Power-law fluid in contact with a Jeffrey fluid in a channel with permeable walls Authors: A.Parandama1, S.Sreenadh1, A.N.S.Srinivas2

Department of Mathematics, Sri Venkateswara University, Tirupati. Department of Mathematics, Annamacharya Institute of Technology and Sciences, Tirupati ABSTRACT: Peristaltic pumping by a sinusoidal traveling wave in the permeable walls of a

two dimensional channel filled with two immiscible fluids is investigated. The core region of the

fluid is occupied by a Power-law fluid where the peripheral region is occupied by a Jeffrey fluid.

The flow is examined in a wave frame of reference moving with the velocity of the wave. The

expressions for the stream function, the velocity and the pressure rise are obtained. The equation

for the interface separating the two fluids is obtained. Numerical results are reported for various

of the physical parameters of interest.


Presenter: Ch. Badari Narayana


_____________________________ Presenter: D. Venkateswarlu Naidu


Presenter: A. Parandama



NON-LINEAR ANALYSIS OF POISEULLIE FLOW OF A JEFFREY FLUID BETWEEN TWO PARALLEL PLATES Authors: S.Sreenadh, R. Madhan Kumar, P.Devaki and E.Sudhakara Department of Mathematics, Sri Venkateswara University, Tirupati. --------------------------------------------------------------------------------------------------------------------- Abstract : The Poiseuille flow of a Jeffrey fluid in a channel bounded by two parallel plates is analyzed. The flow problem is described by means of partial differential equations and the solutions are obtained by an implicit finite technique. The axial and transverse velocities are obtained and their behavior is discussed computationally for different values of Jeffrey parameter at different pressures. ----------------------------------------------------------------------------------------------------------------

EFFECTS OF INDUCED

MAGNETIC FIELD ON

PERISTALTIC FLOW OF

A FOURTH GRADE FLUID IN

AN INCLINED PLANAR

CHANNEL FILLED WITH POROUS MATERIAL Authors: P. Hari Prabhakaran, S. Sreenadh and R. Saravana Department of Mathematics, S.V University, Tirupati, A.P, India.

ABSTRACT: In this paper, we study the effect of induced magnetic field on the peristaltic transport of fourth grade fluid in an inclined symmetric channel filled with porous material under the long wavelength and low Reynolds number assumptions. The flow is analyzed using a perturbation expansion in terms of a variant of the Deborah number. The expressions for the velocity, axial pressure gradient, pressure rise and frictional force over one cycle of wavelength are obtained. The effects of various emerging parameters on pumping characteristics and frictional forces are discussed through graphs. ----------------------------------------------------------------------------------------------------------------

GRAPH THEORY AND ITS INFLUENCE IN VARIOUS FIELDS OF KNOWLEDGE Author: M. Arokiasamy, Andhra Loyola College, Vijayawada, Andhra Pradesh, INDIA. E-MAIL: [email protected]

Abstract: Graphs are among the most ubiquitous models of both natural and human-made

structures. The number of concepts that can be defined on graphs is very large. In particular, many real-world problems of practical interest have been successfully modeled on graphs. Different graph models have been proposed for image analysis, depending on the structures to analyze. Image processing and analysis with graphs is becoming essential for the development of cutting-edge research and applications. Consequently the number of applications based on digital images has drastically increased from multimedia, computer


_____________________________ Presenter: E.Sudhakara


_____________________________ Presenter: R. Saravana

Paper (Oral Presentation) -----------------------------------------------------------------------------------------

PRESENTER: M. AROKIASAMY,



animation, video games, communication and digital arts, medicine, biometry, digital photographs to the medical scans, including satellite images and video films etc.

Graph theory is one of the branches of Mathematics which finds applications in different fields. The recent interest in graph theory has resulted in the emergence of new branches of study like Graph Engineering and Quantum Graph Theory. The graph theoretical problem solving could be followed by web search engines like Google. For example, it is said that the diameter of the World Wide Web (www) is 19. On the web we get from one place to place (pages to pages) by clicking on hypertext links. That is, we travel through such links which are the steps we take in web browsing. Number of researchers had made significant contributions in the area like kinematics chains and mechanisms, tribology, failure analysis, quality, reliability, automobile vehicle design, reinforced polymer composites, electroplating, mechatronic products, thermal power plant, manufacturing systems, total quality management, etc. It is hoped that graph theory will motivate researchers to apply graph theory more effectively in a number of new research areas. In this paper attempt is made to describe some of the topics in graph theory and discuss its applications in various fields. ----------------------------------------------------------------------------------------------------------------

Ideals and Direct Products

of Zero Square Near-Rings

Authors:1Bhavanari Satyanarayana, 2Godloza Lungisile, 3 Munagala Babu Prasad and 4Kuncham Syam Prasad. --------------------------------------------------------------------------------------------------------------------- Abstract: We consider associative near-ring N (not necessarily commutative). In this paper the concepts: zero square near-ring of type-1/type-2, zero square ideal of type-1/type-2, zero square dimension of a near-ring N were introduced and obtained several important results. Finally, some relations between the zero square dimension of the direct sum of finite number of near-rings; and the sum of the zero square dimension of individual near-rings; were obtained. Necessary examples were provided. Key words: zero square near-ring, zero square ideal, direct sum, zero square dimension, uniform ideal, essential ideal.

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting him

References

1. Patricia Jones ‘Zero square Near-rings’, J. Austral Math. Soc. (A) 51 (1991) 497-504. 2. Pilz G “Near-rings”, North Holland, 1983. 3. Ramakotaiah D., ‘Structure of 1-primitive Near-rings’, Math. Z, 110 (1969)15-26. 4. Satyanarayana Bh., “Contributions to Near-Ring Theory”, Ph.D., Thesis, Nagarjuna

Univ., 1984. 5. Satyanarayana Bh., Godloza L, and Vijayakumari A.V., ‘Finite Dimension in Near-

rings’ J. Andhra Pradesh Society for Mathematical Sciences Vol. 1, No. 2(2008) 62-80.

6. Satyanarayan Bh., Godloza L., and Vijayakumari A.V. ‘Some Dimension Conditions in Near-rings with finite Dimension’, Acta Ciencia Indica 34 M (2008) 1397-1404.


Presenter: Munagala Babu Prasad



7. Satyanarayan Bh. and Syam Prasad K., ‘A Result on E-direct Systems in N-Groups’

Indian J. Pure & Appl. Math., 29(3): 285-287, 1998.

8. Satyanarayan Bh. and Syam Prasad K., ‘On Direct and Inverse Systems in N-Groups’ Indian Journal of Mathematics, 42 (2) 183-192, 2000.

9. Satyanarayan Bh. and Syam Prasad K., ‘Linearly Independent Elements in N-Groups with Finite Goldie Dimension’ Bull. Korean Mathematical Society, 42 (3) 433-441, 2005.

10. Satyanarayan Bh. and Syam Prasad K ‘Discrete Mathematics and Graph Theory’ Prentice Hall India Learning Private Limited, 2009.

11. Syam Prasad K., “Contributions to Near-ring Theory II”, Doctoral Thesis, Acharya Nagarjuna University, 2000.

12. Vijayakumari A.V., “Contributions to Near-ring Theory IV” Doctoral Thesis, Acharya Nagarjuna University, 2009.

13 Bhavanari Satyanarayana, Godloza Lungisile, Munagala Babu Prasad, and Kuncham Syam Prasad “ Ideals and Direct Products of Zero Square Near- Rings”, International J. Algebra, Vol.4 (No. 16) (2010) PP 777-789.

----------------------------------------------------------------------------------------------------------------

PSEUDO SYMMETRIC IDEALS OF A SEMIGROUP

Author: Dr. A. Anjaneyulu, Reader & Head, V S R & N V R College, Tenali.

ABSTRACT: In this paper, the terms, ‘pseudo symmetric ideal’ of a semigroup and ‘pseudo symmetric semigroup’ are introduced. It is proved that every completely semiprime ideal of a semigroup is a pseudo symmetric ideal. It is also proved that every prime ideal P minimal relative to containing a pseudo symmetric ideal A in a semigroup is completely prime. Further it is proved that an ideal A of a semigroup is (1) completely prime iff A is prime and pseudo symmetric (2) completely semiprime iff A is semiprime and pseudo symmetric. If A is a pseudo symmetric ideal of a semigroup then it is proved that A1 = A2 = A3 = A4. It is also proved that every left duo semigroup, right duo semigroup, duo semigroup, left pseudo commutative semigroup, right pseudo commutative semigroup, quasi commutative semigroup, normal semigroup, idempotent semigroup, semigroup in which every element is a mid unit are all pseudo symmetric semigroups. REFERENCES: 1. ANJANEYULU A. and RAMAKOTAIAH D., On a class of semigroups – Simon Stevin, Vol.54 (1980) 241-249. 2. ANJANEYULU A., Semigroups in which prime ideals are maximal – Semigroup Form,

Vol.22 (1981) 151-158. 3. CLIFFORD A. H. and PRESTON G. B., The algebraic theory of semigroups – Vol-I,

American Mathematical Society, Providence (1961). 4. CLIFFORD A. H. and PRESTON G. B., The algebraic theory of semigroups – Vol-II,

American Mathematical Society, Providence (1967). 5. LJAPIN E. S., Semigroups, American Mathematical Society, Providence, Rhode Island

(1974). 6. PETRICH. M., Introduction to semigroups - Merril Publishing Company, Columbus,

Ohio, (1973)


Presenter: Dr. A. Anjaneyulu,


12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 123-125 Page. 123

GOLDEN RATIO AND HUMAN BODY

Author: Satyasri Bhavanari Zhejiang University, Hangzhou, Republic of China

In mathematics and the arts, we say that two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

It is also known as the divine proportion.

The golden ratio is an irrational mathematical constant, approximately 1.6180339887. The

golden ratio is denoted by the Greek lowercase letter phi ( ) , while its reciprocal, or ,

is denoted by the uppercase variant Phi( ).

Section-1: Some Natural Examples:

1.1. GOLDEN RECTANGLE: Suppose the rectangle is divided into a square and a smaller rectangle. In a golden rectangle, the smaller rectangle is the same shape as the larger rectangle, in other words, their sides are proportional. In further words, the two rectangles are similar. This can be used as the definition of a golden rectangle. The proportions give us:

a/b = (a+b)/a

Survey Article ----------------------------------------------------

Presenter: Satyasri Bhavanari MBBS IV Yr,

Zhejiang University, Hangzhou,

Republic of China

“VIRTUE IS THE KNOWLEDGE OF GOODNESS”

“SIN IS THE IGNORANCE OF GOODNESS”.



1.2 Many buildings and works of art have the Golden Ratio in them,

such as the Parthenon in Greece.

1.2. FIBONACCI NUMBERS: There is a close relationship between the golden ratio and the fibonacci numbers. The Fibonacci Sequence is the series of numbers. Discovered by Leonardo Fibonacci. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The characteristic of these numbers is each number is formed by the sum of preceding two numbers. The Rule is xn = xn-1 + xn-2.

If you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio. In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation.

1.3. Golden ratio is exhibited by Egypt pyramids, Leonardo DaVinci’s portrait of Monalisa, Sunflower, the snail. A book ‘Universal Principles of design’ also approximates the golden ratio when it is opened.

1.4. Honeybees: A honeybee colony consists of a queen, a few drones and lots of workers. The female bees (queens and workers) all have two parents, a drone and a queen. Drones, on the other hand, hatch from unfertilized eggs. This means they have only one parent. Therefore, Fibonacci numbers express a drone's family tree in that he has one parent, two grandparents, three great-grandparents and so forth.

1.5. The APPLE company’s IPod used the golden ratio.

Section-2: GOLDEN RATIO IN HUMAN BODY Most of our human body parts follow the numbers one, two, three and five. A human being

has one nose, two eyes, three segments to each limb and five fingers on each hand.

2.1. Our fingers have three sections. The proportion of the first two to the full length of the finger gives the golden ratio (with the exception of the thumbs).

2.2. We can also see that the proportion of the middle finger to the little finger is also a golden ratio.



2.3. We have two hands, and the fingers on them consist of three sections. There are five fingers on each hand, and only eight of these are articulated according to the golden number: 2, 3, 5, and 8 fit the Fibonacci numbers.

2.4. The DNA molecule in which all the physical features of living beings are stored, consists of two intertwined perpendicular helices. The length of the curve in each of the helices is 34Amstrong and the Width is 21 ang. 1 angstrom= 100millionth of a centimeter. 21 and 34 are two consecutive fibronacci numbers. Golden ratio applies to idealized human body which scientists and artists agree. The proportions and measurements of the human body can also be divided up in terms of the golden ratio.

2.5. The important example of the golden ratio in the average human body is that when the distance between the navel and the foot is taken as 1 unit, the height of a human being is equivalent to 1.618.

2.6. Some other golden proportions in the average human body are: (i). (The distance between the finger tip and the elbow) / (distance between the wrist and the elbow). (ii). (The distance between the shoulder line and the top of the head) / (head length). (iii). (The distance between the navel and the top of the head) / (the distance between the shoulder line and the top of the head). (iv). (The distance between the navel and knee) / (distance between the knee and the end of the foot). (v). (The total width of the two front teeth on the upper jaw)/(by their length). (vi). (The width of the first tooth and the second tooth of the upper jaw) /(the width of the first tooth of the upper jaw). (vii). (The length of face)/(the width of face). (viii). (The distance between the lips and where the eyebrows meet)/(the length of nose).

(ix). (The length of face)/(the distance between the jaw and where the eyebrows meet). (x). (The length of mouth)/(the width of nose).

(xi). (The distance between the eyes)/(the distance between eyebrows).

Not only the outer parts of the body, but some of the inner structures of the body also follow the golden ratio.

2.7. Example: In a study carried out between 1985 and 1987, the American physicist Bruce West and professor of medicine Aure Gold Burger revealed the existence of golden ratio in the lung. One feature of bronchus that constitute the lung, is that they are not of equal length. The windpipe divides into two unequal bronchi, one long on the left and the other short on the right. It was determined that the proportion of the long bronchus to the short bronchus was 1.618.

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and

Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting

her.

References: Some websites in the net through google search engine.

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 126-127 Page. 126

Shortest Path Problem:- (An Application of Graph Theory) Author: Davuluri Nagamani Assistant Professor, Department of I.T, Sri Chundi Ranganayakulu Engg., College, Ganapavaram, Guntur. -----------------------------------------------------------------------------------------------------------

Let us consider a situation where we need to determine the minimum distance between two place while we travel from one place to another. Here the distance between the two locations is assigned as weights to the edges. Illustration 1: Consider a situation where we want to travel from Chennai to Newyork. We can travel from Chennai to Newyork through London at a cost of “$500” and through Singapore at a cost of “$ 2500”. To travel from Chennai to Newyork we usually want to travel through London, because the “cost of travel is less”. Illustration 2: Consider a graph in which S is the source vertex and D is the destination vertex. The sum of weights of the edges is called the length of the path. The minimum length from S to D determines the shortest Path between the source vertex S and the destination vertex D. A Simple Problem (Model): Consider a weighted graph. The vertices A, B, C, D, E, F, G, H are places/towns/cities and the edges how the Bus/train roots available between the cities. The number on the edges denotes the distance/fare. In the graph Fig-1 A is the source vertex and H is the destination vertex. How to find the shortest path from source and destination? Solution: First we need to determine all the paths that exist between the two nodes A and H. They are (i) Fig-2 represents a path A to H The cost of traveling from A to H on this path is 140 (10+30+40+60).

E

H

20

B

A

G

D

F

C

20 20

30 30

30 40

40

50

60

10

10

Fig-1

H

A

F

C

30

40

60

10 Fig-2


Presenter: D. Nagamani

Proceedings of the National Seminar on ALGEBRA AND ITS APPLICATIONS, JMJ College, Tenali, A.P., (July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 126-127 Page. 127

(ii) Observe that fig-3 and fig-4 represent two other paths available from A to H. The cost of traveling from A to H on the path given is 130(20+20+30+60). The cost of traveling from A to H on the path given in Fig- 4 is 50(20+20+10). Among all the paths, the path given in fig -4 is shortest because the cost 50 is less among all the traveling costs of the paths. Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting her. Conclusion: We discussed how we can solve shortest root problem by using graph theory.

E

H

20

B

A

F

20 30

60 Fig-3

E

H

20

B

A

20

10

Fig-4

If Hard Work is your Weapon Success becomes your Slave.

Secret of Success is concentration. Concentration is essential for every person. Concentration is achieved by continuous striving just as Yogi.

Peace is inner silence filled with the power of Truth.

Let us have noble thoughts from every corner


(July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 128-

134 - 128 -

CONGRUENT RELATED GRAPHS

§ 0. INTRODUCTION:

In artificial intelligence and information organization & retrieval to maintain secrecy of information some coding has to be done. In defence problems in order to keep secrecy confidential, we exhibit relationship between pairs of soldiers. Now, we associate a graph and define adjacency suitably. The type of graph introduced here is very useful in the above areas. The elementary study of this concept is being done by Saradhi, S.V. University under the guidance of Prof. Vanjipuram. This graph is named as congruent related graph. The extended work forms a part of the dissertation work of Mr. K.V.S. Sarma under my guidance.

§ 1.CONCEPTS AND RESULTS:

1.1DEFINITION. Let m, n be positive integers and s be a non negative integer ≤ (m – 1). Then the simple graph G with vertex set

V = V(G) = {1, …, n} and the edge set E = E(G) ={{u,v}: where u, v ∈ V with v ≠ u and u + v ≡ s (mod m)} is called a congruent related graph ( and is denoted by G m , n

(s)). 1.2 OBSERVATIONS.

(i)When n is fixed, for each positive integer m, we get m graphs Gm , n(s)

( s = 0, ... , m -1). (ii) When m = 1, for each positive integer n, we get one and only one graph G1 ,n

(0) = Kn ( the complete graph on n vertices) since for any two vertices u, v, u+v ≡ 0 (mod 1) and thus any two vertices are adjacent. (iii) When n = 1, for any positive integer m, Gm , 1

(s) = K1 for s = 0, ..., m – 1, since there is only one vertex and hence no edges in these graphs. Hence we take m ≥ 2 and n ≥ 2. (iv) When m = 2, we get two graphs namely G2 , n

(0) ,G2 , n(1).

1.3 THEOREM:

Kn/2 + Kn/2 if n is even (n ≥2), G2 , n

(0) = K(n+1)/2 + K(n-1)/2 if n is odd (n≥3).

INVITED TALK __________________________________________________

Prof. I.H. NAGA RAJA RAO. Director & Sr. Prof., G.V.P. College for

P.G. Courses, Visakhapatnam



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-

1.4 THEOREM. Kn/2 , n/2 if n is even (n ≥ 2), G2 , n

(1) = K(n+1)/2 ,(n-1)/2 if n is odd (n ≥ 3).

1.5 OBSERVATIONS (i). From Th.(1.3), the number of edges of G2, n(0)

n/2(n/2 – 1) = n(n – 2)/4 when n is even

i.e. ∈( G2, n(0)) =

½ {(n+1)/2}(n – 1)/2 + ½ {(n – 1)/2}(n – 3)/2 = 1/4(n – 1)2

when n is odd

(ii) From Th.(1.4), it follows that

(n/2)(n/2) = n2/4 if n is even,

i.e. ∈( G2, n(1)) =

{(n+1)/2}(n – 1)/2 = (n2 – 1)/4 if n is odd

1.6 THEOREM. For the (simple) graph G m , n(s) (m, n ≥ 3 and s = 0, 1, 2,

…, (m-1)) with vertex set V = V(G) = {1, 2, 3, …, n } and with

Vi = {v ∈ V: v ≡ i(mod m)} ( i=0, 1, …, m-1) the number of edges ε (G m , n

(s)) of G m , n(s) is

(s/2) – 1 (m/2)-1 (i) ∑ {|Vi| |Vs – i |+|Vs/2| (|Vs/2| –

1)/2 + ∑ |Vs/2 + i | |Vm + s/2 - i | +|V(m+s)/2|(|V(m+s)/2)-1)/2 i = 0 i=(s/2)+1 if both m and s are even; (s-1)/2 (m/2)-1 (ii) ∑|Vi| |Vs – i |+∑ |V(s+1)/2 + i | |Vm + (s-1)/2 - i | i = 0 i=(s+1)/2 if m is even and s is odd; (s/2) – 1 (m-1)/2 (iii) ∑ {|Vi| |Vs – i |+|Vs/2| (|Vs/2| –

1)/2 +∑ |Vs/2 + i | |Vm + s/2 - i | i = 0 i=(s/2)+1 if m is odd and s is even and



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(s-1)/2 (m-3)/2 (iv) ∑ |Vi| |Vs – i |+∑ |V(s+1)/2 + i | |Vm + (s-1)/2 - i |+|V(m+s)/2|(|V(m+s)/2|-1)| /2 i = 0 i=(s+1)/2 if both m and s are odd β where ∑… = 0 if β < α. i=α PROOF: We consider (a) both m and s are even (⇒ m ≥ 4 and s = 0, 2, …, m-2). We can write (s/2)-1 (m/2)-1 V = {∪ (Vi ∪ Vs – i)} ∪ Vs/2 ∪{ ∪ (V(s/2) + i , Vm +(s/2) – i )} ∪ V(m+s)/2 , i = 0 i=(s/2)+1 β where ∪ … stands for empty set if β < α. i = α The decomposition of V is such that any two elements (vertices) of Vs/2 or V(m+s)/2 are adjacent (in G), any element of Vi is adjacent with every element of Vs – i ( i = 0, …, (s/2) – 1) and any element of V(s/2) + i is adjacent with every element of Vm + (s/2) – i ( i = (s/2) + 1, …, (m/2) – 1). So we can write (s/2)-1 (m/2)-1 V = {∪ (Vi ∪ Vs – i)} ∪ Vs/2 ∪{ ∪ (V(s/2) + i ∪ Vm +(s/2) – i )} ∪ V(m+s)/2 , i = 0 i=(s/2)+1 (s/2) – 1 (m/2) – 1 Hence Gm , n

(s) = { ∪ K|Vi| , |Vs – i |}∪ K|Vs/2|∪{( ∪ K|V (s/2) + i | , |Vm + (s/2) – i|)} ∪ K|V (m + s)/2 | i = 0 i =(s/2)+1

β

with the convention ∪ … is empty graph when β < α. i = α

Since ε(Kν) = ν (ν - 1)/2 and ε(Ka,b) = ab, (i) follows.

(b) m is even and s is odd (⇒ m ≥ 4 and s = 1, 3, …, m-1). We can write (s-1)/2 (m/2)-1

V = { ∪ (Vi ∪Vs – i )} ∪ {∪ (V(s+1)/2 + i ,Vm+(s-1)/2 – i)}.

i = 0 i=(s+1)/2



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-

Hence (ii) follows. (c) m is odd and s is even (⇒ m ≥ 3 and s = 0, 2, …, m-1). We can write (s/2)-1 (m – 1)/2

V = {∪ (Vi ∪ Vs – i )} ∪ Vs/2 ∪{∪ (V(s/2) + i , Vm+(s/2) – i)}.

i = 0 i=(s/2)+1

Hence (iii) follows.

Finally,

(d) Both m and s are odd (⇒ m ≥ 3 and s = 1, …, m-2). We can write (s-1)/2 (m – 3)/2 V = {∪ (Vi ∪ Vs – i )} ∪ {∪ (V(s+1)/2 + i ,Vm+(s - 1)/2 – i)} ∪ V (m+s)/2 .

i = 0 i=(s+1)/2

Hence (iv) follows. This completes the proof of the theorem. 1.7 THEOREM. (a) G2, n

(0) is disconnected and G2, n(1) is connected.

(b) Gm, n(s) is disconnected for n ≥ m ≥ 3 and s ∈ {0, 1, …, m – 1}.

1.8 THEOREM. (a) G2, n(0) is bipartite if and only if n = 2, 3, 4 and G2, n

(1) is

bipartite for all n ≥ 2.

(b) For n, m ≥ 3 and s ∈ {0, 1, 2, …, m – 1}, the graph Gm, n(s) is bipartite iff

any one of the following holds:

(i) m is even and s is odd ;

(ii) m is even, s = 0 and n ≤ (5m – 2)/2 ;

(iii) m is odd, s = 0 and n ≤ 3m – 1;

(iv) s assumes even integers from 2 to λ where λ = m – 2 or m – 1

according as m is even or odd and n ≤ (4m + s – 2)/2;

(v) both m and s are odd and n ≤ (5m + s – 2)/2.



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1.9 THEOREM. The (vertex) neighbourhood number n0(Gm , n(s)) of Gm , n

(s)

for n, m ≥ 2, s ∈ {0, 1, …, m – 1} is m.

1.10 THEOREM: The edge neighbourhood number n01(Gm, n

(s)) of Gm, n (s) for n, m ≥ 2,

s ∈ {0, 1, …, m – 1} is as follows:

(i) m is even and

(a) s is even ⇒ = (m+2)/2;

(b) s is odd ⇒ = m/2.

(ii) m is odd ⇒ = (m+1)/2.

1.11 THEOREM. Let m, n be integers ≥ 3 and s ∈ {0, 1, 2,…,m – 1}. Then

the independence number α(Gm, n(s)) of Gm, n

(s) is

(s/2)-1 (m/2)-1

(i) 2 + ∑ max{|Vi|, |Vs-i|} + ∑ max{|V(s/2)+i|, |Vm+(s/2)-i|} i = 0 i=(s/2)+1

when both m and s are even. (s-1)/2 (m/2)-1

(ii) ∑ max{|Vi|, |Vs-i|} + ∑ max{|V(s+1)/2+i|, |Vm+((s-1)/2)-i|} i = 0 i =(s+1)/2

when m is even and s is odd. (s/2)-1 (m-1)/2

(iii) 1+ ∑ max{|Vi|, |Vs-i|} + ∑ max{|V(s/2)+i|, |Vm+(s/2)-i|} i = 0 i=(s/2)+1

when m is odd and s is even. (s-1)/2 (m-3)/2

(iv) 1+∑ max{|Vi|, |Vs-i|} + ∑ max{|V(s+1)/2+i|, |Vm+((s-1)/2)-i|} i = 0 i =(s+1)/2

when both m and s are odd,

1.12. Theorem. Let n ≥ m ≥ 3, n = mq + r and m be even. Then (m/2)q2 +(r – 1)q if 0 ≤ r ≤ m/2 = [m/2], ε(Gm , n

(2)) = (m/2)q2 +(r – 1)q +(r – [m/2] – 1 ) if [m/2] < r ≤ (m-1).



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PROOF. We divide this into three cases. Case(i): m = 4, then from Th.1.6(i), it follows that ε(G4 , n

(2)) = |V0| |V2| + |V1|(|V1| - 1)/2 + 0 + |V3| ( |V3| - 1)/2 . r = 0 ⇒ ε(G4 , n

(2)) = (q)(q) + q(q-1)/2 + q(q-1)/2 = 2q2 – q = (4/2)q2 +(0 – 1)q. r = 1 ⇒ ε(G4 , n

(2)) = (q)(q) + (q+1)(q)/2 + (q)(q-1)/2 = 2q2 = (4/2)q2 +(1 – 1)q. r = 2 ⇒ ε(G4 , n

(2)) = (q)(q + 1)+ (q+1)(q)/2 + (q)(q – 1)/2 = 2q2 + q = (4/2)q2 +(2 – 1)q. r = 3 ⇒ ε(G4 , n

(2)) = q(q + 1) + (q+1)(q)/2 + (q+1)(q)/2 = 2q2 + 2q = (4/2)q2 +(3 – 1)q + (3 – [4/2] – 1). Case(ii): m = 6; then it follows that ε(G6 , n

(2)) = |V0| |V2| + |V1|(|V1| - 1)/2 + |V3| |V5| + |V4| ( |V4| - 1)/2 . r = 0 ⇒ ε(G6 , n

(2)) = (q)(q) + (q)(q – 1)/2 + q(q) + q(q-1)/2 =3q2 - q = (6/2)q2 +(0 – 1)q. r = 1 ⇒ ε(G6 , n

(2)) = (q)(q) + (q + 1)(q)/2 + q(q) + (q)(q-1)/2 = 3q2 = (6/2)q2 +(1 – 1)q. r = 2 ⇒ ε(G6 , n

(2)) = (q)(q + 1) + (q + 1)(q)/2 + q (q) + (q)(q-1)/2 = 3q2 + q = (6/2)q2 +(2 – 1)q. r = 3 ⇒ ε(G6 , n

(2)) = (q)(q + 1) + (q + 1)(q)/2 + (q + 1) (q) + (q)(q-1)/2= 3q2 + 2q = (6/2)q2 +(3 – 1)q. r = 4 ⇒ ε(G6 , n

(2)) = (q)(q + 1) + (q + 1)(q)/2 + (q + 1) (q) + (q + 1)(q)/2 = 3q2 + 3q = (6/2)q2 +(4 – 1)q + (4 – [6/2] – 1). r = 5 ⇒ ε(G6 , n

(2)) = q(q + 1) + (q + 1)(q)/2 + (q + 1) (q + 1) + (q + 1)(q)/2 = 3q2 + 4q + 1 = (6/2)q2 +(5 – 1)q + (5 – [6/2] – 1). Case(iii): m ≥ 8; then from Th.1.6(i), it follows that (m/2)-1

ε(G4 , n(2)) = |V0| |V2| + |V1|(|V1| - 1)/2 +∑ |V1+i| |Vm+1-i| + |V(m+2)/2| ( |V(m+2)/2| - 1)/2 .

i=2

(m/2)-1 r = 0 ⇒ ε(Gm , n

(2)) = (q)(q) + (q)(q – 1)/2 +∑ q(q) + q(q-1)/2 i=2

= q2 + q(q – 1) +m/2) 2)q2 = (m/2)q2 – q = (m/2) q2 + (0 – 1)q. r = 1 ⇒ ε(Gm , n

(2)) = q2 + (q + 1)q/2 + ((m/2) – 2)q2 + q(q-1)/2 = (m/2) q2 = (m/2) q2 + (1 – 1)q. r = 2 ⇒ ε(Gm , n

(2)) = (q)(q +1) + (q + 1)q/2 + ((m/2) – 2)q2 + q(q-1)/2 = (m/2) q2 +q = (m/2) q2 + (2 – 1)q. 3 ≤ r ≤ [m/2] – 1 = (m/2) – 1 r – 1 (m/2)-1

ε(Gm , n(2)) = q(q + 1) + (q + 1) q/2 +∑ (q + 1) q + ∑ (q)(q) + q (q – 1)/2 .

i=2 i=r



- 134

-

= (q2 + q) + q2 +(r – 2) (q2 + q) + ((m/2) – r)q2= (m/2)q2 + (r – 1)q. r = m/2 ⇒

(m/2)-1

ε(Gm , n(2)) = q(q + 1) + (q + 1)q/2 + ∑ (q + 1) (q) + (q)(q - 1)/2

i=2 = (q2 + q) + q2 +((m/2) – 2) (q2 + q) = (m/2)q2 + ([m/2] – 1)q. r = m/2 + 1 ⇒ ε(Gm , n

(2)) = (q2 + q) + (q + 1) (q)/2 + ((m/2) – 2) (q2 + q) + (q + 1) (q)/2 = 2(q2 + q) + ((m/2) – 2) (q2 + q) = (m/2)q2 + (m/2)q. = (m/2)q2 + (m/2+ 1 – 1) q + (m/2 + 1 – [m/2] – 1). ([m/2] + 2) ≤ r ≤ m – 2 = [m/2] + ([m/2] – 2). Now r = [m/2] + t, where 2 ≤ t ≤ [m/2] – 2 ⇒

(m/2) – t (m/2) – 1

ε(Gm , n(2)) = 2(q2 + q) + ∑ (q + 1) q + ∑ (q + 1) (q + 1)

i=2 i = (m/2)– t+1

= 2(q2 + q) + ((m/2) – t – 1) (q2 + q) + (t – 1)(q2 + 2q + 1) = (m/2)q2 + ((m/2) + t - 1)q + (t – 1) = (m/2)q2 + (r – 1)q + (r – [m/2] – 1). r = m – 1⇒

(m/2)-1 ε(Gm , n

(2)) = 2(q2 + q) + ∑ (q + 1)(q + 1) = 2(q2 + q) + ((m/2) – 2) (q2 + 2q + 1) i=2

= (m/2)q2 + (m – 2)q + ((m/2) – 2) = (m/2)q2 + (m – 1 – 1)q + (m – 1 – [m/2] – 1). The proof is completed.

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting her.

They alone live who live for others,

the rest are more dead than alive.



ON LOWER DIFFERENCE GRAPHS Authors: K.V.S. Sarma 1 & I.H.N. Rao2 1Asst. Prof. (Sr.Grade), Regency Institute of Technology, Yanam, 2Director & Sr. Prof., G.V.P. College for P.G. Courses, Visakhapatnam

§ 0. INRODUCTION. We introduce a new type of graphs named as ‘Lower Difference Graphs’. This idea is generated from the concept of difference graphs introduced by Vijaya Saradhi ( dissertation, S.V.U., 1998). Certain useful properties of these graphs are ascertained. The stability nature of these graphs enables us significant applications in the design of networks.

DEFINITION. Let n be any integer ≥ 2. LD2 stands for the complete graph K2 with the vertex set {-1, 1}. For n ≥ 3, LDn stands for the graph with the vertex set V = { -(n-1), -(n-2), …, -1, 1, 2, …, (n-1)} and the edge set E being the two element sets {i, j} where i, j є V with 0 < | i - j | ≤ n. LDn is called a lower difference graph; clearly it is a simple graph and has 2(n-1) vertices. For clear understanding, we give diagrammatic representation of LD5.

-2 -4

-1 -3

LD5:

3 1

4 2

Figure 1(a)

§1. PROPERTIES OF LOWER DIFFERENCE GRAPHS:

1.1 Theorem. LDn (n ≥ 2) has the following properties: (i) The degree of the vertex ‘i’ is 2(n-1) - | i | ; (ii) has (n-1) (3n-4)/2 edges; and (iii) is connected.

1.2 OBSERVATIONS. (i) LD2 is bipartite and LDn is not bipartite for n ≥ 3.

LD2 = K2 and so bipartite. LDn (n ≥ 3) contains K3 (an odd cycle) formed by {-1, -2, 1} and so not bipartite.

(ii) LDn is not Eulerian.


--------------------------------------------------------




136

LD2 = K2 and so not Eulerian. The degree of the vertex ‘-1’ or ‘1’ is 2n-3, an odd integer. Hence by characterization Theorem follows that LDn (n ≥ 3) is not Eulerian.

1.3 THEOREM. The graph LD2 is not Hamiltonian and LDn ( n ≥ 3) is Hamiltonian (Hence has a cycle of length 2(n-1)). Further it has cycles of lengths 3, 4, …, 2(n-1).

PROOF. LD2 =K2 and so not Hamiltonian. When n ≥ 3, the sequence {-1, (n-1), (n-2), …, 1, -(n-1), -(n-2), …, -1} is a cycle of

length 2(n-1). Since LDn has 2(n-1) vertices, follows that this is a Hamiltonian cycle in LDn.

Clearly {-1, 2, 1, -1} is a cycle of length 3 in LD3. Let n ≥ 4; i ∈ {1, 2, …, (n-2)}. Now the sequence {-1, (n-i), (n-(i +1)), …, 1, -(n-1), -(n-2), …, -1} is a cycle of length (n-i) + (n-1) = (2n-1) – i. Thus, there are cycles of length (2n-2), …, (n+1). Clearly {1, -1, -2, …, -(n-1)} forms Kn in LDn and hence LDn has cycles of lengths 3, 4, …, n. This proves the result.

1.4 THEOREM. LDn (n ≥ 2) has a perfect matching. PROOF. Clearly LD2 (= K2) has perfect matching. Let n ≥ 3. Consider the set {{-1, (n-1)}, {-2, (n-2)}, …, {-(n-1), 1}}. It is a subset of the edge set of LDn and has (n-1) elements (edges). These edges saturate all the vertices of LDn and no two elements of this set are adjacent. Hence it is a perfect matching in LDn. This proves the theorem.

1.5 THEOREM. The graph LDn (n ≥ 2) is not regular but contains regular subgraphs of regularity 1, 2, …, (n-1). PROOF. Since 1, 2 are the vertices of LDn and they have different degrees namely, 2(n-1) -1 and 2(n-1) -2 respectively, follows that LDn is not regular. Since LDn contains Kn as a subgraph, namely the subgraph induced by {-1, 1, 2, …, (n-1)} (or by {1, -(n-1), …, -2, -1}) follows that LDn have regular subgraphs of regularity 1, 2, …, (n-1) (In fact each such sub graph is a complete graph as well) .

1.6 THEOREM. The graph LDn is non - planar iff n ≥ 5. PROOF. Since LDn (n ≥ 5) contains K5 and K5 is non-planar, by the characterization theorem of Kuratowski, it follows that LDn is non-planar for n ≥ 5. The fact that LDn (n = 2, 3, 4) is planar follows from the following diagrammatic representations. -1 -2 -1 -3 -1 -2 2 1 2 1 3 1 (a) (b) (c) Fig 2: (a) LD2 (b) LD3 (c) LD4

The proof is completed.

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting him.


11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 137-145

137

ANALYSIS OF LOAD CARRYING CAPACITY IN FINITE POROUS SQUEEZE FILM BEARINGS BY RAPID TECHNIQUE Authors: G. Jaya Chandra Reddy* , C. Eswara Reddy**, and K. Rama Krishna Prasad***

*Department of Mechanical Engineering, K.S.R.M. College of Engineering, KADAPA, A.P. **Department of Mechanical Engineering S.V.U.College of Engineering, TIRUPATI, A.P. ***Department of Mathematics, S.V.U.College of Engineering, TIRUPATI, A.P.

ABSTRACT: Porous bearings have an advantage over conventional sleeve bearings, in which no external supply of lubricant is necessary for their satisfactory operation for long periods. These can operate as self-acting bearings; squeeze film bearings or externally pressurized bearings. Squeeze film is a phenomenon of two lubricated surfaces approaching each other with a normal velocity. The thin film of lubricant between the two surfaces acts as a cushion and prevents the surfaces making instantaneous contact. Generally, the pressure distribution in this fluid film is determined by using numerical methods. In the present analysis, a modified Reynolds equation has been developed by combining the Reynolds equation and the Laplace equation for the finite porous squeeze film bearing. The load carrying capacity is studied by solving this modified Reynolds equation using Rapid Technique. The results are shown graphically for selected parameter values. The results indicate, as the L/D ratio, and or eccentricity ratio increases, load carrying capacity increases but as the permeability parameter increases load carrying capacity decreases. For highly porous bearings, the squeezing effects are not significant. The obtained results are compared with the short bearing and solid bearing analysis.

1. INTRODUCTION Porous metal bearings are widely used in Industry for long time. One of the main advantages of these bearings is that no external supply of lubricant is required for running in. So, these bearings are used, when plain metal bearings are impracticable because of lack of space or inaccessibility for lubrication. Porous bearings are made of porous metals and can operate as self-acting bearings, squeeze film bearings or externally pressurized bearings. Hydrodynamic squeeze films play an important role in engineering practice. The application of squeeze film action is commonly seen in the initial phase of wet clutch engagement, automotive engines and the mechanics of synovial joints in human beings and animals. The squeeze film behavior arises from the phenomenon of two lubricated surfaces approaching each other with a normal velocity. Because of the viscous lubricant present between the two surfaces, it takes a certain time for these to come into contact. Since the viscous lubricant has a resistance to extrusion, a pressure is built up during that interval, and the load is then supported by the lubricant film. If the applied load acts for a short enough time, it may happen that the two lubricated surfaces will not meet at all. Therefore, the analysis of squeeze film action focuses on the load carrying and rate of approach. Analysis of laminar squeezing flow of a non-Newtonian in elastic fluid between parallel disks was studies by Scott[1]. And the first analytical study of porous bearings operating under hydrodynamic conditions was made by Morgon and Cameron[2]. The squeeze film in a non rotating porous journal bearing with a full film of lubricant was studied by J. Prakash and S.K.Vij [3], by assuming the bearing infinitely long in the axial direction; again Prakash and Vij [4] analyzed the porous bearing with Ocvirk’s narrow bearing approximation and solved the problem with the associated boundary conditions. U. Srinivasan [5] examined the influence of velocity slip on the squeeze film action in a non-

Research Paper


Presenter: Prof. Rama Krishna Prasad



138

rotating journal. Murthy [6, 7] Wn [8] and sparrow et al [9] also examined the behavior of slip on porous walled bearings and squeeze films. The squeeze film between two circular plates, one of which has a porous facing, was analyzed by Prakash and Tiwari [10]. Ramanaiah [11] analyzed the squeeze film behavior between finite plates of various shapes lubricated with couple stress fluids. N.M. Bujurke & Jayaraman [12] predicted the characteristics in a squeeze film configuration with reference to synovial joints. J.R. Lin [13, 14, 15] applied the couple stress fluid model to predict the pure squeeze film characteristics of a long partial bearing, short bearing and finite bearing. Recently, Naduvinamani et al [16, 17] have examined the effects & couple stresses on the statistic and Dynamic Effects of couple stresses on the static and dynamic behavior of the squeeze film lubrication of narrow porous journal bearings. K.H. Zaheeruddin [18, 19] investigated the squeeze film behavior of a narrow porous journal bearing lubricated with Micro polar fluid. In the filed of journal bearings, two distinct approaches have been employed with regard to design and performance evaluation. Either the problem has been processed on a computer using numerical techniques or approximate closed-solution methods have been employed. The former, while giving high accuracy has drawback that access to a computer is necessary i.e., the specialized personal are required for programme writing and data processing. Analytical techniques, while being a speedy, low cost method suffer from the distinct disadvantage that the approximations made in the currently available methods effect solution accuracy, offers to an intolerable extent. B.R. Reason and I.P. Narang [20] developed a simple closed solution capable of evaluating any required bearing parameter directly on a calculator, within a low percentage of a digital computer solution. The present paper deals with the application of rapid design technique developed by Reason and Narang to the finite porous squeeze film bearings, to evaluate its pressure distribution and load carrying capacity.

2. ANALYSIS Fig.1 shows a porous squeeze film bearing. The material of the bearing is rigid, homogeneous and isotropic. The lubricant flow in the porous media obeys Darcy’s law and the radial velocity component is zero at the outer surface of the porous wall.

The fluid pressure in the film region satisfies the Reynolds’ equation [3].

∂

∂Φ+=

∂

∂

∂

∂+

∂

∂

∂

∂

=0

33 *12

yy

pV

z

ph

zx

ph

x µµ (1)

Where V= dh/dt is the velocity approach and p* is the pressure in the porous region satisfies the Laplace equation.

∂

∂+

∂

∂+

∂

∂2

2

2

2

2

2 ***

z

p

y

p

x

p = 0 (2)

Integrating equation (2) with respect to y over a bearing wall thickness H,

dyz

p

x

p

y

p

Hy

∂

∂+

∂

∂−=

∂

∂∫

−=

2

2

2

20

0

***

Since 0*

=

∂

∂

−= Hyy

p

~

∂

∂+

∂

∂−

2

2

2

2

z

p

x

pH (3)

If the bearing thickness, H is assumed to be small.



139

Prakash & Vij [4] shown that in the Limit H→ 0 equation (3) is valid exactly. As it is unlikely for a bearing to be almost half as thick as it is long, for practical purposes use of the above approximation is justified to simplify the analysis.

Substituting 0

*

=

∂

∂

yy

p from equation (3) the modified Reynolds equation [21]

becomes

( ) ( )dt

dh

z

pHh

zx

pHh

xµ121212 33 =

∂

∂Φ+

∂

∂+

∂

∂Φ+

∂

∂ (4)

Where h = c (1 + ∈ cos θ)

∴ h3 = c3 (1 + ∈ cos θ)3

≈ (1 + 3∈cosθ) (5)

θCosdt

dc

dt

dh ∈= (6)

Substituting the above equation (5) & (6) in equation (4)

( ){ } ( ){ } θµθθ Cosdt

dc

z

pHc

zx

pHc

x

∈=

∂

∂Φ+∈+

∂

∂+

∂

∂Φ+∈+

∂

∂1212cos3112cos31 33 (7)

(or) by taking c3 out

233

12

12cos3112cos31c

Cosdt

d

z

p

c

H

zx

p

c

H

x

θµ

θθ

∈

=

∂

∂

Φ+∈+

∂

∂+

∂

∂

Φ+∈+

∂

∂ (8)

Let x = Rθ

( )2L

ZZ =

( )ψ121*

+

∈=∈ Where

3c

HΦ=ψ (9)

Substituting and taking (1+12ψ) out Equation (8) becomes

( ) ( )222

.*

12

cos314

cos31c

Cost

z

p

zL

p

R

θµθ

θθ

θ∂

∈∂

=

∂

∂∈+

∂

∂+

∂

∂∈+

∂

∂ ∗∗ (10)

Once again taking ( )θCos∗∈+ 31 ~ h*3

Where ( )θCosh ∗∈+= 1*

And 2

2

*12 R

t

cpp

∂

∈∂=

µ

; D

L=λ ;

θλθθ

Cosz

ph

z

ph =

∂

∂

∂

∂+

∂

∂

∂

∂ 3

2

3 *1

* (11)

Where 3∗h = ( )

Φ=

+

∈=∈∈+ ∗∗

3121;1

c

TandCosc ψ

ψθ

Is the modified Reynolds equation for a porous, squeeze film bearing.

In addition, the boundary conditions are



140

2

0L

Zatp ±==

p = 0 at θ = 0, 2π (12)

πθ ==∂

∂at

Z

p0

3. METHOD OF SOLUTION

RAPID TECHNIQUE: If P, PS and P1 are the pressures in finite, narrow and long bearings respectively, then

by plotting numerically computed pressures P against PS and P1 at extremes of D

L and

eccentricity ratios; the following considerations were observed by Reason and Narang [20]. 1. In general, PS and Pi always exceeds P. 2. The closer one approximation approaches P, the more the other over estimates P.

From the second consideration, at the extremes taken it can be stated.

PS P : P1 α

P1 P: PS α Considering the Reynolds equation, the first derivatives of Poiselulle flow unit length in coordinates (0, Z) are summed together on the left hand side, each containing pressure derivatives. As an initial postulate, a function of the required approximation to P can be assumed as the sum of two separate functions of the approximations.

f1(P) = f2 (PS) + f1 (P1) (13) Applying the above conditions

1

1P

PP

S

+≈ for large value of PS (14)

SPP

P +≈1

1 for large value of P1 (15)

For either alternative to be possible in the same equation, the reciprocal relationship is

1

111

PPP S

+= (16)

∴ Finite bearing pressure is

1

1

pp

ppp

S

S

+=

=

+

1

1p

p

p

S

S (17)

Where P, PS and P1 are the pressures in finite, short and long bearings respectively. Application to Finite Squeeze film bearings:

If the, 5.0≤D

L; it is called short bearing i.e., neglecting the pressure variations in the

X-direction. They equation (11) reduces to



141

θλ

Cosz

ph

z=

∂

∂

∂

∂ 3

2*

1 (18)

and the boundary conditions are p = 0 at Z = 2

1±

0=∂

∂

z

p at Z = 0

The dimension less pressure for short bearing is solving the above equation (18)

( )3

22

*14*6 θ

θλ

µ Cos

Cos

dt

d

cPp S

S∈+

=∈

= (19)

If the, 2≥D

L it is called Long Bearing i.e., neglecting the pressure variation in the Z-

direction; the equation (11) reduces to

θθ

Cosx

ph =

∂

∂

∂

∂ 3* (20)

and the boundary condition are p = 0 at 2

3,

2

ππθ =

0=∂

∂

θ

p at θ = π

Solving the above equation (21) the dimensionless pressure for long bearing in

( ) ( )

∈+−

∈+∈=

∈=

∗∗∗ 2222

2

1

1

1

1

*

11

6 θλµ Cos

dt

dL

cpp l

l (21)

IfD

L ratio is between 0.5 and 2. Then it is called finite bearing

By substituting, the values of short and long bearings in the equation (17) the finite bearing dimensionless pressure becomes

( ) ( ) ( )

( ) ( ) ( )223

2

223

1

1

1

1

1

1

1

1

1

1

1

θθ

λ

θθθ

CosCos

CosCosCos

p

∗∗∗∗∗

∗∗∗∗

∈+∈−

∈+∈+

∈+

∈+−

∈+∈∈+= (22)

Where3121 c

Hand

Φ=

+

∈=∈∗ ψ

ψ

LOAD CARRYING CAPACITY The load carrying capacity of the bearing can be obtained by integrating the pressure components.

dtdRCospCosWW

L

L

x θθφπ

∫∫−

==2

2

0

(23)

dtdRSinpSinWW

L

L

y θθφπ

∫∫−

==2

2

0

(24)



142

and 22

yx WWW += (25)

for short bearing the load carrying capacity is

( )( )2

52

2

3

2

1

21

∗

∗∗

∈−

∈+

∂

∈∂=

πµLR

tcWS (26)

and the dimensionless squeeze load is

( )( )2

52

2

3

2

1

21

∗

∗

•∗

∈−

∈+=

∈

=π

µ RL

CWW S

S (27)

for long bearing the load carrying capacity is

( )2

32

2

3

1

12∗

•∗

∈−

∈=

πµ

C

RLWL (28)

and the dimension less load capacity

( )2

322

3

2

1

3

∗∗

∈−

=∈

=

λ

π

µ RL

CWW L

L (29)

As the load is proportional to the pressure, the load carrying capacity for the finite

bearing isLSf WWW

111+= (30)

( )LS

LS

fWW

WWW

+= Substituting the values of WS and WL and making Non-dimensional

form the squeeze load for the finite bearing is

( )( )( ) ( )2

522

3222

2

3

2

13121

213

∗∗∗

∗

•∗

∈−+∈−∈+

∈+=

∈

=

λ

π

η RL

CWW

ff (31)

Now using eq.no(19) and the above eq.no.(31), the dimensionless pressure and load carrying capacity can be easily solved by using hand calculator.

4.RESULTS AND DISCUSSION Selection of Design Parameters:

(i) L/D Ratio in general, Journal bearings have length to diameter ratio between 0.5 and 2.0; so L/D is a equal to 0.5, 1, 1.5 and 2.0 has been taken for analysis purpose.

(ii) Eccentricity ratio for all finite bearings the minimum eccentricity ratio is 0.2 and maximum is 0.8 in steps & 0.2 has been taken.

(iii) Permeability parameter: Permeability parameter is a dimensionless quantity. It depends on permeability, porous wall thickness and radial clearance. A permeability parameter value of 0.0001, 0.001, 0.01, 0.1 and 1 has been taken for analysis purpose.

Pressure distribution: The variations of the dimensionless pressure to with the circumferential coordinate θ

are depicted in fig.2, fig.3 and fig.4. The variation of the film pressure with θ is shown in

fig.2 for different values of permeability parameter. It is found that the effect of ψ is to reduce the squeeze film pressure. Fig.3 represents the pressure variation for different values of eccentricity ratio at constant permeability parameter. As the eccentricity ratio increases the



143

pressure also increases. Fig.4 gives the pressure variation for different bearings. The finite bearing pressure lies in between short and long bearing pressures. Load carrying capacity:

The variation of the dimensionless load carrying capacity W with the permeability

parameter is depicted in figs. 5,6,7&8. It is observed that ψ - dependence of W is not

significant for value of ψ, 0.001. However load decreases rapidly with increasing the value

of ψ>0.001. Thus there exists a critical value of ψ critical of ψ, such that both p and ω are

insensitive to the variations in ψ for ψ < ψ critical and reduce for ψ > ψ Critical.. The theoretical

design parameter responsible for the effect of porosity in thin walled bearings. 3

c

HΦ=ψ is

very sensitive even to small clearance changes. The results obtained by the rapid technique are compared with the porous short bearing analysis[4]. Further, it is also observed that load carrying capacity increases with the increase in eccentricity ratio. Load carrying capacity is almost constant at low eccentricity ratio for all permeability parameters. Fig.7 represents the load carrying capacity variation with respect eccentricity ratio for various permeability parameters. The results obtained are in good agreement with solid bearing [15]. Fig.8 illustrates the load carrying capacity variation for various L/D ratio’s. As the L/D ratio increases the load carrying capacity decreases for a constant eccentricity ratio.

5. CONCLUSIONS A modified Reynolds equation is derived for finite porous squeeze film bearing by combining Reynolds equation and Laplace equation. And a simple equation has been derived for a finite porous squeeze film bearing by using rapid technique, which can be solved very easily by using hand calculator for the dimensionless pressure and dimensionless load carrying capacity. Load carrying capacity decreases with the increase in porosity, but increases with the increase in eccentricity ratio. And also as the L/D ratio increases the load carrying capacity decreases.

6. NOMENCLATURE c = Journal bearing radial clearance, D = Journal diameter, e = Eccentricity,

h = Oil film Thickness c + e Cos θ, h* = Modified oil film thickness,

*h = Dimensionless oil film thickness (Modified), P = Fluid film pressure,

p = Dimensionless fluid film pressure, V = Shaft normal velocity

W = Bearing External load, W = Dimensionless bearing load, ∈ = Eccentricity ratio c

e

∈* = Modified eccentricity ratio,

θ = Bearing circumferential angle measured from the line of centers in the direction of rotation,

Φ = Permeability, H = Porous wall Thickness, ψ = Permeability parameter 3

c

HΦ,

L = Bearing length, µ = Viscosity coefficient. Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting him. 7. REFERENCES

[1]. Scott, J.R. “Theory and application of the parallel plate plastic meter”, trans inst.

Rubber, Ind. 7 (1931) 435. [2] Margan V.T, Cameron, A. “Mechanism of Lubrication in porous metal bearings”.

Proceedings of the conference on lubrication and wear, London, Institution of

Mechanical Engineers, (1957) 151-157.



144

[3] Prakash, J. and Vij, S.K. “Squeeze films in porous metal bearings” J. Lubrication

Technology, 94 (1972) 302-305. [4] Prakash, J and Vij, S.K. “Squeeze films in porous bearings” Wear, 27 (1974) 359-

366. [5] Srinivasan, V “Effect & Velocity slip on squeeze films in porous bearings” Wear 51,

(1978) 39-47. [6] Murthy, P.R.K. “Some aspects of slip flow in porous bearings”, Wear 19 (1972) 123-

129. [7] Murthy, P.R.K. “Effect & slip in narrow porous bearings”, J. Lubrication Technology,

95 (1973) 518-523. [8] Wn, H “Effect & Velocity Slip in squeeze film between porous rectangular plates”,

Wear, 20 (1972) 67-71. [9] Sparrow, E.M, Beavers G.S., and Hwang, L.T. “Effect of velocity slip on porous

walled squeeze films”, J. Lubrication Technology, 94 (1972) 260-265. [10] Prakash, J and Tiwari, K. “An Analysis of the squeeze film between rough porous

rectangular plates with arbitary porous wall thickness”, Trans, ASME, 106 (1984) 218-226.

[11] Ramanaiah, G. “ Squeeze films between finite plates lubricated by fluids with couple stress”, Wear 54 (1979) 315-320.

[12] Bujurke, N.M. and Jayaraman, G “The Influence of couple stresses in squeeze films”, Int. J. Mech. Sci., 24 (1982) 369-376.

[13] Lin, J.R. “Squeeze film characteristics of a long partial bearing lubricated with couple stress fluids”, Tribology International, (1997) 30 (1) 53-58.

[14] Lin, J.R. “Static and Dynamic behaviour of pure squeeze films in couple stress fluid-lubricate short journal bearings”, Proc. Instn., Mechanical Engineering, (1997) 62(1), 175-184.

[15] Lin, J.R. “Squeeze film characteristics of finite journal bearing : Couple stress fluid model”, Tribology, International, 31 No.4 (1998) 201-207.

[16] Naduvinamani, N.B, Hiremath.P.S, Gurubasavaraj G, “Statistic and Dynamic behavior of squeeze film lubrication of narrow porous journal bearings with couple stress fluid”, proceedings of institution of Mechanical Engineers, Part J, 205 (2001) 45-62.

[17] Naduvinamani, N.B, Hiremath, P.S. & Gurubasavaraj, G. “Squeeze film lubrication of a short porous journal bearing with couple stress fluids”, tribology International, Vol. 34 (2001) 739-747.

[18] Zaheeruddin, K.H. “Squeeze film narrow porous journal bearings lubricated with a micro polar fluid”, wear 64 (1980) 163-174.

[19] Zaheeruddin, K.H. “The Dynamic behavior of squeeze films in one Dimensional porous journal bearings lubricated by a Micro polar fluid”, wear 7 (1981) 139-152.

[20] Reason, B.R. and Narang, I.P. “Rapid design and performance evaluation of steady-state journal bearing – a Technique amenable to programmable hand calculator”, Trans. ASME. Vol.25, No:4; 429-444.

[21] Sneck, H.J. “A Mathematical analog for determination of porous metal bearing performance characteristics’, J. Lubr. Techo, ASME, April (1967) 220-226.

[22] Pincus, O. and Sternlicht, B, “Theory of Hydro dynamic lubrication”, McGraw-Hill,

New York, 1961.



145

Fig. 1 Geometry and Coordinates of the finite porous squeeze film bearing

e R

h

H

y θ

W

Fluid Film Region

Porous Wall Solid Housing

Line of Centers

z

L/2 L/2

A Lecture on Pseudo-complemented Almost Distributive

Lattices

G. Nanaji Rao

Department of Mathematics,

Andhra University,

Visakhapatnam-530003, INDIA.

[email protected]

Abstract. The concept of pseudo-complementation ∗ on an Almost Distributive

Lattice (ADL) with 0 is introduced and it is proved that it is equationally defin-able. A one to one correspondence between the pseudo-complementation on an

ADL L with 0 and maximal elements of L is obtained. It is also proved that the set

L∗ = {a∗ | a ∈ L} is a Boolean algebra which is independent (up to isomorphism) of

the pseudo-complementation ∗ on L. The concept of ∗−Almost Distributive Lattice,

Stone Almost Distributive Lattice are introduced. Some necessary and sufficient

conditions for an ADL to become a ∗− ADL (Stone ADL) in topological and alge-braic terms are proved. Characterization of ∗−ADLs and Stone ADLs in terms of

prime ideals and minimal prime ideals are established.

Key words and Phrases: Almost Distributive Lattice (ADL), Prime ideal, Primefilter, Pseudo-complementation, Boolean algebra, maximal element, dense element,

Hull-kernel topology, ∗−ADL, Stone ADL .

1. Introduction

A pseudo-complemented lattice is a lattice L with 0 such that to each a ∈ L,there exists a∗ ∈ L such that for all x ∈ L, a ∧ x = 0 if and only if x ≤ a∗.Here a∗ is called the pseudo-complement of a. For each element a of a pseudo-complemented lattice L, a∗ is uniquely determined by a, so that “ ∗ ” can beregarded as a unary operation on L. Moreover, each pseudo-complemented latticecontains the unit element 0∗ (0 ∧ x = 0 ⇔ x ≤ 0∗ for all x). It follows that everypseudo-complemented lattice L can be regarded as an algebra (L,∨,∧, ∗, 0, 1) oftype (2, 2, 1, 0, 0). The fact that class of pseudo-complemented distributive latticesis equationally definable was first observed by P. Ribenboin [9 ]. Also, K.B. Lee [6]proved that the class of pseudo-complemented distributed lattice is generated byits finite members and a complete description of the lattice of equational classes ofpseudo-complemented distributive lattices is given. G. Gratzer [5 ] studied aboutthe pseudo-complemented semi-lattice L, and proved that the set L∗ = {a∗ | a ∈ L},

2000 Mathematics Subject Classification: 06D99.

Received: dd-mm-yyyy, accepted: dd-mm-yyyy.

1

2 G. Nanaji Rao

where ∗ is a pseudo-complementation on L, becomes a Boolean algebra.

After the Boole’s aximitization of the two valued propositional calculus, intoa Boolean algebra, many generalizations of the Boolean algebras have come intobeing. The concept of an Almost Distributive Lattice (ADL) was introduced bySwamy and Rao [ 14 ]as a common abstraction of almost all the lattice theoretic andring theoretic generalization of a Boolean algebra like, P−rings, bi-regular rings,associative rings, p1−rings triple system, Bair-rings and m−domains rings.

In [15], we introduced the concept of pseudo-complementation on Almost Dis-tributive Lattice (ADL). It is well known that in the case of distributive lattice, thepseudo-complementation ∗ if exists, is unique. But, in the case of ADLs, there canbe several pseudo-complementations. In fact, if there is a pseudo-complementationon an ADL L, then we prove that each maximal element of L, gives rise to a pseudo-complementation and that this correspondence between the set of maximal elementsof L, and the set of pseudo-complementations on L is one-to-one. Also, we provethat if ∗ is a pseudo-complementation on an ADL L, then the set L∗ = {a∗ | a ∈ L}becomes a Boolean algebra and if ∗ and ⊥ are two pseudo-complementations on L,then we prove that the corresponding Boolean algebras L∗ and L⊥ are isomorphic.In other words, the Boolean algebra L∗ is independent (up to isomorphism) of thepseudo-complementation ∗. Also, we prove that the pseudo-complementation ∗ onan ADL is equationally definable.

The problem of the characterizing pseudo-complemented distributive latticessatisfying x∗ ∨ x∗∗ = 1 (called Stone lattice) has been studied by several mathe-maticians like O. Frink [ 3 ], G, Gratzer [ 4], T.P.Speed [11 ], etc,. In [15 ], wehave proved that if ∗ and ⊥ are two pseudo-complementations on an ADL L, thena∗ ∨ a∗∗ = 0∗ ⇔ a⊥ ∨ a⊥⊥ = 0⊥. for all a ∈ L. This motivates us, in [ 16], we haveintroduced a more generalized class of ADLs like ∗−ADLs and Stone ADLs whichproperly contains the class of ADLs with pseudo-complementation as a generaliza-tion of results of T.P. Speed [11]. Also, we have characterized the Stone ADLs, bothalgebraically and topologically in terms of their prime ideals and minimal primeideals with hull-kernel topology and dual hull-kernel topology. Also, the ∗−ADLswas characterized by means of dense of elements.

2. Preliminaries

First we recall certain definitions and properties of Almost Distributive Lat-tice from [7],[14], that are required in the text.

Definition 2.1. An algebra (L,∨,∧, 0) of type (2,2,0) is called an Almost Dis-tributive Lattice(ADL) if , for any x, y, z ∈ L,

(1) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)(2) (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z)

A Lecture on Pseudo-complemented ADLs 3

(3) (x ∨ y) ∧ x = x

(4) (x ∨ y) ∧ y = y

(5) x ∨ (x ∧ y) = x

(6) 0 ∧ x = 0

It is observed that in [14], that any non-empty set A can be regarded as anADL with some arbitrary fixed element a0 ∈ A as zero where the operation ∨ and∧ are defined as follows.

Definition 2.2. Let A be a non-empty set and a0 ∈ A. For any a, b ∈ A, define,

(a ∨ b) =

{

a if a 6= a0

b if a = a0

(a ∧ b) =

{

b if a 6= a0

a0 if a = a0

This is called a discrete ADL with 0.

Lemma 2.3. Let L be an ADL. Then we have the following:(1)a ∨ b = a ⇔ a ∧ b = b

(2)a ∨ b = b ⇔ a ∧ b = a

(3)a ∧ b = b ∧ a whenever a ≤ b

(4) ∧ is associative in L

(5) a ∧ b ∧ c = b ∧ a ∧ c

(6)(a ∨ b) ∧ c = (b ∨ a) ∧ c

(7)a ∧ b = 0 ⇔ b ∧ a = 0(8)a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)(9) a ∧ (a ∨ c) = a, (a ∧ b) ∨ b = b and a ∨ (b ∧ a) = a

(10) a ≤ a ∨ b and a ∧ b ≤ b

(11) a ∧ a = a and a ∨ a = a

(12) a ∨ 0 = a = 0 ∨ a and a ∧ 0 = 0(13) if a ≤ c, b ≤ c for some c ∈ L, then a ∧ b = b ∧ a and a ∨ b = b ∨ a.

Definition 2.4. A homomorphism between ADLs L and L′ is a mapping f : L → L′

satisfying the following:(1)f(a ∨ b) = f(a) ∨ f(b)(2)f(a ∧ b) = f(a) ∧ f(b)(3)f(0) = 0.

A nonempty subset I of L is called an ideal of L if x ∨ y ∈ I and x ∧ a ∈ I

whenever x, y ∈ I and a ∈ L. For any A ⊆ L, the ideal generated by A is(A] = {(

∨

n

i=1ai) ∧ x | ai ∈ A, x ∈ L, n ∈ Z+}. If A = {a} ,then we write (a] for

(A]. The set of all principal ideals of L is a distributive lattice and it is denoted byPI(L). A proper ideal P of L is called prime if for any x, y ∈ L, x∧y ∈ P then x ∈ P

or y ∈ P . For any x, y ∈ L ,define x ≤ y if and only if x = x ∧ y or equivalentlyx∨y = y. Then ≤ is a partial ordering on L in which 0 is the least element. For any

4 G. Nanaji Rao

x, y ∈ L with x ≤ y, [x, y] = {t ∈ L | x ≤ t ≤ y} is a bounded distributive latticewith respect to the operations in L. If, in addition, [x, y] is a Boolean algebra, thenL is relatively complemented ADL and in this case, the operation ∨ is associative.An element m is maximal in (L,≤) if and only if m ∧ x = x for all x ∈ L.For anyA ⊆ L,A∗ = {x ∈ L | a ∧ x = 0 ∀ a ∈ A} is an ideal of L. We write [a]∗ for {a}∗.We don’t know, so far , whether ∨ is associative in an ADL or not. In this talk, Ldenotes an ADL in which ∨ is associative.

Lemma 2.5. Let L be an ADL and a ∈ L. Then (a] = L if and only if a is amaximal element.

Lemma 2.6. Let L be an ADL and I an ideal of L. Then, for any a, b ∈ L, wehave the following:(1)(a] = {a ∧ x | x ∈ L}(2)a ∈ (b] ⇔ b ∧ a = a

(3)a ∧ b ∈ I ⇔ b ∧ a ∈ I

(4)(a] ∩ (b] = (a ∧ b] = (b ∧ a](5)(a] ∨ (b] = (a ∨ b] = (b ∨ a].

Lemma 2.7. A ring R is called a regular ring if,to each a ∈ [R], there exists x ∈ [R]such that axa = a

3. Pseudo-Complementation

In this section, we give the definition of pseudo-complementation on an al-most distributive lattice with 0 and prove some basic properties of an ADL withpseudo-complementation and we prove that an ADL with Pseudo-complementatinis equationally definable.

Definition 3.1. Let(L,∨,∧, 0) be an ADL with 0. Then a unary operation a 7→ a∗

on L is called a pseudo-complementation on L if, for any a, b ∈ L, it satisfies thefollowing conditions:(P1) a ∧ b = 0 ⇒ a∗ ∧ b = b;(P2) a ∧ a∗ = 0;(P3) (a ∨ b)∗ = a∗ ∧ b∗.

We observe that p1, P2 and P3 are independent.

Example 3.2. Let X be a discrete ADL with 0 and with at least two elements,say 1, 2 other than 0. Then (X3,∨,∧, 0) is an ADL with 0, where ∨,∧ are definedcoordinatewise. Now, for any x ∈ X3, we write p x p for the number of non-zeroentries in x. Define ∗ on x3 as follows: For any x ∈ X3 and i = 1, 2, 3

x∗

i=

0 if xi 6= 0

1 if xi = 0 and |x| = 1 and 0∗ = (2, 2, 2)

2 if xi = 0 and |x| > 1


then (X3,∨,∧, 0) is an ADL with (0, 0, 0) as 0 which satisfies P1 and P2 but not(P3) (if a = (1, 0, 0) and b = (0, 1, 0), then a∗ = (0, 1, 1), b∗ = (1, 0, 1), and a ∨ b =(1, 1, 0), so (a ∨ b)∗ = (0, 0, 2) and a∗ ∧ b∗ = (0, 0, 1))

Example 3.3. Let L be an ADL with 0 with at least two elements. Define a∗ = 0for all a ∈ L. Then L satisfies (P2) and (P3) but not (P1) (if 0 6= b ∈ L, then0 ∧ b = 0 and 0∗ ∧ b = 0 6= b.)

Example 3.4. Let L be a bounded distributive lattice with bounds 0 6= 1. Now forany a ∈ L, define a∗ = 1 for all a ∈ L. Then L satisfies (P1) and (P3) but not(P2).

Now, we give certain examples of pseudo-complementation on ADLs.

Example 3.5. Let (R,+, ·, 0) be a commutative regular ring. To each a ∈ R, Leta0 be the unique idempotent element in R such that aR = a0R. Define, for anya, b ∈ R,(1) a ∧ b = a0b;(2) a ∨ b = a+ (1− a0)b;(3) a∗ = 1− a0;then (R,∨,∧, 0) is an ADL with 0 and ∗ is a pseudo-complementation on R.

In the case of a distributive lattice with 0, it is well known that the ele-ment a∗ satisfying the properties (P1) and(P2) is unique (if it exists) and that(P3) is a consequence of (P1) and (P2) and hence, there can be at most onepseudo-complementation. However, in an ADL with 0, there can be several pseudo-complementations; for, consider the following examples.

Example 3.6. Let (X,∨,∧, 0) be a discrete ADL with 0. For any x 6= 0 in X,define

a∗ =

{

0 if a 6= 0

x if a = 0

then ∗ is a pseudo-complementation on X. Here, with each x 6= 0 in X, we obtaina pseudo-complementation on (X,∨,∧, 0). More generally, we have the following:

Example 3.7. Let X be a non-empty set with at least two elements and let Y beany set and f0 ∈ XY . Now, for any a, b ∈ XY , define

(1) (a ∨ b)(y) =

{

a(y) if a(y) 6= f0(y)

b(y) if a(y) = f0(y);

(2) (a ∧ b)(y) =

{

b(y) if a(y) 6= f0(y)

f0(y)if a(y) = f0(y).

Then(XY ,∨,∧, f0) is an ADL with f0 as zero element. Now, let f ∈ XY

such that f(y) 6= f0(y) for all y ∈ Y . For any a, b ∈ XY , define

af (y) =

{

f0(y) if a(y) 6= f0(y)

f(y) if a(y) = f0(y)

6 G. Nanaji Rao

Then a 7→ af is a pseudo-complementation on XY and, conversely, if a 7→ a∗ is apseudo-complementation on XY , then there exists f ∈ XY such thatf(y) 6= f0(y)for all y ∈ Y and a∗ = af for all a ∈ XY ( take f = f∗

0)

In the following, we prove that any relatively complemented ADL with 0 is pseudo-complemented,and hence, the examples given in Examples 3.5−3.7 are special casesof this.

Theorem 3.8. Let L be a relatively complemented ADL with 0 and with a maximal

element m0. For any a ∈ L, define a∗ to be the complement of a in [0, a ∨ m0].Then ∗ is a pseudo-complementation on L.

Now, we give the some properties of pseudo-complementation in the following.

Lemma 3.9. Let L be an ADL with 0 and ∗ a pseudo-complementation on L.

Then, for any a, b ∈ L, we have the following:

(1) 0∗ is maximal;

(2) if a is maximal, then a∗ = 0;(3) 0∗∗ = 0;(4) a∗ ∧ a = 0;(5) a∗∗ ∧ a = a;

(6) a∗ = a∗∗∗;

(7) a∗ = 0⇔ a∗∗ is maximal;

(8) a∗ ≤ 0∗;(9) a∗ ∧ b∗ = b∗ ∧ a∗;

(10) a ≤ b⇒ b∗ ≤ a∗;

(11) a∗ ≤ (a ∧ b)∗ and b∗ ≤ (a ∧ b)∗;(13) a = 0⇔ a∗∗ = 0;


Then for any a, b ∈ L, the following are equivalent:

(1) a ∧ b = 0;(2) a∗∗ ∧ b = 0;(3) a∗∗ ∧ b∗∗ = 0;(4) a ∧ b∗∗ = 0;


Then, for any a, b ∈ L, the following hold:

(1) (a ∧ b)∗∗ = a∗∗ ∧ b∗∗;

(2) (a ∧ b)∗ = (b ∧ a)∗;(3) (a ∨ b)∗ = (b ∨ a)∗;

In [6], Lee derived certain sets of identities characterizing pseudo-complementeddistributive lattice. In the following theorems, we obtain a set of of identities whichcharacterize a pseudo-complementation on an ADL with 0.


Theorem 3.12. Let (L,∨,∧, 0) be an ADL with 0. Then a unary operation∗ : L −→ L is a pseudo-complementation on L if and only if it satisfies the follow-ing equations:(1) a ∧ a∗ = 0;(2) a∗∗ ∨ a = a∗∗;(3) (a ∨ b)∗ = a∗ ∧ b∗;(4) (a ∧ b)∗∗ = a∗∗ ∧ b∗∗;(5) 0∗ ∧ a = a;

Theorem 3.13. Let (L,∨,∧, 0) be an ADL with 0. Then a unary operation∗ : L −→ L is a pseudo-complementatin on L if and only if it satisfies the fol-lowing equations:(1) a∗ ∧ b = (a ∧ b)∗ ∧ b;(2) 0∗ ∧ a = a;(3) 0∗∗ = 0;(4) (a ∨ b)∗ = a∗ ∧ b∗

4. one to one correspondence

Here, we prove that, for any ADL with 0 and with pseudo-complementation∗, the set L∗ = {a∗|a ∈ L} becomes a Boolean algebra. In sec.3,it is remarked thatan ADL with 0 can have more than one pseudo-complementation and exampleswere given to this effect. In fact, if L is an ADL with a pseudo-complementation∗, then to each maximal element m ∈ L, we obtain a pseudo-complementation∗m. we prove that this correspondence between the maximal elements of L andpseudo-complementations on L is one-to-one and that the Boolean algebra L∗ isindependent (upto isomorphism) of the pseudo-complementation ∗.

Theorem 4.1. Let L be an ADL with 0 and ∗ a pseudo-complementation on L.For any a∗, b∗ ∈ L∗, define a∗ ≤ b∗ if and only if a∗ ∧ b∗ = a∗. Then (L∗,≤) is aBoolean algebra, in which (a∗∗ ∧ b∗∗)∗ is the least upper bound (l.u.b) and (a∗ ∧ b∗)is the greatest lower bound (g.l.b) of a∗and b∗.

We remarked that an ADL with 0 can have more than one pseudo-complementation.Now, we prove that for any two pseudo-complementations ∗, ⊥ on L, the Booleanalgebras L∗ and L⊥ are isomorphic. First we prove the following lemma.

Lemma 4.2. Let L be an ADL with 0 and let ∗ and ⊥ be two pseudo-complementationson L. Then for any a, b ∈ L, we have the following:(1) a∗ ∧ a⊥ and a∗ ∨ a⊥ = a∗;(2) a∗⊥ = a⊥⊥;(3) a∗ = b∗ ⇔ a⊥ = b⊥;(4) a∗ = 0⇔ a⊥ = 0⇔ (a ∧ b = 0⇒ b = 0);

8 G. Nanaji Rao

(5) a⊥ = a∗ ∧ 0⊥;(6) a∗ ∨ a∗∗ = 0∗ ⇔ a⊥ ∨ a⊥⊥ = 0⊥;

Theorem 4.3. Let L be an ADL with 0 and ∗ a pseudo-complementation on L. Let

M be the set of all maximal elements in L and let PC(L) be the set of all pseudo-

complementations on L. For any m ∈ M , define ∗m : L −→ L by a∗m = a∗ ∧m

for all a ∈ L. Then m 7→ ∗m is a bijection of M onto PC(L).

In the following theorem, we prove that the Boolean algebra L∗ is indepen-dent, upto isomorphism of the pseudo-complementations ∗ on L

Theorem 4.4. Let L be an ADL with 0 and let ∗,⊥ be two pseudo-complementation

on L .Then the map f : L∗−→ L⊥ defined by f(a∗) = a⊥ is an isomorphism of

Boolean algebras.

References

1. Birkhofdf, G.: Lattice Theorey, Vol.25,American Mathematical Society Colloquium Publica-

tion,1967.

2. Burris, S, Sankappanavar, H.P.: A course in universal algebra, Springer Verlag.1981.

3. Frink,O: Pseudo-complements in semilattices, Duke Math. Journal,20, 505-515(1962).4. Gratzer, G.: A generalization on Stone’s representation theorem for Boolean algebra,Duke.

Math.Journal 30,469-474((1963).

5. Gratzer, G: Lattice Theorey: First Concepts and Distributive Lattices, W.H.Freeman and

Company, Sanfransisco (1971).

6. Lee, K.B.: Equational class of distributive pseudo-complemented lattice, Cand. J.Math., 22,

881-891(1970)7. Rao, G.C.: Almost distributive lattices, Doctoral Thesis, Andhra University,Waltair, 1980.

item[8.] Rao. G.C.: and S. Ravi Kumar.:Minimal prime ideals in almost distributive lattices

, Int. Journal of Contemp. Math. Sciences, Vol. 4, no. 9-12, 475-484(2009).

9. Ribenboim, P.: Characterization of the pseudo-compliment in distributive lattice with least

element, Summa Brasil. Math., 2(1949), No.4, 43-49.MR 11,75.

10. Speed, T.P: Some remarks on a class of distributive lattices, J.Austral.Math. Soc.9,289-296(1969).

11. Speed, T.P: On stone lattices, J.Austral.Math. Soc.9,297-307(1969).12. Swamy, U.M., Mankyamba, P.: Prime ideal characterization of stone lattices, Maths Seminar

Note 7, Kobe University, 25-31(1979).

13. Swamy,U.M., Murti, G.S.: Boolean centre of a universal algebra, Algebra Universals 13,202-205(1981)

14. Swamy, U.M. and Rao.G.C.: Almost Distributive Lattices, Journal of Australian

Math.Soc,(Series A) 31, 77-91(1981).15. Swamy, U.M., Rao.G.C.: and Nanaji Rao.G.:Pseudo-Complementation on Almost Distribu-

tive Lattices, Southeast Asian Bulletin of Mathematics, 24:95-104(2000).

16. Swamy, U.M., Rao.G.C.:and Nanaji Rao.G.:Stone Almost Distributive Lattices, SoutheastAsian Bulletin of Mathematics, 27: 513-526(2003).


17. Swamy, U.M., Rao.G.C.: and Nanaji Rao.G.:Dense Elements in Almost Distributive Lattices,

Southeast Asian Bulletin of Mathematics, 27: 1081-1088(2004).

18. Swamy, U.M., Rao.G.C.: and Nanaji Rao.G.:On Characterizations of Stone Almost Distribu-tive Lattices, Southeast Asian Bulletin of Mathematics, 32, 1167-1176(2008).

19. Swamy, U.M., Rama Rao.V.V.: Triple and sheaf representation of Stone Lattices, AlgebraUniversalis 5, 104-113(1975).


Some Results on Normal fuzzy Ideals of MΓΓΓΓ-Modules Authors: Nagaraju Dasari, Satyanarayana Bhavanari , Babu Prasad Munagala and Venkatachalam Abstract: The notion of Γ-near-ring, a concept which is a generalization of both near-

ring and Γ-ring was introduced by Satyanarayana [5]. In 2004, Satyanarayana [6]

introduced MΓ-module. Later Fuzzy Ideals of MΓ-modules were studied by Satyanarayana, Vijaya Kumari, Godloza & Nagaraju [7]. In this present paper we

introduced the concept of Normal fuzzy ideals in MΓ-modules and prove some interesting results Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting him. References: [1]. Y. B. Jun, M. Sapanci & M. A. Ozturk “Fuzzy ideals in gamma near-rings”, Turk. J. Math 22 (1998), 449-459. [2]. Y. B. Jun, K. H. Kim & M. A. Ozturk “On Fuzzy Ideals of Gamma Near-rings”, J.

Fuzzy Math, 9(1), (2001), 51-58. [3]. Y. B. Jun, K. H. Kim & M. A. Ozturk “Fuzzy Maximal Ideals of Gamma

Near-rings”, Turk. J. Math 25 (2001), 457-463. [4]. J. N. Mordeson & D. S. Malik “Fuzzy Commutative Algebra”, World Scientific

Publishing Co. Pvt. Ltd, 1998. [5]. Bh. Satyanarayana “Contributions to near-ring theory”, Ph.D., thesis, Nagarjuna University, 1984. Published by VDM Verlag Dr Mullar, Germany, 2009 (ISBN: 978-3-639-22417-7). [6]. Bh. Satyanarayana “Modules over Gamma Near-rings”, ANIJMIT, 01 (2) (2004),

109-120. [7]. Bh. Satyanarayana, A. V. Vijaya Kumari, L. Godloza & D. Nagaraju “Fuzzy Ideals

of Modules over Gamma near-rings”, accepted. [8]. H. J. Zimmermann “Fuzzy Set Theory and its Applications”, Kluwer, Boston, 1985. [9]. Zadeh L. A “Fuzzy Sets”, Information and Control 8 (1965) 338-353.

Paper (Oral Presentation) _____________________________ Presenter: A. Venkatachalam

Periyar Maniammai University Thanjavur, Tamilnaidu.



N(A) – SEMIGROUPS

Author: D. Madhusudhana Rao, V S R & N V R College, Tenali

ABSTRACT: In this paper, the terms, ‘A-potent’, ‘left A-divisor’, ‘right A-divisor’, ‘A-divisor’ elements and ideals, ‘N(A)-semigroup’ for an ideal A of a semigroup are introduced. If A

is an ideal of a semigroup S then it is proved that (1) )()()( 012 ANANANA ⊆⊆⊆ (2) N0(A) =

A2, N1(A) is a semiprime ideal of S containing A, N2(A) = A4 are equivalent, where No(A) = The set of all A-potent elements in S, N1(A) = The largest ideal contained in No(A), N2(A) = The union of all A-potent ideals. If A is a semipseudo symmetric ideal of a semigroup then it is proved that N0(A) = N1(A) = N2(A). It is also proved that if A is an ideal of a semigroup such that N0(A) = A then A is a completely semiprime ideal. Further it is proved that if A is an ideal of semigroup S then R(A), the divisor radical of A, is the union of all A-divisor ideals in S. In a N(A)-semigroup it is proved that R(A) = N1(A). If A is a semipseudo symmetric ideal of a semigroup S then it is proved that S is an N(A)-semigroup iff R(A) = N0(A). It is also proved that if M is a maximal ideal of a semigroup S containing a pseudo symmetric ideal A then M contains all A-potent elements in S or S\M is singleton which is A-potent. REFERENCES: 1. ANJANEYULU A. and RAMAKOTAIAH D., On a class of semigroups – Simon Stevin, Vol.54 (1980) 241-249. 2. ANJANEYULU A., Semigroups in which prime ideals are maximal – Semigroup Form, Vol.22

(1981) 151-158. 3. CLIFFORD A. H. and PRESTON G. B., The algebraic theory of semigroups – Vol-I,

American Mathematical Society, Providence (1961). 4. CLIFFORD A. H. and PRESTON G. B., The algebraic theory of semigroups – Vol-II,

American Mathematical Society, Providence (1967). 5. LJAPIN E. S., Semigroups, American Mathematical Society, Providence, Rhode Island (1974). 6. PETRICH. M., Introduction to semigroups - Merril Publishing Company, Columbus, Ohio, (1973).

----------------------------------------------------------------------------------------------------------------

PSEUDO INTEGRAL SEMIGROUPS Author: A. Gangadhara Rao,

V S R & N V R College, Tenali, Guntur Dt, A.P.

ABSTRACT: In this paper, the term, ‘pseudo intergral semigroup’ is introduced. It is

proved that (1) every pseudo symmetric semigroup with nonempty kernel is a pseudo integral semigroup (2) If S is a semigroup with empty kernel such that S has no nontrivial K-divisor elements then S is a pseudo integral semigroup. It is also proved that an ideal A of a semigroup S is pseudo symmetric iff S\A is a pseudo integral semigroup. If S is a pseudo integral semigroup then it is proved that S is strongly archimedean, S is archimedean, S has no proper completely prime ideals, S has no proper completely semiprime ideals, S has no proper prime ideals, S has no proper semiprime ideals, every element in S is a K-potent element are equivalent. It is proved that if T is a maximal subsemigroup of a pseudo integral semigroup S such that ∅=∩ KT then S\T is a minimal prime ideal in S. REFERENCES: [The references are same ([1] to [5]) as in the above paper]

Paper (oral Presentation) ----------------------------------------------------

Presenter: D. Madhusudhana Rao,


A. Gangadhara Rao,


(July 11-12, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana and Dr A.V. Vijaya Kumari) PP: 155

SECOND ORDER RESPONSE SURFACE MODEL WITH NEIGHBOR EFFECTS Authors: B.Re.Victor Babu and K. Rajyalakshmi, Department of Statistics, Acharya Nagarjuna University.

ABSTRACT: Box and Wilson (1951) introduced Second Order Response Surface Model. Box and Hunter (1957) introduced rotatable designs. Das and Narasimham (1962) constructed Rotatable designs through Balanced Incomplete Block Designs.Box and Draper (1987) analysed the concept of Response Surface Methodology. Myers and Montogomery (1995) discussed the concept Response Surface Designs. Khuri and Cornell (1996) analysed the Response Surfaces. Jaggi, Gupta and Ashraf (2006) introduced the concept of block designs partially balanced for neighboring competition effects. Sarika et.al (2009) introduced second order response surface model with neighbor effects. It is shown that these designs have the neighbor effects from immediate left and right neighboring plots assuming the plots to be place adjacent linearly with no gaps. The variance of the estimated response at different values of ‘α’ varying from 0 to 1 for a second order model (v = 2, 3, 4) with neighbor effects. In this paper, we extend the work of Sarika et.al (2009) to get the variance of the estimated response at different levels of ‘α’ varying from 0 to 1 for a second order model (v = 5) by using fractional replication with neighbor effects. References: [1] Azias, J.M., Bailey, R. A., Monod, H. (1993). A catalogue of efficient neighbor designs with border plots. Biometrics 49:1252-1261. [2] Box, G.E.P., Draper, N. R. (1987). Emperical Model Building and Response surfaces. New york: John Wiley and Sons. [3] Draper, N. R., Guttman, I. (1980). Incorporating overlap effects form neighboring units into response surface models. Appl. Statisti. 29(2):128-134. [4] Jaggi, S., Gupta, V.K. Ashraf, J. (2006). On block designs partially balanced for neighboring competition effects. J.Ind. Statist. Assoc. 44:27-41. [5] Khuri, A.I., Cornell, J.A. (1996). Response Surfaces_ Dsigns and Analysis. New York: Marcel Dekker. [6] Myers, R.H. Montogomery, D.C.(1995).Response Surface Methodology-Process and Product Optimization Using Designed Experiments. New york: John Wiley Publication. [7] Tomar, J.S., Jaggi, s., Varghese, C. (2005). On totally balanced block designs for competition effects. J. Appl. Statist. 32(1):87-97. [8] Sarika, Jaggi, Seema and Sharma, V.K. (2009) ‘Second Order Response Surface Model with Neighbor Effects’, Communication in Statistics – Theory and Methods, 38:9, 1393-1403.


Presenter: K. RajyaLakshmi

THEY ALONE LIVE

WHO LIVEFOR OTHERS, THE REST ARE MORE DEAD THAN ALIVE



Page: 156

Some types of Prime ideals in Gamma Near – Rings Abstract: In this paper we considered the algebraic system Γ-near-ring that was introduced by

Satyanarayana in 1984. Γ-near-ring is a more generalized system than both the near-ring and the Γ-ring.

The aim of this short talk is to study some types of ideals in Γ-near-ring.

1. Introduction: The concept Γ-ring, a generalization of ‘ring’ was introduced by Nobusawa [4] and generalized by Barnes [ 1 ]. Later Satyanarayana [8], Satyanarayana, Pradeep Kumar & Srinivasa Rao

[12 ] also contributed to the theory of Γ-rings. A generalization of both the concepts near-ring and the Γ-

ring, namely Γ-near-ring was introduced by Satyanarayana [ 9 ] and later studied by several authors like: Booth [ 2 ], Booth & Godloza [3], Syam Prasad [15], Satyanarayana, Pradeep kumar, Sreenadh, and

Eswaraiah Setty [13]. If I is an ideal of a nearring N then we denote it by I ⊴ N. For fundamental

definitions and results in near-rings, we refer Pilz [5], Satyanarayana & Syam Prasad [14].

5. Definition: Let M be a Γ-near-ring and γ ∈ Γ. A subset A of M is said to be a γ-ideal of the Γ-near-

ring M if A is an ideal of the near-ring (M, +, *γ).

6. Definition: Let γ ∈ Γ. A γ-ideal I of M is said to be

(i) γ-completely prime if a, b ∈ M, aγb ∈ I ⇒ a ∈ I or b ∈ I.

(ii) γ-completely semiprime if a ∈ M, aγa ∈ I ⇒ a ∈ I.

7. Definitions: (i). A γ-ideal P of a Γ-near-ring M is said to be a γ-prime γ-ideal of M

(with γ ∈ Γ) if AγB ⊆ P for any two γ-ideals A, B of M implies A ⊆ P or B ⊆ P.

(ii).A γ-ideal S of a Γ-near-ring M is said to be a γ-semiprime γ-ideal of M

(with γ ∈ Γ) if AγA ⊆ S for any γ-ideal A of M implies A ⊆ S. 8. Theorem: If S is a semi-prime ideal of M, then the following are equivalent:

(i) If xΓx ⊆ S, then <x>Γ<x> ⊆ S. (ii) S is completely semi-prime ideal of M.

(iii) If xΓy ⊆ S, then <x>Γ<y> ⊆ S. 9. Theorem (Corollary 5.1.10 of Satyanarayana [ 9 ] Let N be a near-ring and A an ideal of N. Then A is completely semiprime ideal if and only if A is the intersection of completely prime ideals of N containing A.

10. Corollary: Let M be a Γ-near-ring, γ ∈ Γ and A be a γ-ideal of M. Then A is γ-completely

semiprime γ-ideal if and only if A is the intersection of γ-completely prime γ-ideals of M containing A. Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and

Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting

him.

REFERENCES [1] Barnes W.E. “On the Gamma-rings of Nobusawa”, Pacific J. Math 18 (1966) 411- 422.

[2] Booth G.L.“A note on Γ-Near-rings”,Stud.Sci.Math.Hunger 23 (1988) 471-475. [3] Booth G.L. and Godloza L. “ On Primeness and Special Radicals of -rings”, Rings and

Radicals, Pitman Research notes in Math series (contains selected lectures presented at the international conference on Rings and Radicals, held at Hebei, Teachers University, Shijazhuang, Chaina, August 1994) pp 123–130.

15 min talk ---------------------------------------------------

Dr. T.V. Pradeep Kumar



Page: 157

[4] Nobusawa “On a Generalization of the Ring theory”, Osaka J. Math. 1 (1964) 81-89 [5] Pilz .G “Near-rings”, North Holland, 1983. [6] Pradeep Kumar T.V “Contributions to Near-ring Theory - III”, Doctoral Dissertation, Acharya

Nagarjuna University, 2006 [7] Ramakotaiah Davuluri “Theory of Near-rings”, Ph.D. Diss., Andhra univ., 1968.

[8] Satyanarayana Bh. "A Note on Γ-rings", Proceedings of the Japan Academy 59-A(1983) 382-83.

[9] Satyanarayana Bh. “Contributions to Near-ring Theory”, Doctoral Dissertation, Acharya Nagarjuna University, 1984.

[10] Satyanarayana Bh. “A Note on Γ-near-rings”, Indian J. Mathematics (B.N. Prasad Birth Centenary commemoration volume) 41(1999) 427-433.

[11] Satyanarayana Bh. “Contributions to Near-ring Theory”, VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-639-22417-7).

[12] Satyanarayana Bh., Pradeep Kumar T.V. and Srinivasa Rao M. “On Prime left ideals in Γ-rings”, Indian J. Pure & Appl. Mathematics 31 (2000) 687-693.

[13] Satyanarayana Bh., Pradeep kumar T.V., Sreenadh S., and Eswaraiah Setty S. “On Completely

Prime and Completely Semi- Prime Ideals in Γ-Near-Rings”, International Journal of Computational Mathematical Ideas Vol. 2, No 1(2010) 22 – 27.

[14] Satyanarayana Bhavanari & Syam Prasad Kuncham “Discrete Mathematics & Graph Theory”, Prentice Hall of India, New Delhi, 2009 (ISBN: 978-81-203-3842-5).

[15] Syam Prasad K. “Contributions to Near-ring Theory II”, Doctoral Dissertation Acharya Nagarjuna University, 2000.

----------------------------------------------------------------------------------------------------------------------------

Some Concepts of

Graph Theory

Applied in Electronics

The main object of this paper is how the Graph Theory will be used in electronics. This paper presents some important terminology used in graphs, digital representation of electronic circuits using matrices, connectivity of graphs, Euler and Hamilton paths. Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and

Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting him.

------------------------------------------------------------------------------------------------------------------------

Paper(Oral Presentation) --------------------------------------- Mr. Bhavanari Mallikarjun

Department of Nanotechnology,

School of Electronics, VIT University,

Vellore, Tamilnadu.

COMING TOGETHER IS A BEGINNING,

KEEPINGTOGETHER IS PROGRESS,

WORKING TOGETHER IS SUCCESS.

----------------------------------------------------------------------------------------------

WHO EVER ACQUIRES KNOWLEDGE AND DOES NOT PRACTICE IT,

RESEMBLES HIM WHO PLOUGHS HIS LAND LEAVES UNSOWRN



EFFECTS OF PERMEABILITY, ELASTICITY ON VISCOUS FLOWS I N A CIRCULAR TUBE

Introduction: In recent years, a great deal of interest has been generated to study the viscous flow of biofluids through elastic tubes because of their important applications in biology, engineering and medicine. Among many discoveries in the area of fluid dynamics, Poiseuille law is considered to be very significant as it describes the relation between the flux and the pressure gradient. According to Poiseuille’s law, the flux of a viscous incompressible fluid through a rigid tube is a linear function of the pressure difference between the ends of the tube. However, in the vascular beds of mammals, the pressure flow relation is always non-linear. This non-linearity has been attributed to the elastic nature of the wall of the duct. In numerous technological and geophysical systems, the presence of a porous medium cannot be ignored. Such media substantially alter the hydrodynamic characteristics of the system and usually for the case of slow, viscous-dominated flows, are simulated using the Darcy linear model, which relates the global impedance effect of the porous matrix fibers to pressure drop across the medium. Metallic foams, ceramics, geomaterials astrophysical debris, cosmic dust, perfused tissue, pulmonary alveoli and foodstuffs are just a few of the common examples of porous media. In this lecture, the effects of elasticity, permeability and yield stress on the flows of Newtonian and non-Newtonian fluids through an elastic tube are discussed.

MATHEMATICAL MODEL FOR A HERSHEL-BULKLEY FLUID FLOW IN AN ELASTIC TUBE Mathematical Formulation and Solution of the Problem

Consider the Poiseuille flow of a Herschel-Bulkley fluid in an elastic tube (see Figure 1) of radius ( ).a z

The flow is axisymmetric: The axisymmetric geometry facilitates the choice of the cylindrical co-ordinate

system ( ), , .R zθ The fluid pressures at the inlet and outlet are ( )1 2 1andp p p< , respectively. The pressure

outside the tube is assumed to be 0p . z denotes the distance along the tube from the inlet end. The pressure

difference at z inside the tube is ( ) 0p z p− . Due to the pressure change at inside and outside of the tube, the

tube may expand or contract due to the elastic property of the wall.

Using the following non-dimensional quantities: 1

0

2 2

0 0 0 0

, , , , , , ,n

n

ar z u q q ar z u q Q a P P

a L U a U a U a L Uπ π µ

+

= = = = = = =

( ) ( ) ( )0

0

0 0 0

, , ,rzrzn n n

TT

U a U a U a

τ ττ τ

µ µ µ= = = (3.1)

where 0a is the radius of the tube in the absence of elasticity, L is the length of the tube and U is the average

velocity of the fluid, the governing equations (after dropping the bars) become

0

1.

nu p

rr r r z

τ ∂ ∂ ∂

− + = − ∂ ∂ ∂ (3.2)

The non-dimensional boundary conditions are:

is finite at 0rz

rτ = , (3.3a)

( )0 .u at r a z= = (3.3b)

Solving equation (3.2) subjected to the conditions (3.3) we obtain the velocity field as

( )

1 1

0 0

2.

1 2 2

k kP P

u a rP k

τ τ+ +

= − − − +

(3.4)

Fig 1:Physical Model

Invited talk --------------------------------------

Prof. S. Sreenadh Department of Mathematics

Sri Venkateswara University Tirupati



Here, 1

.kn

= Using the boundary condition 00 atu

r rr

∂= =

∂, the upper limit of the plug flow region is

obtained as 00

2.r

P

τ= Also by using the condition at

yx ar aτ τ= = (Bird et al. [16]) we obtain

2.aP

a

τ=

Hence 0 0 , 0 1.a

r

a

ττ τ

τ= = < < (3.5)

Using relation (3.5) and taking 0r r= in equation (3.4), we obtain the plug flow velocity as

( )1

1

01 for 0 .2 1

k kk

p

P au r r

kτ

++

= − ≤ ≤ +

(3.6)

The volume flux Q through any cross-section is given by

( )( )

( ) ( )( )( )

0

0

1

3

1

0

1 2 1 21 .

2 1 2 3

kr a

k k

p k

r

kQ u r dr u r dr a p

k k k

τ τ τ+

+

+

− − + += + = −

+ + + ∫ ∫ (3.7)

Equation (3.7) gives the volume flux for a tube of varying radius ( ).a z Now, we use the fact that this

variation occurs due to elasticity of the tube wall: So we assume Poisseuille law for Herschel-Bulkley fluid

flow in elastic tube as ( )0 ,

kdp

Q p pdz

σ

= − −

(3.8)

( ) 3

0where .kp p Faσ +− = Note that when 1 and 0k τ= = equation (3.8) reduce to the assumption of

Rubinow and Keller [17] for a Newtonian fluid flow in an elastic tube. Integrating equation (3.8) with

respect to from 0z z = and using the inlet condition ( ) 10p p= , we obtain

( )( )

1 0

0

11

,p p

kk

p z pQ z p dpσ

−

−′= ∫ (3.9)

where ( ) ( ) 0p p z pσ ′ = − . This equation determines ( )p z implicitly in terms of andQ z . To find Q , we set

1z = and ( ) 21p p= in equation (3.9) to obtain ( )( )

1 0

0

11

1.

p pkk

p pQ p dpσ

−

−′= ∫ (3.10)

The above equation (3.10) can also be written as ( )( )

1 0

0

3 1

01

.np p

n n

p pQ F a p p dp

+−

−= − ∫ (3.11)

Equation (3.11) can be solved if we know the form of the function ( )0a p p− . If the stress or tension T(a) in

the tube wall is known as a function of ' 'a then ( )a p can be found using the equilibrium condition

( )0

T ap p

a= − . (3.12)

Roach & Burton [18] determined the static pressure-volume relation of a 4 cm long piece of the human external iliac artery, and converted it into a tension versus length curve. Using least squares method Rubinow and Keller [17] gave the following equation:

( ) ( ) ( )5

1 21 1 ,T a t a t a= − + − where 1 213 and 300t t= = . (3.13)

By substituting (3.13) into (3.12) and on simplification we obtain

3 2122 2

14 15 20 10 .

tdp t a a a da

a a

= + − + − +

(3.14)

Using (3.14), (3.11) can be written as 1 0

2 0

3 1 3 2122 2

14 15 20 10 .

p pn n n

p p

tQ F a t a a a da

a a

−+

−

= + − + − +

∫ (3.15)

On further simplification, we get ( ) ( )( )1 2 ,Q F g a g a= −



where ( ) ( )

( ) ( )

11

1 1 0 2 2 01

12 1 2

11 , , ,

1 112 32 1

n

n

nF a a p p a a p p

n nn

τ ττ

+

− + + − = − = − = − + ++

( )

1

3 3 5 3 4 3 3 3 2 3

1 2

4 15 20 10.

3 3 5 3 4 3 3 3 2 3

n n n n n n na a a a a ag a t t

n n n n n n

+ + + + = + − + − +

+ + + +

We note that 1 2anda a are determined by solving equation (3.9) with 1p p= and 2p p= , respectively. We

observe that equation (3.15) reduce to the corresponding results of Rubinow and Keller [17] for the flow of

Newtonian fluid ( )1, 0n τ= = in an elastic tube.

The variation of flux with radius for different fluids.

The variation of flux with radius for different values of the elastic parameter t2 when t1 = 13, n = 3 & τ = 0.2.

HAGEN POISEUILLE FLOW THROUGH AN ELASTIC CIRCULAR PIPE HAVING INTERNAL CIRCULAR POROUS LINING

Mathematical Formulation and Solution: Consider the Poiseuille flow of a viscous fluid through a long, straight pipe with porous lining. The flow region, which is surrounded by the porous bed, is governed by Navier-Stokes equations. The flow in the porous lining outside the nominal surface is governed by the Darcy’s law. A set of non dimensional variables be introduced as follows :

*

av

ww

u= *

0

rr

a= *

2

av

pp

uρ=

*

0

aa

a=

*

2

0 av

TT

a uρ= (1)

Where uav is the average velocity, a0 is the radius of the inelastic tube, ρ is the density, L is the length of the tube. Now the non dimensional governing equations are (dropping the asterisks):

0w

z

∂=

∂ (6)

0p

r

∂=

∂ (7)

0p

θ

∂=

∂ (8)

Fig1 : Physical Model



Where , where is the Reynolds number.

The boundary conditions are : (10)

(11)

Where , is the Darcy velocity (12)

Where , σ - permeability parameter α - Slip parameter ε - Thickness of the porous lining

Using (10) and (11) in (9) we have (13)

Now the flux is : (14)

Using (13) in (14) we have

(15)

This is the flux for the flow of a viscous fluid in as tube in the absence of the elastic nature of the tube.

It is assumed that the flux F is related to pressure gradient, by the relation

(16)

Where σ1, Known as the conductivity of the tube , will be a function of the pressure difference, ,

that is (17)

Integrating (15) with respect to z from z=0 and using the inlet condition p(0)=p1, we obtain

Where

This equation determines p(z) implicitly in terms of Q and z . To find Q, we set z=1 and p(1)=p2 in (18) to obtain

In the special case of a circular porous lining with radius (a-ε), a function of *'P , the conductivity is given

by (20)

Using (20) in (19) we obtain (21)

The stress or tension T(a) in the tube wall is known as a function of ‘a’ , then a(p’) can be found using the

equilibrium condition (22)

Then , Rubinow and Keller [56] used the least squares method to fit this curve. The non dimensional form is : (23)

Using the relation (23) in (22) we have, (24)

Differentiating this with respect to ‘a’ we have,

(25)



Now, using (25) in (21), we get the flux in an elastic tube as :

[ ]0 1 2Re ( ) ( )F k a k a= −

(26)

Where

(27) 3 2 2 2 3 4 3

2

1 3 3 1 3 2( ) ( ) ( ) ( ) log ( )

48 4 4 2 8 4 2 4 4 16 4

a ag a a a

a

ε ε ε ε ε ε ε

ασ ασ σ ασ ασ= + − + − + + − − − (28)

8 7 2 6

2

2 2 53

2

4 3 2 2 43

2

4 3 2 23

1 15 3 3 15 15 2 5( ) ( )

32 16 7 2 4 4 4 4 6

3 45 9 5 15 55

8 4 2 8 5

45 15 9 6 10 104

16 4 2 4 4 4 4

3 15 15 156

16 4 4

a a ah a

a

a

ε ε εε

ασ ασ ασ σ

ε ε εε ε

ασ ασ ασ σ

ε ε ε ε ε εε

ασ ασ ασ σ ασ

ε ε ε εε

ασ ασ

= + − − + − + − + +

+ − − + + + − −

+ − + − + + − + −

+ − + − + − −3

2

4 3 2 23

4 3 2 2 3 4 33

2

3 5 1

2 16 3

5 15 9 5 1( 4 )

8 2 4 4 4 2

5 3 3 1 3 2( ) ( ) log ( )

8 2 4 8 2 4 4 16 4

a

a

a aa

εε

ασ σ

ε ε ε ε εε

ασ ασ ασ ασ

ε ε ε ε ε ε ε εε ε

ασ ασ σ ασ ασ

+ − +

+ − + − + − +

+ + − + − + + + − − −

(29)

Effect of α on flux with varying radius . The relationship between the flux and radius for different σ.

Conclusions 1) The flux decreases with increasing values of power law index , yield stress and permeability parameter; 2) The flux increases with increasing values of the elastic parameters and Slip parameter; 3) The flux increases with the decreasing values of the Porous thickness; and 4) The inlet and outlet pressures have opposite effects on the flux.

Acknowledgements: The author thank Dr A V Vijaya Kumari (Organizing Secretary), and

Prof. Dr Bhavanari Satyanarayana (Academic Secretary) of the National Seminar, for inviting him.

TRUTH AND LOVE ARE TWO SIDES OF THE SAME COIN

Invitation

NATIONAL SEMINAR ON

“PRESENT TRENDS IN ALGEBRA AND ITS APPLICATIONS”,

(U.G.C – SPONSORED)

11th and 12th July, 2011,

ORGANIZING COMMITTEE OF THE SEMINAR JMJ College for Women, Tenali, Guntur Dt., AP.

INAUGURAL FUNCTION At 10 AM on Monday, July 11th, 2011

Venue: J.M.J. Seminar Hall

Chief Guest:

Prof. Dr K.L.N. Swamy, FAPAS Former Professor, Andhra University, Waltair.

Guests of honors:

Prof. Dr. I.H. Nagaraja Rao Director, GVP College for PG Courses, Waltair.

Prof. Dr L. Nagamuni Reddy

Principal, JBW Inst. Tech., Tirupathi.

Chair Person: Prof. Dr Bhavanari Satyanarayana,

AP Scientist Awardee, Fellow, AP Akademy of Sciences

TEACHER SESSION 2 pm to 3.30 pm: Discussion and Problem Solving Session

(for School Teachers) Chair Person: Sri Pokala Chandar, Executive Engineer, Warangal. Guests of Honor: 1. Sri R. Jesupadam,

Director, A.P. Government Text Book Press, Hyd. 2. Mrs. P. Parvathi,

RJD & DEO, School Education, Guntur Division. Organized by: Mr. Ch. V. Narasimha Rao, Director, AIMEd, Vijayawada. Speakers: 1. Sri. G.V. Chalapathi Rao, Rtd. HM., Hon.President, Teachers Association, Narasaraopet 2. Mr. Khasim, Guntur

3. Mrs. T. Madhavi Latha, M.Sc., M.Phil., Vice-Principal, APRSW School, Jangareddygudem.

VALEDICTORY FUNCTION TUESDAY, July 12th, 2011 (at 3.45 pm to 5 pm)

Venue: J.M.J. Seminar Hall

Chief Guest: Sri R. Jesupadam, Director, A.P. Government Text Book Press, Hyderabad.

Guests of Honor:

Rev. Dr Sr Mary Thomas, Correspondent, JMJ College. Rev. Dr Sr Jacintha, Principal, JMJ College.

Rev. Dr Sr K. Mareelu, Vice-Principal, JMJ College.

Response by the Participants.

Concluding Remarks: Prof. Dr Bhavanari Satyanarayana, Academic Secr. Vote of thanks: Dr. A.V. Vijaya Kumari, Organizing Secretary.

NATIONAL SEMINAR ON “PRESENT TRENDS IN ALGEBRA AND ITS APPLICATIONS”,

(U.G.C – SPONSORED) 11th

and 12th

July, 2011, JMJ College for Women, Tenali.

PROGRAM on Monday, July 11, 2011

08:30 to 10:00 am Registration 10 am to 11:15 am INAUGARAL FUNCTION 11:15AM TO 11:30 am Tea Break

Technical Session – 1 Chair Person: Prof. Dr I.H.Nagaraja Rao,

Director, GVP College for PG Courses, Waltair. 11:30 am to 12 noon Key Note Speaker : Prof. Dr. Bhavanari Satyanarayana Topic : Gamma Near- Rings

12:00 noon to 12:30 pm Invited talk by: Prof. Dr L . Nagamuni Reddy Principal, JBW Inst. of Tech., Tirupathi 12:30 pm to 1:00 pm Invited talk by: Prof. Dr K. Suvarna Sri Krishna Devaraya University, Ananthapur

Technical Session – 2 Chair Person : Prof. L. Nagamuni Reddy Tirupathi

02:00 pm to 02:30 pm : Invited talk by Dr. K. Shobha Latha Sri Krishna Devaraya University, Ananthapur

15 minute talks 02:30 pm to 03:30 pm 1) Satyasri Bhavanari, Zhejiang Univ., China. 2) Mohiddin Shaw Shaik (ANU) 3) Jagadeesha Bhat, Manipal University 4) Dr D. Nagaraju, Hindustan Univ., Chennai 3.30 to 3.45 pm Tea Break

Technical Section – 3 Chair Person : Prof. K. Suvarna, Ananthapur. 03:45 pm to 04:15 pm Invited talk by: Dr. L. Madhavi, Yogi Vemana Univ., Kadapa. 04:15 pm to 04:45pm Invited talk by: Dr. K.N.S. Kasi Viswanadham, NIT, Warangal. 04:45 pm to 05:30pm Paper Reading Session

NATIONAL SEMINAR ON “PRESENT TRENDS IN ALGEBRA AND ITS APPLICATIONS”,

(U.G.C – SPONSORED) 11th

and 12th

July, 2011, JMJ College for Women, Tenali.

PROGRAM on Tuesday, July 12, 2011

Technical section -4 Chair Person: Prof. Dr S. Sreenadh, SVU, Tirupathi.

10:00 am to 10:30 am: Invited talk by: Prof. Dr K.L.N. Swamy, Waltair 10:30 am to 11:00 am : Invited talk by: Dr Nanaji Rao Galla , AU, Waltair. 11:00am to 11:15am : Invited talk by: Dr T.V. Pradeep Kumar, ANU-Coll.E&T. 11:15 am to 11:30 am : Tea Break.

Technical Section – 5 Chair Person: Prof. Dr K.L.N. Swamy, Waltair

11:30 pm to 12:00 noon : Prof. Dr V. Sitaramaiah, Pondicherry. 12:00 noon to 12:30 pm : Prof. Dr Sreenadh, SVU, Tirupathi. 12:30 pm to 12:50 pm : Prof. Dr Mariya Doss, Mangalore 12:50 pm to 01:00 pm : Short Talk by Sk.Shakeera, Waitair. 01: 00 pm to 02:00 pm : LUNCH BREAK

Technical Section – 6 (Parallel Sessions) 2 pm to 3.30 pm: Discussion and Problem Solving Session (for School Teachers)

Chair Person: Sri Pokala Chandar, Executive Engineer, Warangal. Guests of Honor: Sri R. Jesupadam, Director, A.P. Government Text Book Press, Hyd.

Mrs. P. Parvathi, RJD, School Education, Guntur Division.

Organized by: Mr. Ch. V. Narasimha Rao, Director, AIMEd, Vijayawada.

Speakers: Sri. G.V. Chalapathi Rao, Rtd. HM., Hon.President, Teachers Association, Narasaraopet

Mr. Khasim, Guntur

Mrs. T. Madhavi Latha, M.Sc., M.Phil., Vice-Principal, APRSW School, Jangareddygudem.

Technical Section : 7 Chair Person: Prof. T. Srinivas, KU, Warangal.

02 pm to 02:30 pm Invited talk by: Prof. K. Rama Krishna Prasad, i/c Principal , SVU College of Sciences, Tirupathi. 2.30 pm to 3.30 pm Paper Reading Session 3.30 pm to 3.45 pm Tea Break 3.45 pm to 5 pm VALIDICTORY FUNCTION & DISTRIBUTION OF CERTIFICATES FOR PARTICIPANTS.

Greetings to Prof. Dr Bhavanari Satyanarayana on getting the Honor: one of "TOP 100 PROFESSIONALS - 2011" (from IBC, Cambridge, England). ------------------------------------ Monday, 4 July, 2011 9:21 AM Respected Professor, Congratulations Sir. I am very happy to see that. Sincerely yours

Dr Venku Naidu, ISI, Bangalore. ----------------------------------- Monday, 4 July, 2011 9:04 AM Congrats sir keep smile for ever, Murali Krishna Hyderabad ------------------------------------ Monday, 4 July, 2011 8:54 AM Good morning sir, Congrajulations for selected as "Top 100 professionals of 2011" . Yours sincierly, Jagadeesha B, Manipal University, Karnataka, India. ------------------------------------- Monday, 4 July, 2011 3:41 PM Dear Sir, congrats for the honor (top 100 professionals in 2011) that you received from the International Biographical centre, England. I hope you will receive more (International) honors in the years to come by. with regards and best wishes,

Yours, Dr Dasari Nagaraju Hindustan University,Chennai - 603 103. -------------------------------------------- Monday, 4 July, 2011 6:27 PM congratulations. let it be a motivation for all teachers.we will forward among the students.with prayers, Narayanan, AYURVEDIC TRUST, Coimbatore. ------------------------------------ Monday, 4 July, 2011 4:31 PM From: "Dr. Bala subramanian" <[email protected]> Congradulations Profesor for attaing such prestigeour Prize. Yours Sincerely, S. Balasubramanian

------------------------------------- Monday, 4 July, 2011 1:14 AM Dear Sir,Hearty congratulations and wish you many more awards in the future. I am very happy to know that you have received the award of one of the top 100 professionals in

the world. Dr Babushri Srinivas, Institute of Ring Theory, USA. ------------------------------------- Tuesday, 5 July, 2011 8:24 PM Respected Professor, "HEART CONGRATULATIONS FOR YOUR SUCCESS IN OCCUPYING THE PLACE IN TOP 100 PROFESSIONALS LIST IN THE INTERNATIONAL LEVEL" With regards, Dr NV Ramana murty, Dept. of Mathematics, Andhra Loyola College, Vijayawada-520008

Greetings to Prof. Dr Bhavanari Satyanarayana on getting the Honor: one of "TOP 100 PROFESSIONALS - 2011" (from IBC, Cambridge, England). Wednesday, 6 July, 2011 9:24 PM Dear Dr. Satyanarayana:

Congratulations on your Top 100 Professionals -2011 honor! It is quite a distinguished honor and one of which you can always be most proud! Sincerely, Carol A. Mitchell, Editor-in-Chief, American Biographical Institute, 5126 Bur Oak Circle, Raleigh, NC 27612

------------------------------------

Monday, 4 July, 2011 7:36 AM Dear Dr Satyanarayana Congrats.

This is a great recognition of your acievement. I have seen the news in the download. I admire your getting applauds from Our CM. May God Bless you to win further honours in the academic endeavours

…………….PV Arunachalam, Former Founder Vice-Chancellor, Dravidian University, Kuppam, A.P., India.

------------------------------------ Monday, 4 July, 2011 2:30 AM

Congratulations. Sounds like a great honor, making all of us proud. Regards, Sreekanth Malladi, USA ------------------------------------- Monday, 4 July, 2011 11:50 AM Congratulations Professor Satyanarayana That is a great news

Regards, Aastha Sarma, Taylor and Francis, New Delhi ------------------------------------- Monday, 4 July, 2011 10:42 AM congratulations for continuous awards, all the best Beldi Sridhar, Former President, AP State AVOPA, Hyderabad, Beldi & Associates, Charted Accountant.

July 2011 Dear prof.Dr.Bhavanari Satyanarayana:

Kindly accept my hearty congrats for the International award for your meritorious work. I am glad to say I have an INTERNATIONALLY famous friend like you... accept my wishes and .I ask God to bless you with BEST HEALTH and WEALTH and

HAPPINESS. with regards, Prof. Dr Mariadoss, Mangalore ------------------------------------- Monday, 4 July, 2011 9:36 AM, Respected sir, We (our family members) are delighted on seeing your mail that you have recieved another international award. we saw the pictures of you with the CM Mr. kiram kumar reddy. Really awesome. Yours truly, Dr.A.V.VIJAYA KUMARI, Head of the Department of Mathematics, JMJ College,

Greetings to Prof. Dr Bhavanari Satyanarayana on getting the Honor: one of "TOP 100 PROFESSIONALS

- 2011" (from IBC, Cambridge, England).

Hi sir i am happy congrates you and feel proud of your student. Now i

am course co-ordinater and recently my paper was presented in

symbiasis operation management

Topic: Role of quantitative analysis in Operations

B.V.MANIKANTA.

HEAD OF THE DEPARTMENT IN MATHEMATICS, & STATISTICS

DEPT.OF B.B.SC,ACHS, ASMARA, ERITRIA

-------------------------------------

Tuesday, 5 July, 2011 8:19 AM,

Well done and congratulations! The fruit of

your labor is sweet, and I must say you deserve

it. I know you are the greatest professor. U. Surya Kameswari, Department of Computer Science, ANU. ------------------------------------------------------

Tuesday, 5 July, 2011 3:05 AM dear satyanarayana, congratulations for the honour bestowed

on you, hope that you will be achieving more honours like

this, best wishes, -radhakrishna, Rtd. Professor, Kakatiya

University, Warangal.

------------------------------------

Wednesday, 6 July, 2011 12:35 PM

Dear Sir ,

May each day in your life be the New Show and new achievements . Keep

up , congratulations, With regards. Jayashree., Studied in Guawahathi University

----------------------------------------


Dear professor,

CONGRATULATIONS FOR YOUR GREAT ACHIEVEMENT

THIS IS NOT ONLY FOR U BUT OUR ENTIRE MATHEMATICS

FAMILY AND ALL OVER INDIA.HOPE THE SAME THING IN UR

FUTURE, WITH REGARDS, PROF.I.H.NAGARAJARAO, Waltair.

------------------------------


Dear sir, congratulations for ur greatest achievement .This is inspiring news

for young scholors like me. But our beloved propfessor I.H.N.RAO sir

also felt very happy becaue u are getting awards continiously for

every quarter month.This is result for ur hardwork sir. Once again congrts. yours

sincerly, K.V.S SARMA, DEPARTMENT OF MATHEMATICS, REGENCY

INSTITUTE OF TECHNOLOGY, YANAM - 533 464, U.T.OF PONDICHERRY

----------------------------------------

Dr. BHAVANARI SCHOLARS / STUDENTS ASSOCIATION

Dr Kuncham Syam Prasad: He awarded Gold Medal for his first rank in M. Sc., Mathematics in 1994. He is a recipient of CSIR-Senior Research Fellowship. He got awarded M. Phil., (Graph Theory) in 1998 and Ph.D., (Algebra - Nearrings) in 2000 under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee). He published Nineteen research papers in reputed journals and presented research papers in thirteen National Conferences and six International

Conferences in which five of them were outside India: U.S.A (1999), Germany (2003), Taiwan (2005), Ukraine (2006), Austria (2007), Bankok (2008), Indonesia (2009). He also visited the Hungarian Academy of Sciences, Hungary for joint research work with Dr Bhavanari Satyanarayana (2003) and the National University of Singapore (2005) for Scientific Discussions. He authored nine books (UG/PG level). He is also a recipient of Best Research Paper Prize for the year 2000 by the Indian Mathematical Society for his research in Algebra. He received INSA Visiting Fellowship Award (2004) for the collaborative Research Work. Presently working as Associate Professor of Mathematics, Manipal University, Karnataka, India. E-mail: [email protected]

Dr. Tumurukota Venkata Pradeep Kumar: He got awarded M.Phil., (ΓΓΓΓ-ring theory) and Ph. D., (Near-ring Theory) under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee). He published five research papers in Indian and Abroad International Journals. He attended three National Conferences and one International Conference. At present he is working as Assistant Professor in ANU College of Engineering & Technology.

Dr. Dasari Nagaraju: He completed his Ph. D., (Ring Theory) He is a Project Associate in UGC-Major Research

Project (2004-2007) under the Principal Investigatorship of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee). He published eight research papers. He Worked in Rajiv Gandhi University (AP), Periyar Maniammai University (Tanjavur). Presently working in Hindusthan University, Chennai.

Dr. Kedukodi Babushri Srinivas: He is an Associate Professor in Mathematics, Manipal University, Karnataka. His educational qualifications are DOEACC ‘O’ LEVEL from DOEACC Society, Department of Electronics, Govt.

India, M. Sc., and P.G.D.C.A. from Goa University. He qualified in the Joint CSIR-UGC JRF(JRF-NET), Maharashra State Eligibility Test (SET) for Lectuership (accredited by UGC) and GATE in Mathematics. He got Ph.D., (Fuzzy and Graph Theoritic aspects of Near-rings, 2009) under the

guidance of Dr Kuncham Syam Prasad and Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee). He attended a number of workshops/Seminars in Mathematics and pubished four research papers in

international Journals like: Soft Computing, Communications in Algebra. He presented papers/delivered Lectures in International Conferences held at Ukraine (2006), Austria (2007), Bankok (2008), Indonesia (2009). He is now visiting Institute of Ring Theory, USA. E-mail: [email protected]

Dr Arava Venkata Vijaya Kumari: She completed her M.Sc., (Mathematics) from ANU with third rank. She got Awarded her Ph.D., (Nearrings) under the guidance of Dr Bhavanari Satyanrayana (AP SCIENTIST AWARDEE, by DST, New Delhi, 2009, Fellow-AP Akademy of Sciences, 2010). She published 3 research papers in National and International Journals. Presently Heading the Department of Mathematics, JMJ College, Tenali.

Mr. M. B. V. Lokeswara Rao: He completed his M. Sc., (Mathematics) from ANU with third rank. He got awarded M. Phil., (Matrix Near-rings) under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee) with A grade. He is an elected General Secretary of “Association for Improvement of Maths Education (AIMEd., Vijayawada)”. He published one research paper in Matrix Near-rings.

Mr. Sk. Mohiddin Shaw: He completed his M. Phil., (Module Theory) under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee). He visited Institute of Mathematical Sciences (Chennai), IIT (Chennai), ISI (Calcutta), IIT (Guwahati) and Burdwan University (West Bengal) for his research purpose. He attended eight Conferences/Seminars/ Workshops. He worked as a faculty in the ANU P.G Centre at Ongole. He published four research papers in Ring Theory.

Mr. J. L. Ramprasad: Awarded with Kavuru Gold Medal for College first in B. Sc., Course and with JCC Gold Medal for Town first. Qualified in GATE-2001 Examination with Percentile score of 85.73. Awarded with M. Phil., (Module Theory) in May 2005 under the guidance of Sri. Dr Bhavanari Satyanarayana

(AP SCIENTIST Awardee). He authored two books at PG level. He published a research paper in USA. Presently working as a Lecturer in P.G. Department of P. B. Siddhartha College, Vijayawada. E-mail: [email protected]

Mr. K. S. Balamurugan: He got First Rank in B. Sc., and Second Rank in M. Sc., course. He awarded with M.

Phil., (Ring Theory) under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee) in 2006. He is

working as Sr. Lecturer in RVR & JC College of Engineering. He published one research paper in Ring Theory.

Mrs. T. Madhavi Latha: Her educational qualifications are B. Sc., B. Ed., M. Sc., Ed., M. Phil., PGDCA and IELTS: 7.5. She was a NCERT scholarship holder during 1992-94. She got Visista Acarya Puraskar Award in 1997 by Amalapuram Educational Society. She was the author of 3 books. She attended various National and International seminars both on Education and Mathematics. She worked as a resource person for various academic programmes. Presently she is working as a PGT in APSWR JC.

Dr. BHAVANARI Mrs. Sk. Shakeera: She got M. Phil., degree (2007) in

Satyanarayana (AP SCIENTIST Awardee)

Mr. D. Srinivasulu: He got his M. Phil., degree (

Satyanarayana (AP SCIENTIST Awardee)

Brief Biodata of Prof. Dr

• Got 2nd Rank securing 75% of marks in

• Got 1st Rank in Certificate Course in Statistics, ANU.

• Undergone Certificate Courses in Electronic Computers (i). Calcutta (1986); and (ii) Annamalai University

• Awarded CSIR-JRF (1980-82), OFFICER (1988), INSA Visiting Fellowship Award 2005, AP State Scientist-2009 AwardSciences (2010), Shiksha Rattan Puraskar

• International Awards: Glory of India & International Achievers’ Award• One of the “Top 100 Professionals • Awarded Five Ph.D., degrees and

• One Research Student (Dr. Kuncham Syam Prasad, working in Manipal Academy of Higher Education, Deemed University) got the National Award: IMS Award

• Life member of Eight Mathematics Associations• Elected President (2005-2007, 2007

(AIMEd.,), Vijayawada.

• Director of the National Seminar on Algebra andANU, Jan 05-06, 2006.

• Published 29 General Articles

• Authored / Edited 36 books (for B.Com. / M.A. (Eco.) / B.C.A / M.Sc.(Maths) (including a book on Discrete Mathematics & GT, published by Prentice Hall of India, New Delhi))DR MULLER, GERMANY.

• Honorary Editor for the two Mathematical Periodicals (in Telugu Language): ““Ganitha Vahini” Published from Andhra Pradesh.

• Member Secretary and Managing Editor& Information Technology”, Acharya Nagarjuna University.

• Got Paul Erdos No. 3.

• Attended 13 International Conferences• Principal Investigator of 3 Major Research Projects• Published 65 research papers

Journals.

• Introduced the algebraic system “

• Visiting Fellow at Tata Institute of Fundamental Research, Bombay, May 1989.

• Visiting Professor at Walter Sisulu University• Visited Austria (1988), Hongkong

(2005), Singapore (2005), Hungarydeliver lectures / Collaborative research work).

• Selected Scientist (By Hungarian Academy of SciencesNew Delhi, 2003) to work withAcademy of Sciences) during June 05published with the co-authorship ofNetherlands, 2005, pp.293-299

• Selected Sr. Scientist (By Hungarian Academy of Sciences, Budapest; Academy, New Delhi), Aug. 16

Name : Dr Bhavanari SatyanarayanaDesignation : Professor Date of Birth : 12Place of Birth : Madugula (a Village in PalnaduMother : B. Ansuryamma (Late)Father : B. Ramakotaiah (Retired Teacher) (Late)Elementary School Edu: : Reddypalem (Near pedakodamagundla), Adigoppula, Madugula of Palnadu.High School Education : St. Joseph’s Boys High School, Rentachintala, Guntur (Dt)Inter + B.Sc. : S.S.N. College, Narasaraopet, Guntur (Dt)M.Sc. + Ph.D : Acharya Nag Ph: 0863E-mail : [email protected]

Dr. BHAVANARI SCHOLARS / STUDENTS ASSOCIATION

She got M. Phil., degree (2007) in ΓΓΓΓ-ring theory under the guidance of Dr Bhavanari

(AP SCIENTIST Awardee).

: He got his M. Phil., degree (Graph Theory) under the guidance of Dr Bhavanari

(AP SCIENTIST Awardee).

Dr SATYANARAYANA BHAVANARI, ANU

securing 75% of marks in M.Sc., Maths (1977-79), ANU.

in Certificate Course in Statistics, ANU.

Undergone Certificate Courses in Electronic Computers (i). Indian Statistical Institute, Annamalai University.

82), CSIR-SRF (1982-85), UGC-Research AssociateshipINSA Visiting Fellowship Award 2005, and ANU – Best Research Paper Award2009 Award (by DST New Delhi & APCOST Hyderabad), Fellow iksha Rattan Puraskar (2011)

Glory of India & International Achievers’ Award (Thailand, March 26, 2011)

Top 100 Professionals – 2011” (by IBC, Cambridge, England)

and Ten M.Phil., degrees under his supervision.

One Research Student (Dr. Kuncham Syam Prasad, working in Manipal Academy of Higher Education, National Award: IMS Award - 2000) for best research paper in Algebra.

Life member of Eight Mathematics Associations.

2007, 2007-2009) of the Association for Improvement of Maths Education

National Seminar on Algebra and its Applications, organized by the Department of Maths,

in periodicals.

(for B.Com. / M.A. (Eco.) / B.C.A / M.Sc.(Maths) (including a book on Discrete GT, published by Prentice Hall of India, New Delhi)), Five books published by VDM VERLAG

for the two Mathematical Periodicals (in Telugu Language): “” Published from Andhra Pradesh.

Managing Editor of “Acharya Nagarjuna International Journal of Mathematics ”, Acharya Nagarjuna University.

. Collaborative Distance with Einstein = 5International Conferences (INCLUDING ICM-2010) and 24 National Conferences

Major Research Projects (Sponsored by U G C, New Delhi).

research papers (in Algebra / Fuzzy Algebra / Graph Theory) in National an

Introduced the algebraic system “Gamma near-ring” in 1984.

at Tata Institute of Fundamental Research, Bombay, May 1989.

Walter Sisulu University (WSU), Umtata, South Africa, March 26

Hongkong (1990), South Africa (1997), Germany (2003) Hungary (2005), Ukraine (2006), and South Africa (2007) on official wo

deliver lectures / Collaborative research work).

Hungarian Academy of Sciences, Budapest; and University Grants CommissionNew Delhi, 2003) to work with Prof. Richard Wiegandt at A.Renyi Institute of Mathematics (

during June 05- Sept. 05, 2003. A research paper on Radical theory of Nearauthorship of Prof. Wiegandt (in the Book: Nearrings and Near

299).

Hungarian Academy of Sciences, Budapest; and , New Delhi), Aug. 16 – Sept. 05, 2005.

Dr Bhavanari Satyanarayana Professor 12-11-1957 Madugula (a Village in Palnadu region) B. Ansuryamma (Late) B. Ramakotaiah (Retired Teacher) (Late) Reddypalem (Near pedakodamagundla), Adigoppula, Madugula of Palnadu.St. Joseph’s Boys High School, Rentachintala, Guntur (Dt)S.S.N. College, Narasaraopet, Guntur (Dt) Acharya Nagarjuna University, Nagarjuna Nagar Ph: 0863-2232138 (R); Cell: 98480 59722.

[email protected], [email protected]

SCHOLARS / STUDENTS ASSOCIATION

under the guidance of Dr Bhavanari

) under the guidance of Dr Bhavanari

, ANU

Indian Statistical Institute,

Research Associateship (1985), CSIR-POOL Best Research Paper Award-2006,

Fellow – AP Akademi of

(Thailand, March 26, 2011)

One Research Student (Dr. Kuncham Syam Prasad, working in Manipal Academy of Higher Education, ) for best research paper in Algebra.

2009) of the Association for Improvement of Maths Education

, organized by the Department of Maths,

(for B.Com. / M.A. (Eco.) / B.C.A / M.Sc.(Maths) (including a book on Discrete books published by VDM VERLAG

for the two Mathematical Periodicals (in Telugu Language): “Ganitha Chandrica” &

Acharya Nagarjuna International Journal of Mathematics

Collaborative Distance with Einstein = 5

National Conferences.

(Sponsored by U G C, New Delhi).

) in National and International

(WSU), Umtata, South Africa, March 26 – April 10, 2007.

(2003) Hungary (2003), Taiwan (2007) on official works (to

University Grants Commission, at A.Renyi Institute of Mathematics (Hungarian

Radical theory of Near-rings was Nearrings and Near-fields, Springer,

Indian National Science

Reddypalem (Near pedakodamagundla), Adigoppula, Madugula of Palnadu. St. Joseph’s Boys High School, Rentachintala, Guntur (Dt)

[email protected]

Bio-Data of Editors

Prof. Dr Bhavanari Satyanarayana have 28 yrs Teaching ex-

perience in Acharya Nagarjuna Univ. Authored 36 books

(including a book by Prentice Hall of India, New Delhi, and Five

books by VDM Verlag Dr Muller, Germany). Published 65 Re-

search papers (Algebra/Fuzzy Algebra/Graph Theory) in Inter-

national Journals. He is a Member of several Editorial Boards,

Mathematical Journals. He is an AP SCIENTIST–2009 Awardee,

a Fellow AP Akademi of Sciences. He received Shiksha Rattan

Puraskar Award (IIFS, New Delhi, 2011), Glory of India Award

and International Achievers Award (Indo-Thai Friendship Ban-

quet, Thailand, 2011). Top 100 Professionals - 2011

(International Biographical Centre, Cambridge, England). Col-

laborative Distance with Einstein is 5. Got Paul Erdos No. 3,

Scientist UGC-HAS (Hungarian Academy of Sciences), 2003.

Sr Scientist INSA–HAS 2005. Principal Investigator of 3

MAJOR Research Projects (UGC). He Introduced an algebraic

system “Gamma near-ring”. Awarded Five Ph.D., and 10

M.Phil., Degrees. Visiting Professor, Walter Sisulu University,

South Africa (2011). Visited Austria (1988), Hongkong (1990),

South Africa (1997), Germany (2003) Hungary (2003),

Taiwan (2005), Singapore (2005), Hungary (2005), Ukraine

(2006), South Africa (2007), and Thailand (2008, 2011) on of-

ficial works (to deliver lectures/Collaborative research work).

Dr Arava Venkata Vijaya Kumari has 29 years of Teaching

experience in JMJ College for Women, She completed her M.Sc.,

(Mathematics) from ANU with third rank. She got Awarded her

Ph.D., (Nearrings) under the guidance of Dr Bhavanari

Satyanrayana (AP SCIENTIST AWARDEE, by DST, New Delhi, 2009,

Fellow-AP Akademy of Sciences, 2010). She published 3 research

papers in National and International Journals. Presently Heading

the Department of Mathematics, JMJ College, Tenali (Andhra

Pradesh).

Mr. Sk. Mohiddin Shaw completed his M. Phil., (Module Theory)

under the guidance of Dr Bhavanari Satyanarayana (AP SCIEN-

TIST Awardee). He visited Institute of Mathematical Sciences

(Chennai), IIT (Chennai), ISI (Calcutta), IIT (Guwahati) and

Burdwan University (West Bengal) for his research purpose. He

attended Nine Conferences/Seminars/ Workshops. He worked as

a faculty in the ANU P.G Centre at Ongole for 5 years. He pub-

lished Six research papers in National and International Jour-

nals. He is the co-author of the book “Fuzzy Dimension of Mod-

ules over rings” published by VDM Verlag Dr Muller, Germany.

Thailand, March 26, 2011

proceedings of the national seminar on present trends in algebra and its applications

Documents