Research ArticleSome Properties of Multiplicative119867V-Rings of Polynomials overMultiplicative Hyperrings

Utpal Dasgupta

Department of Mathematics Sree Chaitanya College Habra-Prafullanagar 24 Parganas (North) West Bengal 743268 India

Correspondence should be addressed to Utpal Dasgupta dasgupta utpalyahoocoin

Received 22 May 2014 Accepted 30 September 2014 Published 27 October 2014

Academic Editor Andrei V Kelarev

Copyright copy 2014 Utpal Dasgupta This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The set of all polynomials119877 [119909] over a multiplicative hyperring (119877 + sdot) form a commutative group with respect to the component-wise addition (+) of the polynomials For polynomials 119891 119892 in 119877 [119909] 119891 lowast 119892 is a set of polynomials whose (119896 + 1)th components(119896 isin N cup 0) are chosen from the set sum

119894+119895=119896119886119894sdot 119887119895 where 119886

119894and 119887

119895are the (119894 + 1)th and the (119895 + 1)th components of 119891 and 119892

respectively Amultiplicative hyperring is polynomially structured if the hyperstructure (119877 [119909] + lowast) is amultiplicative119867V-ringThepurpose of the paper is to study the properties of the multiplicative119867V-ring (119877 [119909] + lowast) corresponding to those of a polynomiallystructured multiplicative hyperring 119877

1 Introduction

119867V-structures [1] are introduced by Vougiouklis at theFourth AHA congress in the year of 1990 Since then thestudy of119867V-structure theory has been approached in severaldirections by many researchers (see [2ndash5]) The essence ofthe notion of 119867V-structures is to generalize the well-knownalgebraic hyperstructures (such as hypergroup hyperringand hypermodule) simply by replacing some or all axiomsof the respective hyperstructures by the corresponding weakaxiomsThe119867V-structure of our initial concern is multiplica-tive119867V-ring studied in [6 7] which is a commutative group(119877 +) along with a hyperoperation ∘ such that (i) (119877 ∘) is an119867V-semigroup [3 8] (ie a hyperstructure in which ∘ is weakassociative in the sense that 119886 ∘ (119887 ∘ 119888) cap (119886 ∘ 119887) ∘ 119888 = 120601 for all119886 119887 119888 isin 119877) and (ii) ∘ is weak distributivewith respect to + (ie119886∘(119887+119888)cap(119886∘119887+119886∘119888) = 120601 and (119886+119887)∘119888cap(119886∘119888+119887∘119888) = 120601 for all119886 119887 119888 isin 119877) A multiplicative119867V-ring (119877 + ∘) is commutativeif 119886 ∘ 119887 = 119887 ∘ 119886 for all 119886 119887 isin 119877 The identity element 0

119877of

the group (119877 +) is said to be absorbing in the multiplicative119867V-ring (119877 + ∘) if 119886 ∘ 0119877 = 0119877 ∘ 119886 = 0119877 for all 119886 isin 119877 Anonempty finite subset E = 119890

1 1198902 119890

119899 of a multiplicative

119867V-ring (119877 + ∘) is called an identity set (or 119894-set in short) [9]

of 119877 if (i) 119890119894notin ⟨0119877⟩ for at least one 119894 = 1 2 119899 and (ii) for

any 119886 isin 119877 119886 isin (sum119899119894=1119890119894∘ 119886) cap (sum

119899

119894=1119886 ∘ 119890119894) An element 119890 of 119877

is called a hyperidentity of 119877 if the set 119890 is an 119894-set of 119877Unlike a ring the equality of the set-expressions (minus119886) ∘

119887 119886 ∘ (minus119887) and minus(119886 ∘ 119887) does not hold in general on amultiplicative 119867V-ring (119877 + ∘) for any 119886 119887 isin 119877 In fact if(Z + sdot) is the ring of integers and if ∘ is a hyperoperation onZ defined by 119886 ∘ 119887 = 119886 + 119887 minus119886 minus 119887 119886 119887 for all 119886 119887 isin Z then(Z + ∘) is a commutative multiplicative 119867V-ring in which1 ∘ (minus2) = (minus1) ∘ 2 (minus1) ∘ 2 = minus(1 ∘ 2) and minus(1 ∘ 2) = 1 ∘ (minus2)

A multiplicative 119867V-ring (119877 + ∘) is said to satisfy thecondition (R) if the set equality (minus119886) ∘ 119887 = 119886 ∘ (minus119887) = minus(119886 ∘ 119887)(called the condition (R) [9]) holds true for any two elements119886 and 119887 of 119877 Let (119877 + sdot) be a ring and ∘ a hyperoperation on119877 defined by 119886 ∘ 119887 = 119886 119887 minus119886 minus119887 for all 119886 119887 isin 119877 Then(119877 + ∘) is a multiplicative119867V-ring with condition (R)

We consider now an 119867V-structure (119877 + ∘) in which (i)(119877 +) is a commutative group (ii) (119877 ∘) is an119867V-semigroupand (iii) ∘ is semidistributive across the operation + (ie119886 ∘ (119887 + 119888) sube 119886 ∘ 119887 + 119886 ∘ 119888 and (119886 + 119887) ∘ 119888 sube 119886 ∘ 119888 + 119887 ∘ 119888for all 119886 119887 119888 isin 119877)This119867V-structure is clearly a multiplicative119867V-ring and we thus call it a semidistributive multiplicative119867V-ring Henceforth throughout the paper a multiplicative

Hindawi Publishing CorporationAlgebraVolume 2014 Article ID 392902 8 pageshttpdxdoiorg1011552014392902

2 Algebra

119867V-ring wherever considered will always be assumed to bea semidistributive multiplicative 119867V-ring with the condition(R)

In 1982 the notion ofmultiplicative hyperring is inductedin the study on hyperring theory by Rota which is subse-quently investigated in [10ndash14] A commutative group (119877 +)endowed with an associative hyperoperation ∘ is called amultiplicative hyperring [15] if (i) ∘ is semidistributive acrossthe operation + on 119877 and (ii) ∘ satisfies the condition (R) forelements in 119877 An associative hyperoperation is eventuallyweakly associative and thus a multiplicative hyperring iseventually a semidistributive multiplicative 119867V-ring withcondition (R)

Procesi Ciampi and Rota define in [16] polynomialsover multiplicative hyperring as follows let (119877 + sdot) be amultiplicative hyperring with absorbing zero 0

119877and let 119909 be

any symbol out of 119877Then a polynomial in 119909 is an expressionof the form 119891(119909) = 119886

01199090+ 11988611199091+ 11988621199092+ sdot sdot sdot = sum 119886

119896119909119896

(119896 isin N 119886119896isin 119877) in which + is a connective and only a

finite number of the 119886119896rsquos (called the coefficients of 119909119896 in 119891(119909))

are different from zero (0119877) of 119877 The degree of a polynomial

119891(119909) = sum119886119896119909119896 (in short deg 119891(119909)) is a nonnegative integer 119899

such that 119886119899= 0119877and 119886119896= 0119877 for all 119896 gt 119899 A polynomial

119891(119909) = sum119886119896119909119896 over a multiplicative hyperring 119877 will be

written as 119891(119909) = sum119899119896=1119886119896119909119896 when and only when deg119891(119909) =

119899 For an integer 119898 isin N and any 119886 119886119896 119887 isin 119877 (119886 = 0

119877

119887 = 0119877 119896 = 1 2 119898 minus 1) we write the polynomials

119886(119909) = 01198771199090+sdot sdot sdot+0

119877119909119898minus1+119886119909119898 and119891(119909) = 1198861199090+119886

11199091+sdot sdot sdot+

119886119898minus1119909119898minus1+119887119909119898 simply as 119886119909119898 and 119886+119886

1119909+ sdot sdot sdot + 119886

119898minus1119909119898minus1+

119887119909119898 respectively Denote by 119877[119909] the set of all polynomials

in 119909 over 119877 and define on 119877[119909] a binary operation + and ahyperoperation lowast as follows for any two polynomials 119891(119909) =sum119886119896119909119896 and 119892(119909) = sum 119887

119896119909119896 from 119877[119909] 119891(119909) + 119892(119909) =

sum(119886119896+ 119887119896)119909119896 and 119891(119909) lowast 119892(119909) = sum 119888

119896119909119896 119888119896isin sum119894+119895=119896119886119894119887119895

where for any 119886 119887 isin 119877 the juxtaposition 119886119887 means theset 119886 sdot 119887 The purpose of the present paper is to study theproperties of the hyperstructure (119877[119909] + lowast) in connection tothose of a particular class of multiplicative hyperrings calledpolynomially structured multiplicative hyperrings which wedescribe formally in the following section

2 Polynomially StructuredMultiplicative Hyperring

It is asserted in [16] that for a multiplicative hyperring 119877 thehyperstructure (119877[119909] + lowast) is always a multiplicative hyper-ring But we note here that given a multiplicative hyperring119877 with absorbing zero the hyperoperation lowast (as is definedin Section 1) does not necessarily induce a multiplicativehyperring structure over the group of polynomials (119877[119909] +)In fact we have the following example

Example 1 Let (Z + sdot) be the ring of integers and (Z + ∘)the multiplicative hyperring where for any 119886 119887 isin Z 119886 ∘ 119887 =2119886119887 3119886119887 (denoting the product 119886 sdot 119887 of elements in thering (Z + sdot) simply by the juxtaposition 119886119887) Consider three

polynomials 119891(119909) = 1 + 2119909 119892(119909) = 2 + 1119909 and ℎ(119909) = 1 + 3119909over the multiplicative hyperring (Z + ∘)

Then the set of coefficients of 119909 in the polynomialsbelonging to (119891(119909) lowast 119892(119909)) lowast ℎ(119909) is 119872 = 4 6 ∘ 3 +

10 11 14 15 ∘ 1 Again the set of coefficients of 119909 in thepolynomials belonging to 119891(119909) lowast (119892(119909) lowast ℎ(119909)) is 119873 = 1 ∘14 15 20 21+2∘4 6 By a tedious but routine calculationone can see that 57 74 82 sube 119872 whereas 57 74 82 cap 119873 =120601 So there are some polynomials in (119891(119909) lowast 119892(119909)) lowast ℎ(119909)the coefficient of 119909 in each of which is an element of the set57 74 82These polynomials do not belong to119891(119909)lowast(119892(119909)lowastℎ(119909)) (since 57 74 82cap119873 = 120601)Thus (119891(119909)lowast119892(119909))lowastℎ(119909) sube119891(119909) lowast (119892(119909) lowast ℎ(119909)) So there is no question of claiming(Z[119909] + lowast) to be a multiplicative hyperring

However we observe that it is possible to constructa multiplicative hyperring 119877 corresponding to which thehyperstructure (119877[119909] + lowast) turns out to be a multiplicative119867V-ring if not a multiplicative hyperring at all

Let us consider a commutative group (119866 +) Suppose that120601 = 119860 sube Hom(119866Hom(119866)) is such that for any 119886 119887 isin 119860and 119909 119910 isin 119866 (119909

119886(119910))119887= 119909119886119910119887 where 119909

119886denotes the image

of 119909 isin 119866 under 119886 isin 119860 and 119909119886119910119887is simply the mapping

composition Define a hyperoperation ∘119860on119866 by stating that

119909∘119860119910 = 119909

119886(119910) 119886 isin 119860 Then we have the following

Lemma2 (119866 + ∘119860) is amultiplicative hyperring with absorb-

ing zero

Proof Let 119909 119910 119911 isin 119866 Then 119901 isin 119909∘119860(119910∘119860119911) rArr 119901 isin 119909∘

119860119904 (for

some 119904 isin 119910∘119860119911) rArr 119901 = 119909

119886(119904) and 119904 = 119910

119887(119911) (for some

119886 119887 isin 119860) rArr 119901 = 119909119886(119910119887(119911)) = (119909

119886119910119887)(119911) = (119909

119886(119910))119887(119911) rArr

119901 isin (119909119886(119910))∘119860119911 rArr 119901 isin (119909∘

119860119910)∘119860119911 (since 119909

119886(119910) isin 119909∘

119860119910)

So 119909∘119860(119910∘119860119911) sube (119909∘

119860119910)∘119860119911 The reverse inclusion can also

be shown to be true by adopting similar arguments Hence119909∘119860(119910∘119860119911) = (119909∘

119860119910)∘119860119911 Now119901 isin 119909∘

119860(119910+119911) rArr 119901 isin 119909

119886(119910+119911)

(for some 119886 isin 119860) = 119909119886(119910)+119909

119886(119911) isin 119909∘

119860119910+119909119886∘119860119911 So 119909∘

119860(119910+

119911) sube 119909∘119860119910 + 119909119886∘119860119911 Again 119901 isin (119910 + 119911)∘

119860119909 rArr 119901 isin (119910 + 119911)

119886119909

(for some 119886 isin 119860)= (119910119886+119911119886)119909 (since 119886 isin Hom(119866Hom(119866))) =

119910119886(119909) + 119911

119886(119909) sube 119910∘

119860119909 + 119911∘

119860119909 So (119910 + 119911)∘

119860119909 sube 119910∘

119860119909 + 119911∘

119860119909

Moreover 119909∘119860(minus119910) = 119909

119886(minus119910) 119886 isin 119860 = minus(119909

119886(119910)) 119886 isin 119860

(since 119909119886isin Hom(119866)) = minus119909

119886(119910) 119886 isin 119860 = minus(119909∘

119860119910) and

(minus119909)∘119860119910 = (minus119909)

119886(119910) 119886 isin 119860 = (minus(119909

119886))(119910) 119886 isin 119860

(since 119886 isin Hom(119866Hom(119866))) = 119909119886(minus119910) 119886 isin 119860 = 119909∘

119860(minus119910)

Thus (119866 + ∘119860) is a multiplicative hyperring Finally if 0

119866

denotes the identity element of the group (119866 +) then 119909∘0119866=

119909119886(0119866) 119886 isin 119860 = 0

119866 (since for 119886 isin 119860 119909

119886isin Hom(119866)) and

also 0119866∘ 119909 = (0

119866)119886(119909) 119886 isin 119860 = 0

119866 (since for 119886 isin

Hom(119866Hom(119866)) (0119866)119886is the zero homomorphism from 119866

to 119866) Thus 0119866is absorbing in the multiplicative hyperring

(119866 + ∘119860)

The multiplicative hyperring (119866 + ∘119860) defined in

Lemma 2 is called a multiplicative 119860-hyperring (of course ifsuch a set 119860 exists for the group (119866 +)) That a multiplicative119860-hyperring exists is evident in the following example

Example 3 Let (119877 + sdot) be a ringThen as is shown in [9 13](119877 +

lowast

119875) is a multiplicative hyperring with absorbing zero

Algebra 3

where 119875 isin 119875(119877) with |119875| ge 2 andlowast

119875 is the 119875-hyperoperation[8 17] on the semigroup (119877 sdot) that is119909

lowast

119875 119910 = 119909sdot119886sdot119910 119886 isin 119875

for all 119909 119910 isin 119877 Now for each 119886 isin 119875 and 119909 isin 119877 we define amapping 119909

119886 119877 rarr 119877 by stating that 119909

119886(119910) = 119909 sdot 119886 sdot 119910 for all

119910 isin 119877 Then 119909119886isin Hom(119877 +) Thus corresponding to each

119886 isin 119875 we have a mapping 119886 (119877 +) rarr Hom(119877 +) givenby 119886(119909) = 119909

119886 for any 119909 isin 119877 Then 119875 sube Hom(119877Hom(119877))

Moreover for any 119909 119910 119911 isin 119877 and 119886 119887 isin 119875 (119909119886(119910))119887(119911) = (119909 sdot

119886sdot119910)sdot119887sdot119911 = 119909sdot119886sdot(119910sdot119887sdot119911) = 119909sdot119886sdot(119910119887(119911)) = 119909

119886(119910119887(119911)) = (119909

119886119910119887)(119911)

that is (119909119886(119910))119887= 119909119886119910119887 Thus the hyperoperation ∘

119860(for

119860 = 119875) is defined on the group (119877 +) Note that for any119909 119910 isin 119877 119909∘

119860119910 = 119909

119886(119910) 119886 isin 119860 = 119909 sdot 119886 sdot 119910 119886 isin 119875 = 119886

lowast

119875 119910Thus (119877 +

lowast

119875) is a multiplicative 119860-hyperring for 119860 = 119875

Proposition4 For amultiplicative119860-hyperring (119877 + ∘119860) the

hyperoperation lowast induces a multiplicative119867V-ring structure onthe group (119877[119909] +) of polynomials over 119877

Proof Let 119891(119909) = sum119886119896119909119896 119892(119909) = sum 119887

119896119909119896 and ℎ(119909) = sum119889

119896119909119896

be three polynomials in 119877[119909] Then (119891(119909) lowast 119892(119909)) lowast ℎ(119909) =sum 119905119896119909119896 119905119896isin sum119906+V=119896(sum119894+119895=119906 119886119894∘119860119887119895)∘119860119889V and 119891(119909) lowast (119892(119909) lowast

ℎ(119909)) = sum 119904119896119909119896 119904119896isin sum119906+V=119896 119886119906∘119860(sum119894+119895=V 119887119894∘119860119889119895) Now we

choose and fix an element 120572 isin 119860 Then for each 119896 isin N0

119901119896= sum

119906+V=119896( sum

119894+119895=119906

(119886119894)120572(119887119895))

120572

(119889V)

isin sum

119906+V=119896( sum

119894+119895=119906

119886119894∘119860119887119895)∘119860119889V

119902119896= sum

119906+V=119896(119886119906)120572( sum

119894+119895=V(119887119894)120572(119889119895))

isin sum

119906+V=119896119886119906∘119860( sum

119894+119895=V119887119894∘119860119889119895)

(1)

Again 119901119896

= sum119906+V=119896(sum119894+119895=119906(119886119894)120572(119887119895))120572(119889V) =

sum119906+V=119896sum119894+119895=119906((119886119894)120572(119887119895))120572(119889V) (since120572 is a homomorphism) =sum119906+V=119896sum119894+119895=119906((119886119894)120572(119887119895)120572)(119889V) (since (119909120572(119910))120572 = 119909120572119910120572) 119902119896 =sum119906+V=119896(119886119906)120572(sum119894+119895=V(119887119894)120572(119889119895)) = sum119906+V=119896sum119894+119895=V(119886119906)120572((119887119894)120572(119889119895))

(since (119886119906)120572

is a homomorphism) =sum119906+V=119896sum119894+119895=V((119886119906)120572(119887119894)120572)(119889119895) Clearly then for each 119896 isin N0

119901119896= 119902119896isin ( sum

119906+V=119896( sum

119894+119895=119906

119886119894∘119860119887119895)∘119860119889V)

⋂( sum

119906+V=119896119886119906∘119860( sum

119894+119895=V119887119894∘119860119889119895))

(2)

Hence (119891(119909) lowast 119892(119909)) lowast ℎ(119909) cap 119891(119909) lowast (119892(119909) lowast ℎ(119909)) = 120601It is shown in [16] that for any multiplicative hyperring 119877the hyperoperationlowast defined on119877[119909] is semidistributive overthe operation + on 119877[119909] and also satisfies the condition (R)for any two polynomials in 119877[119909] Thus for the multiplicative119860-hyperring (119877 + ∘

119860) the hyperstructure (119877[119909] + lowast) is a

multiplicative 119867V-ring (with absorbing zero 0119877[119909]= 01198771199090+

01198771199091+ 01198771199092+ sdot sdot sdot )

We call a multiplicative hyperring (119877 + sdot)with absorbingzero polynomially structured if (119877[119909] + lowast) is a multiplicative119867V-ringThe class ofmultiplicative119860-hyperrings is a subclassof the class of polynomially structured multiplicative hyper-rings (by Proposition 4) Throughout the rest of the paper119877 will stand for a polynomially structured multiplicativehyperring

3 Polynomials over Integral Hyperrings

An element 119886 ( = 0119877) of a multiplicative 119867V-ring (119878 + ∘) is

a left (resp right) divisor of zero in 119878 if there exists 119887 isin 119878lowast(resp 119888 isin 119878lowast = 119878 0

119878) such that 0

119878isin 119886 ∘ 119887 (resp 0

119904isin 119888 ∘ 119886)

and a divisor of zero in 119878 if it is either a left or a right divisorof zero in 119878 An element 119886 ( = 0

119878) of 119878 is a left (resp right)

strong divisor of zero in 119878 if there exists 119887 isin 119878lowast = 119878 0119878

(resp 119888 isin 119878lowast) such that 119886 ∘ 119887 = 0119878 (resp 119888 ∘ 119886 = 0

119878) and

a strong divisor of zero in 119878 if it is either a left or a right strongdivisor of zero in 119878

Definition 5 A multiplicative 119867V-ring is called an integral119867V-ring if there is no strong divisor of zero in it A commuta-tive integral119867V-ring is an119867V-domain A strong integral119867V-ring is a multiplicative119867V-ring in which there is no divisor ofzero A strong 119867V-domain is a commutative strong integral119867V-ring We call an integral119867V-ring (resp an119867V-domain) 119878simply an integral hyperring (resp a hyperdomain) [13]when the119867V-ring 119878 is a multiplicative hyperring

Before entering into the study of the multiplicative 119867V-ring 119877[119909] of polynomials over integral hyperring and hyper-domain let us go through the following useful observations

Remark 6 (a) For any polynomially structuredmultiplicativehyperring119877 the identity element 0

119877[119909]= 01198771199090+01198771199091+01198771199092+

sdot sdot sdot of the group (119877[119909] +) is absorbing in the multiplicative119867V-ring (119877[119909] + lowast)

(b) 119877[119909] is commutative if 119877 is a commutative multiplica-tive hyperring

(c)Themapping120595 119877 rarr 119877[119909] defined by for all 119903 isin 119877lowast120595(119903) = 119903119909

0 and 120595(0119877) = 0119877[119909]

is a strong monomorphism Infact for any 119903 119904 isin 119877120595(119903119904) = 120595(119888) 119888 isin 119903119904 = 1198881199090 119888 isin 119903119904 =(1199031199090) lowast (119904119909

0) = 120595(119903) lowast 120595(119904) Thus 119877 can be identified with its

isomorphic image in 119877[119909] and for any 119903 isin 119877lowast we can writethe polynomial 1199031199090 simply as 119903 and the zero polynomial 0

119877[119909]

as 0119877

(d) If E = 1198901 1198902 119890

119899 is an 119894-set of 119877 then the set

E119909sube 119877[119909] is an 119894-set in the multiplicative 119867V-ring 119877[119909]

where for any119860 isin 119875lowast(119877)119860119909denotes the set 1198861199090 119886 isin 119860In

4 Algebra

fact for any 119891(119909) = sum119886119896119909119896isin 119877[119909] we have that sum119899

119894=1119890119894lowast

119891(119909) = sum119899

119894=1(119890119894lowast sum119886

119896119909119896) = sum

119899

119894=1sum 119886119894119896119909119896 119886119894119896isin 119890119894119886119896 =

sum(sum119899

119894=1119886119894119896)119909119896 119886119894119896isin 119890119894119886119896 = sum 119887

119896119909119896 119887119896isin sum119899

119894=1119890119894119886119896

Then since 119886119896isin sum119899

119894=1119890119894119886119896 we have that119891(119909) isin sum119899

119894=1119890119894lowast119891(119909)

identifying 1198901198941199090 with 119890

119894 Similarly one can see that 119891(119909) isin

sum119899

119894=1119891(119909) lowast 119890

119894

On the other hand if for some 119899 isin N E119877[119909]= 120598119894(119909) =

sum 119890119894119896119909119896isin 119877[119909] 119894 = 1 2 119899 is an 119894-set in 119877[119909] then

E = 1198901198940 119894 = 1 2 119899 is an 119894-set in 119877 In fact for any

119886 isin 119877 1198861199090 isin sum119899119894=1(1198861199090) lowast 120598119894(119909) = sum

119899

119894=1sum 119886119894119896119909119896isin 119877[119909] 119886

119894119896isin

119886119890119894119896 = sum(sum

119899

119894=1119886119894119896)119909119896isin 119877[119909] 119886

119894119896isin 119886119890119894119896 = sum 119886

119896119909119896isin

119877[119909] 119886119896isin sum119899

119894=1119886119890119894119896 rArr 119886 isin sum

119899

119894=11198861198901198940 Similarly from

1198861199090isin sum119899

119894=1120598119894(119909) lowast (119886119909

0) one may arrive at 119886 isin sum119899

119894=11198901198940119886

This is clear fromRemark 6 (d) that for any hyperidentity119890 of 119877 the polynomial 1198901199090 isin 119877[119909] is a hyperidentity in themultiplicative119867V-ring 119877[119909] Is every hyperidentity of 119877[119909] ofthe form 1198901199090 for some hyperidentity 119890 of 119877 Following is anexample of a multiplicative hyperring 119877 such that 119877[119909] has ahyperidentity 120598(119909) = 1198901199090 for any hyperidentity 119890 of 119877

Example 7 Let (Z + sdot) be the ring of integers and 119875 = 0 1Then (Z +

lowast

119875) is a commutative polynomially structuredmultiplicative hyperring (as is shown in Example 3) Denotethe multiplicative hyperring (Z +

lowast

119875) by Z119875 Consider a

polynomial 1 + 1119909 isin Z119875[119909] Then for any 119891(119909) = sum119886

119896119909119896isin

Z119875[119909] we see that (1+1119909)lowast119891(119909) = sum 119887

119896119909119896 1198870isin 1lowast

119875 1198860 and119887119896isin 1lowast

119875 119886119896 + 1lowast

119875 119886119896minus1 (for 119896 isin N) Clearly 1198860 isin 1lowast

119875 1198860 andfor any 119896 isin N 119886

119896isin 1lowast

119875 119886119896 + 1lowast

119875 119886119896minus1 (since 0 isin 1lowast

119875 119886119896minus1)Thus 119891(119909) isin (1 + 1119909) lowast 119891(119909) and so 1 + 1119909 is a hyperidentityin Z119875[119909] which is not in the form 1198901199090 for any hyperidentity

119890 of Z119875

Remark 8 Let 120598(119909) = sum 119890119896119909119896isin 119877[119909] be a hyperidentity in the

multiplicative119867V-ring 119877[119909] Then from Remark 6 (d) 1198900is a

hyperidentity in the multiplicative hyperring 119877

Proposition 9 Let 119877 be a strong integral hyperring Thenevery hyperidentity in the multiplicative119867V-ring 119877[119909] is of theform 1198901199090 for some hyperidentity 119890 of 119877

Proof Suppose that 120598(119909) = sum 119890119896119909119896isin 119877[119909] is a hyperidentity

in the multiplicative 119867V-ring 119877[119909] Then by the Remark 8119890 = 1198900is a hyperidentity in119877 Now let 119886 isin 1198770

119877 be arbitrary

and 119891(119909) = sum119886119896119909119896isin 119877[119909] where 119886

0= 119886 and 119886

119896= 0119877 for all

119896 isin N Then 119891(119909) isin 120598(119909) lowast 119891(119909) cap 119891(119909) lowast 120598(119909) whereby0119877isin 119890119896119886 cap 119886119890

119896 for all 119896 isin N whence 119890

119896= 0119877for all 119896 isin N

(since 119877 is a strong integral hyperring and 119886 = 0119877) Thus

120598(119909) = 1198901199090+01198771199091+01198771199092+sdot sdot sdot where 119890 = 119890

0is a hyperidentity

in 119877

Definition 10 If 0119877[119909]

= 119891(119909) = sum119886119896119909119896isin 119877[119909] then the

smallest integer 119899 such that 119886119899= 0119877is called the order of119891(119909)

and is denoted by ord119891(119909) The order of 0119877[119909]

is defined to bezero For a nonempty set119860 isin 119875(119877[119909]) the smallest element inthe set ord119891(119909) 119891(119909) isin 119860 does exist and is called the orderof 119860 being denoted by ord119860 that is ord119860 = minord119891(119909) 119891(119909) isin 119860

In the next proposition we will find some propertiesof ord119860 for some 119860 isin 119875lowast(119877[119909])(= 119875(119877[119909]) 120601) Forthat it is necessary at this point to frame some suitablenotations corresponding to different types of hyperproductsof elements in the multiplicative119867V-ring 119877[119909] Indeed in anymultiplicative 119867V-ring (119878 + ∘) (which is not a multiplicativehyperring) the expression like 119904

1∘1199042∘1199043∘sdot sdot sdot∘119904

119899(119904119894isin 119878 119899 ge 3)

bears no connotation in 119878 unless the parentheses ldquo(rdquo andldquo)rdquo are meaningfully inserted in Note that the following twoexpressions

1199041∘ (11199042∘ (21199043∘ (31199044∘ sdot sdot sdot ∘ (

119899minus3119904119899minus2∘ (119899minus2119904119899minus1∘ 119904119899)119899minus2)119899minus3sdot sdot sdot )3)2)1

(1(2(3sdot sdot sdot (119899minus3(119899minus21199041∘ 1199042)119899minus2∘ 1199043)119899minus3∘ sdot sdot sdot ∘ 119904

119899minus3)3∘ 119904119899minus2)2∘ 119904119899minus1)1∘ 119904119899

(3)

are meaningful called the finite hyperproducts of type 119897 andtype 119903 andwritten in notations respectively as [119904

1∘1199042∘sdot sdot sdot∘119904

119899]119897

and [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903

Proposition 11 (i) For any 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]

119890119894119905ℎ119890119903 0119877[119909]isin 119891 (119909) lowast 119892 (119909)

119900119903 ord (119891 (119909) lowast 119892 (119909)) ge ord 119891 (119909) + ord 119892 (119909) (4)

(ii) If 119877 is a strong hyperdomain then the implication that

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119903

) = ord 119892 (119909) (5)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119903holds true for any

119899 isin N with 119899 ge 2

Proof (i) Let ord119891(119909) = 119899 and ord119892(119909) = 119898 Then 119891(119909) =sum119886119896119909119896 where 119886

119899= 0119877 119886119896= 0119877for 119896 lt 119899 and 119892(119909) = sum 119887

119896119909119896

where 119887119898= 0119877 119887119896= 0119877for 119896 lt 119898 So for anysum119888

119896119909119896isin 119891(119909)lowast

119892(119909) we see that whenever 119896 lt 119899 + 119898 119888119896isin sum119894+119895=119896119886119894119887119895=

0119877 (since 0

119877is absorbing in 119877 and 119886

119896= 0119877(119896 lt 119899) 119887

119896=

0119877(119896 lt 119898)) Thus if 0

119877[119909]notin 119891(119909) lowast 119892(119909) then for any ℎ(119909) =

sum 119888119896119909119896isin 119891(119909) lowast 119892(119909) we have that ord ℎ(119909) ge 119899 + 119898 So

ord(119891(119909) lowast 119892(119909)) ge 119899 + 119898 = ord119891(119909) + ord119892(119909)(ii) Now suppose that 119877 is a strong hyperdomain with

absorbing zero Consider two polynomials 119891(119909) 119892(119909) isin119877[119909] 0

119877[119909] Let ord119891(119909) = 119901 and ord119892(119909) = 119898 Then

119891(119909) = sum119886119896119909119896 where 119886

119901= 0119877 119886119896= 0119877for 119896 lt 119901 and

Algebra 5

119892(119909) = sum 119887119896119909119896 where 119887

119898= 0119877 119887119896= 0119877for 119896 lt 119898

So 0119877notin 119886119901119887119898(since 119877 is a strong hyperdomain) and also

119886119901119887119898= sum119894+119895=119901+119898

119886119894119887119895(since 119886

119896= 0119877for 119896 lt 119901 and 119887

119896= 0119877

for 119896 lt 119898) Thus for any ℎ(119909) = sum 119888119896119909119896isin 119891(119909) lowast 119892(119909)

119888119901+119898

= 0119877(since 119888

119901+119898isin sum119894+119895=119901+119898

119886119894119887119895= 119886119901119887119898) and 119888

119896= 0119877

for all 119896 with 0 le 119896 lt 119901 + 119898 Thus ordℎ(119909) = 119901 + 119898 forany ℎ(119909) isin 119891(119909) lowast 119892(119909) and so ord(119891(119909) lowast 119892(119909)) = 119901 + 119898Hence ord119891(119909)+ord119892(119909) = ord(119891(119909)lowast119892(119909)) = ord ℎ(119909) forall ℎ(119909) isin 119891(119909) lowast 119892(119909) So the implication is true for 119899 = 2(noting that [119891(119909) lowast 119892(119909)]

119903= 119891(119909) lowast 119892(119909)) Suppose that for

some integer 119896 ge 2 the implication holds true for each valueof 119899 ranging from 2 to 119896minus1 and take any119891

119894(119909) isin 119877[119909]0

119877[119909]

for 119894 = 1 2 119896 Thensum119896minus1119894=1

ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast

sdot sdot sdotlowast119891119896minus1(119909)]119903) = ord ℎ(119909) for all ℎ(119909) isin [119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast

119891119896minus1(119909)]119903 Now let 119892(119909) isin [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903be

arbitraryThen 119892(119909) isin ℎ(119909) lowast119891119896(119909) for some ℎ(119909) isin [119891

1(119909) lowast

1198912(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903 So ord119892(119909) = ord(ℎ(119909) lowast 119891

119896(119909)) =

ord ℎ(119909) + ord119891119896(119909) = sum

119896minus1

119894=1ord119891119894(119909) + ord119891

119896(119909)(since ℎ(119909) isin

[1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903) = sum

119896

119894=1ord119891119894(119909) Thus

sum119896

119894=1ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119896(119909)]119903) = ord119892(119909)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903

Hence by strong induction the implication follows forany 119899 isin N with 119899 ge 2

Corollary 12 If 119877 is a strong hyperdomain then the implica-tion

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119897

) = ord 119892 (119909) (6)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897holds true for any

119899 isin N with 119899 ge 2

Proof Since themultiplicative hyperring119877 is a hyperdomainit is commutative and so 119877[119909] is also a commutative multi-plicative 119867V-ring Hence for any 119891119894(119909) isin 119877[119909] 0119877[119909] (119894 =1 2 119896) we have that [119891

1(119909)lowast119891

2(119909)lowast119891

3(119909)lowastsdot sdot sdotlowast119891

119896(119909)]119897=

[119891119896(119909) lowast 119891

119896minus1(119909) lowast sdot sdot sdot lowast 119891

2(119909) lowast 119891

1(119909)]119903 Hence the assertion

follows straight from Proposition 11

Proposition 13 If the multiplicative hyperring 119877 is a stronghyperdomain the multiplicative 119867V-ring 119877[119909] is a strong 119867V-domain

Proof 119877 being a strong hyperdomain is a commutativemultiplicative hyperring Thus 119877[119909] is a commutative mul-tiplicative 119867V-ring Again since 0119877 is absorbing in 119877 0

119877[119909]

is also absorbing in 119877[119909] Thus we take 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]Then by Proposition 11 ord(119891(119909)lowast119892(119909)) = ord119891(119909)+

ord119892(119909) gt 0 Hence 0119877[119909]notin 119891(119909) lowast 119892(119909) So 119877[119909] is a strong

119867V-domain

4 C-Ideals in 119877[119909]

A subgroup 119868 of the group (119878 +) is called a left (resp right)119867V-ideal of a multiplicative 119867V-ring (119878 + ∘) if for any 119904 isin 119878

and 119886 isin 119868 119904 ∘ 119886 sube 119868 (resp 119886 ∘ 119904 sube 119868) 119868 is an119867V-ideal of 119878 if itis both a left and a right119867V-ideal of 119878

We call an 119867V-ideal of a multiplicative 119867V-ring 119878 simplya hyperideal when 119878 is a multiplicative hyperringThe notionof a typical hyperideal in a multiplicative hyperring calledC-ideal is introduced in [18] to study prime and primaryhyperideals of multiplicative hyperrings A hyperideal 119868 of amultiplicative hyperring 119878 is a C-ideal if for any 119860 isin C 119860 cap119868 = 120601 rArr 119860 sube 119868 where C = 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899 119904119894isin 119878

119899 isin N sube 119875lowast(119878) Following is the definition of aC-ideal in anarbitrary multiplicative119867V-ring

Definition 14 A left (resp right) 119867V-ideal 119868 of a multiplica-tive 119867V-ring (119878 + ∘) is called a left (resp right) C-ideal iffor any type 119897 hyperproduct [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897(resp type 119903

hyperproduct [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903) of elements 119904

119894isin 119878 we have

that [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897cap 119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897sube 119868 (resp

[1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903cap119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903sube 119868) An119867V-ideal

119868 of a multiplicative119867V-ring 119878 is called a C-ideal if it is a leftas well as a rightC-ideal in 119878

We writeC-ideal(119878) (respC-ideallowast(119878)) to denote the setof all (resp proper)C-ideals of a multiplicative119867V-ring 119878 Ina commutative multiplicative119867V-ring 119878 every left C-ideal isa right C-ideal and vice versa since commutativity impliesthe equality [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903= [119904119899∘ 119904119899minus1∘ sdot sdot sdot ∘ 119904

1]119897 for any

119904119894isin 119877 Note that in amultiplicative hyperring (even if it is not

commutative) a hyperideal is a leftC-ideal (as an119867V-ideal) ifand only if it is a rightC-ideal

Proposition 15 If 119877 is a strong hyperdomain then the setC-119868119889119890119886119897lowast(119877[119909]) is nonempty

Proof For each 119899 isin N cup 0 let C119899= 119891(119909) isin 119877[119909]

ord119891(119909) ge 119899 cup 0119877[119909] Then C

119899is a subgroup of the group

(119877[119909] +) (since for any 119891(119909) 119892(119909) isin 119877[119909] ord(minus119891(119909)) =ord119891(119909) and ord(119891(119909) + 119892(119909)) = minord119891(119909) ord119892(119909)when 119891(119909)+119892(119909) = 0

119877[119909]) Let 119892(119909) isin 119877[119909] and 119891(119909) isin C

119899 If

119892(119909) = 0119877[119909]

or 119891(119909) = 0119877[119909]

then 119892(119909) lowast119891(119909) = 0119877[119909] sube C119899

(since 0119877[119909]

is absorbing in 119877[119909]) So let 119892(119909) = 0119877[119909]

and119891(119909) = 0

119877[119909] Then by Proposition 11(ii) for all ℎ(119909) isin 119892(119909)lowast

119891(119909) ord ℎ(119909) = ord(119892(119909) lowast 119891(119909)) = ord119891(119909) + ord119892(119909) gt119899 Thus 119892(119909) lowast 119891(119909) sube C

119899 Hence C

119899is an 119867V-ideal of

119877[119909] (since 119877[119909] is a commutative multiplicative 119867V-ring)Let [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897be a type-119897 hyperproduct of

elements of the119867V-ring 119877[119909] such that [1198911(119909) lowast 1198912(119909) lowast sdot sdot sdot lowast119891119899(119909)]119897sube C119899 Then 119891

119894(119909) = 0

119877[119909]for each 119894 (since 0

119877[119909]isin C119899

is absorbing in119877[119909])Thus ord([1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897) =

ord119892(119909) for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 Now

since [1198911(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119899(119909)]119897sube C119899 so there exists ℎ(119909) isin

[1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897such that ℎ(119909) notin C

119899Then for any

119892(119909) isin [1198911(119909) lowast119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 ord119892(119909) = ord ℎ(119909) lt 119899

Hence by definition ofC119899 [1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897capC119899=

120601 and soC119899is a leftC-ideal and thus aC-ideal of 119877[119909] (since

119877[119909] is commutative) Hence C119899isin C-Ideallowast(119877[119909]) for all

119899 gt 0Since the intersection of left C-ideals of a multiplicative

119867V-ring 119878 is also a left C-ideal of 119878 and 119878 is itself a left C-ideal so the smallest left C-ideal containing a subset 119860 of 119878

6 Algebra

being naturally called the left C-ideal generated by 119860 existsand is in fact the intersection of all left C-ideals containing119860 The left C-ideal generated by a left 119867V-ideal 119868 of 119878 iscalled the leftC-closure of 119868 and is denoted byC

119897(119868) Clearly

C119897(C119897(119868)) = C

119897(119868) for a left 119867V-ideal 119868 C119903(119868) and C(119868)

respectively denote the right C-closure of a right 119867V-ideal 119868and theC-closure of an119867V-ideal 119868 of 119878 For an119867V-ideal 119868 of acommutativemultiplicative119867V-ring 119878C119897(119868) = C119903(119868) = C(119868)The following lemma presents a description of the set C

119897(119868)

for a left119867V-ideal 119868 of amultiplicative119867V-ring 119878The setC119903(119868)

for a right119867V-ideal 119868 can be described dually

Lemma 16 Let 119878 be a multiplicative 119867V-ring with an 119894-set Eand let L denote the set of all left 119867V-ideals of 119878 119865 119875(119878) rarr119875(119878) is a mapping defined by

119865 (119860) = ⋃

119899

sum

119894=1

[1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897 119899 119898

119894isin N 119903

1198941isin 119878

119904119886119905119894119904119891119910119894119899119892 [1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897cap 119860 = 120601

(7)

for any 119860 isin 119875(119878) Then for any 119868 1198681 1198682isin L we have the

following (i) 119865(119868) isin L (ii) 119868 sube 119865(119868) (iii) 1198681sube 1198682rArr 119865(119868

1) sube

119865(1198682) (iv) 119865(119868) = 119868 if and only if 119868 is a left C-ideal of 119878 and

(v) left C-closure of 119868 is C119897(119868) = ⋃

infin

119896=1119865119896(119868) where for any

119896 isin N 119865119896 denotes the (119896 minus 1)-times mapping composition of 119865

Proof All the assertions made in this lemma can be estab-lished by adopting the arguments that are applied in provingwell-known analogous results on ldquocomplete closure of a setrdquoin semihypergroup theory (see [3 8])

Proposition 17 Let 119877 be a strong hyperdomain with an 119894-setThen for any 119867V-ideal 119868 of the multiplicative 119867V-ring 119877[119909]ord(C(I)lowast) ge ordIlowast where for any 119867V-ideal 119869 of 119877[119909] 119869lowast =119869 0119877[119909]

Proof Since 119877 is a (strong) hyperdomain the multiplicative119867V-ring is commutative So for any 119867V-ideal 119868 of 119877[119909]C(119868) = C

119897(119868) = ⋃

infin

119896=1119865119896(119868) (by Lemma 16(v)) For any

119899 isin N 119891(119909) isin 119865119899(119868)lowast rArr 119891(119909) isin sum119898

119894=1119860119894 for some

type-119897 hyperproducts 119860119894= [1198911198941(119909) lowast 119891

1198942(119909) lowast sdot sdot sdot lowast 119891

119894119898119894(119909)]119897

of elements of 119877[119909] satisfying 119860119894cap 119865119899minus1(119868) = 120601 Since

here 119877[119909] is a strong 119867V-domain (by Proposition 13) withabsorbing zero we may assume that 0

119877[119909]notin 119860119894for each 119894

(since 0119877[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119877[119909])) Then for

each 119894 (= 1 2 119898) and 119895119894(= 1 2 119898

119894) 119891119894119895119894

(119909) = 0119877[119909]

Thus by Corollary 12 ord119860

119894= ord119892

119894(119909) for any 119892

119894(119909) isin 119860

119894

Now since for each 119894 119860119894cap 119865119899minus1(119868) = 120601 so there exists

119892119894(119909) isin 119860

119894such that 119892

119894(119909) isin 119865

119899minus1(119868)lowast Then for any 119891(119909) isin

119865119899(119868)lowast ord119891(119909) ge sum119898

119894=1ord119860119894= minord119892

119894(119909) 119894 ge

ord(119865119899minus1(119868)lowast) Consequently ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast)for any 119899 isin N Now let 119891(119909) isin C(119868)

lowast be arbitrary Then119891(119909) isin 119865

119899(119868)lowast for some 119899 isin N So we have that ord119891(119909) ge

ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast) ge sdot sdot sdot ge ord(119865(119868)lowast) ge ord119868lowastThus ord(C(119868)lowast) ge ord119868lowast

5 Polynomials over Multiplicative Hyperfield

A nonzero element 119886 of a multiplicative119867V-ring (119878 + ∘) withan 119894-set E = 119890

1 1198902 119890

119899 is referred to be an E-invertible

element (or anE-unit) of 119878 if for each 119894 = 1 2 119899 (119899 isin N)there exist 119860

119894119895isin H119897(119886) = [119904

119901∘ 119904119901minus1∘ sdot sdot sdot ∘ 119904

1∘ 119886]119897 119904119894isin 119878

119901 isin N and 119861119894119896isin H119903(119886) = [119886 ∘ 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119902]119903 119904119894isin 119878

119902 isin N such that 119890119894isin (sum119898

119895=1119860119894119895) cap (sum

119897

119896=1119861119894119896) An element

119886 ( =0119878) of the multiplicative119867V-ring 119878with a hyperidentity 119890

is said to be 119890-hyperinvertible (or an 119890-hyperunit) in 119878 if thereexist 119860 isinH

119897(119886) and 119861 isinH

119903(119886) such that 119890 isin 119860 cap 119861

If 119886 isin 119878 is an E-unit (resp 119890-hyperunit) in a multiplica-tive 119867V-ring (119878 + ∘) with two 119894-sets E and E1015840 (resp withtwo hyperidentities 119890 and 1198901015840) then one can easily verify that119886 is also an E1015840-unit (resp an 1198901015840-hyperunit) in 119878 We thuscall an E-unit (resp an 119890-hyperunit) of a multiplicative 119867V-ring 119878 simply a unit (resp a hyperunit) Denote by 119880(119878)and 119880

ℎ(119878) respectively the sets of units and hyperunits of a

multiplicative119867V-ring 119878An 119867V-ideal 119868 ( = 119878) of a multiplicative 119867V-ring 119878 is

maximal in 119878 if for any 119867V-ideal 119869 of 119878 119868 ⊊ 119869 sube 119878 rArr 119869 = 119878For a commutative multiplicative119867V-ring 119878with an 119894-set thisis immediate to observe that 119886 isin 119880(119878) if and only if 119886 notin 119872for any maximal119867V-ideal119872 of 119878

Proposition 18 Let themultiplicative hyperring119877 be commu-tative and contain a hyperidentity 119890 Then for a polynomial119891(119909) = sum119886

119896119909119896isin 119877[119909] 119891(119909) isin 119880

ℎ(119877[119909]) if and only if

1198860isin 119880ℎ(119877)

Proof Since 119890 is a hyperidentity in the multiplicative hyper-ring 119877 1198901199090 isin 119877[119909] is a hyperidentity in the multiplicative119867V-ring 119877[119909] Now let 119891(119909) = sum119886

119896119909119896isin 119880ℎ(119877[119909]) Then there

exist 119892119894(119909) = sum 119887

119894119896119909119896isin 119877[119909] (119894 = 1 2 119899 119899 isin N) such that

1198901199090isin [119891(119909) lowast 119892

1(119909) lowast sdot sdot sdot lowast 119892

119899(119909)]119903 So 119890 isin 119886

0(1198871011988720 1198871198990)

and thus there exists 119887 isin 1198871011988720 1198871198990sube 119877 such that 119890 isin 119886

0119887

whence 1198860isin 119880ℎ(119877) (since 119877 is commutative)

Conversely let 1198860be hyperinvertible in 119877 with respect to

the hyperidentity 119890 Then there exists 1198870isin 119877 such that 119890 isin

11988601198870 We assert that there is a sequence 119887

119896119896isinN in 119877 whose

119899th term 119887119899(119899 isin N) is inductively defined so as to satisfy the

relation that

0119877isin 1198860119887119899+ 1198861119887119899minus1+ 1198862119887119899minus2+ sdot sdot sdot + 119886

1198991198870sdot sdot sdot () (8)

In fact we see that 0119877isin 01198771198870= (minus119886

1+ 1198861)1198870sube (119890(minus119886

1) +

1198861)1198870sube ((119886

01198870)(minus1198861) + 1198861)1198870sube (119886

01198870)(minus1198861)1198870+ 11988611198870=

1198860((minus1198870)11988611198870) + 11988611198870rArr there exist 119887

1isin (minus119887

0)11988611198870such that

0119877isin 11988601198871+ 11988611198870(the relation for 119899 = 1) Suppose for

some 119898 ge 1 the terms 119887119896(1 le 119896 le 119898 minus 1) are defined

in such a way that each 119887119896(1 le 119896 le 119898 minus 1) satisfies the

relation () for 119896 Then (1198861119887119898minus1+1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+1198861198981198870)

is defined to be a nonempty subset of 119877 Let 119905 isin 1198861119887119898minus1+

1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870 Then 0

119877= minus119905 + 119905 isin 119890(minus119905) + 119905 sube

(11988601198870)(minus119905) + 119905 sube 119886

0(minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+

1198861198981198870) + (1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) rArr there exists

119887119898isin (minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) such that

0119877isin 1198860119887119898+ 1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

1198981198870(the relation

for 119896 = 119898) Hence the assertion is true for all 119899 isin N Thus

Algebra 7

consider the polynomial 119892(119909) = sum 119887119896119909119896isin 119877[119909] Then from

the definition of lowast 119891(119909) lowast 119892(119909) = sum 119888119896119909119896 1198880isin 11988601198870and

119888119896isin 1198860119887119896+ 1198861119887119896minus1+ 1198862119887119896minus2+ sdot sdot sdot + 119886

1198961198870for 119896 isin N Thus

1198901199090isin 119891(119909) lowast 119892(119909) (due to relation and since 119890 isin 119886

01198870) So

119891(119909) is hyperinvertible in 119877[119909] that is 119891(119909) isin 119880ℎ(119877[119909])

The (left right) 119867V-ideal of a multiplicative 119867V-ring 119878generated by 119860 isin 119875(119878) is the smallest (resp left right) 119867V-ideal of 119878 containing119860which is denoted by (resp ⟨119860⟩

119897 ⟨119860⟩119903)

⟨119860⟩ The principal (left right) 119867V-ideal of the multiplicative119867V-ring 119878 generated by an element 119886 of 119878 denoted by (resp⟨119886⟩119897 ⟨119886⟩119903) ⟨119886⟩ is the (resp left right)119867V-ideal (resp ⟨119886⟩119897

⟨119886⟩119903) ⟨119886⟩ of the multiplicative119867V-ring 119878

If the multiplicative 119867V-ring 119878 has an 119894-set then for any119886 isin 119878 ⟨119886⟩

119897= H119897(119886) = ⋃sum

119899

119894=1119860119894 119860119894isin H119897(119886) 119899 isin N and

⟨119886⟩119903=H119903(119886) = ⋃sum

119899

119894=1119860119894 119860119894isinH119903(119886) 119899 isin N

Definition 19 A commutative multiplicative 119867V-ring 119878 withan 119894-setE is called amultiplicative119867V-field (resp an inversivemultiplicative 119867V-field) if 119878 ⟨0

119878⟩ sube 119880(119878) (resp 119878

⟨0119878⟩ sube 119880

ℎ(119878)) If a multiplicative119867V-field (resp an inversive

multiplicative 119867V-field) is a multiplicative hyperring thenwe call it a multiplicative hyperfield (resp an inversivemultiplicative hyperfield)

Proposition 20 Let 119865 be a polynomially structured inversivemultiplicative hyperfield Then a polynomial 119891(119909) = sum119886

119896119909119896isin

119865[119909]0119865[119909] is hyperinvertible in119865[119909] if and only if ord 119891 = 0

Proof If 119865 is an inversive multiplicative hyperfield then byProposition 18 any polynomial119891(119909) = sum119886

119896119909119896isin 119865[119909]0

119865[119909]

is hyperinvertible in 119865[119909] if and only if 1198860= 0119865 Hence the

result follows

Definition 21 A commutative multiplicative 119867V-ring 119878 iscalled a principal C-ideal 119867V-ring if every C-ideal of119878 is a principal 119867V-ideal A principal C-ideal 119867V-ringwhich is a (strong) 119867V-domain is called a principal C-ideal(strong) 119867V-domain

Proposition 22 Let 119865 be a polynomially structured inversivemultiplicative hyperfieldThen themultiplicative119867V-ring119865[119909]of polynomials over 119865 is a principalC-ideal119867V-ring

Proof Let 119890 be a hyperidentity of the inversive multiplicativehyperfield 119865 Then the polynomial 1198901199090 is a hyperidentity in119865[119909] and thus 119865[119909] = ⟨1198901199090⟩ (since 119865 is commutative) Solet 119868 be any proper C-ideal of 119865[119909] If 119868 = 0

119865[119909] then 119868 is

the principal hyperideal ⟨0119865[119909]⟩ (since for119865 having absorbing

zero 0119865[119909]

is absorbing in 119865[119909]) Suppose that 119868 = 0119865[119909]

Then take a nonzero polynomial 119891(119909) = sum119886119896119909119896isin 119868 such

that ord119891(119909) le ord 119905(119909) for any 119905(119909) isin 119868lowast Let us writeord119891(119909) = 119898 Then 119886

119898= 0119865and 119886119896= 0119865for any 0 le 119896 lt 119898

Consider then the polynomial 119892(119909) = sumlowast119896isinN0119887119896119909119896isin 119865[119909]

where 119887119896= 119886119898+119896

Then clearly 119891(119909) isin (119890119909119898) lowast 119892(119909) Alsoby Proposition 18 119892(119909) is hyperinvertible in 119865[119909] Thus thereexists ℎ(119909) isin 119865[119909] such that 1198901199090 isin 119892(119909) lowast ℎ(119909) Now119891(119909) isin (119890119909

119898) lowast 119892(119909) rArr 119891(119909) lowast ℎ(119909) sube [119890119909

119898lowast 119892(119909) lowast ℎ(119909)]

119903

Again 119891(119909) lowast ℎ(119909) sube 119868 (since 119891(119909) isin 119868) So [119890119909119898 lowast 119892(119909) lowastℎ(119909)]119903cap 119868 = 120601 Hence [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868 (since

119868 is a C-ideal and every C-ideal is a right C-ideal) Now[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap [119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119903= 120601 whence

[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap 119868 = 120601 (since [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868)

Consequently [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]119897sube 119868 (since 119868 is a left C-

ideal) Then 119890119909119898 isin (119890119909119898) lowast (1198901199090) sube 119890119909119898 lowast (119892(119909) lowast ℎ(119909)) =[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897sube 119868 Thus ⟨119890119909119898⟩ sube 119868

Now let ℎ(119909) = sum 119887119896119909119896isin 119868lowast be arbitrary Suppose that

ord ℎ = 119899 Then 119887119899= 0119865and 119887119896= 0119865for any 0 le 119896 lt 119899

By choice of 119891(119909) from 119868 here 119898 le 119899 So one can definea polynomial 119892(119909) = sum 119888

119896119909119896isin 119865[119909] where 119888

119896= 0119865for all

0 le 119896 le 119899 minus 119898 minus 1 and 119888119896= 119887119898+119896

for all 119896 ge 119899 minus 119898 Clearlythen ℎ(119909) isin 119890119909119898 lowast 119892(119909) sube ⟨119890119909119898⟩ whence 119868 sube ⟨119890119909119898⟩ Thus119868 = ⟨119890119909

119898⟩

Remark 23 In a ring an invertible element can never be adivisor of zero This not true in general for a multiplicativehyperring In fact on the commutative group of integers(Z +) if we define a hyperoperation ∘ by stating that 119909 ∘ 119910 =0 119909119910 for all 119909 119910 isin Z then (Z + ∘) is a commutative mul-tiplicative hyperring with a hyperidentity 1 Every nonzeroelement of (Z + ∘) is a zero divisor and 1 minus1 isin Z are inparticular hyperunits of (Z + ∘) To get a parity with thering theory in this regard we perceive the notion of stronghyperinvertibility of an element of amultiplicative hyperring

Definition 24 A hyperinvertible element of a multiplicativehyperring with a hyperidentity is said to be strongly hyperin-vertible (or a strong hyperunit) if it is not a zero divisor inthat multiplicative hyperring A commutative multiplicativehyperring with absorbing zero and a hyperidentity 119890 is said tobe a strongly inversive multiplicative hyperfield if each of itsnonzero elements is a strong hyperunit

Example 25 Let 119877 = 119886radic2 + 119887radic3 119886 119887 isin Q and 119860 =radic2radic3 Then with respect to usual addition + of reals(119877 +) is a commutative group with identity 0 On 119877 ∘ is ahyperoperation defined by

(119886radic2 + 119887radic3) ∘ (119888radic2 + 119889radic3)

= (119886radic2 + 119887radic3) sdot 119905 sdot (119888radic2 + 119889radic3) 119905 isin 119860

= 119901radic2 + 2119902radic3 3119902radic2 + 119901radic3

(9)

where 119901 = 2119886119888 + 3119887119889 and 119902 = 119887119888 + 119886119889Then (119877 + ∘) is a strongly inversivemultiplicative hyper-

field which is polynomially structured

Definition 26 A local (C-local) multiplicative 119867V-ring is acommutative multiplicative119867V-ring with an 119894-set which hasa unique maximal119867V-ideal (respC-ideal)

Proposition 27 Let 119865 be a polynomially structured stronglyinversive multiplicative hyperfieldThen (i) 119865[119909] is a principalC-ideal strong 119867V-domain (ii) for any 119891 isin 119865[119909] 0

119865[119909]

there exist a hyperinvertible element 119892 isin 119865[119909] and 119898 isin

N cup 0 such that 119891 isin (119890119909119898) lowast 119892 where 119890 is a hyperidentity

8 Algebra

in 119865 and (iii) 119865[119909] is a local as well as aC-local multiplicative119867V-ring

Proof (i) Here 119865 is a strongly inversive multiplicative hyper-field So 119865 is a strong hyperdomain Thus by Proposition 13the multiplicative 119867V-ring 119865[119909] is a strong 119867V-domainAgain by Proposition 22 119865[119909] is a principal C-ideal 119867V-ring (since 119865 is an inversive multiplicative hyperfield) So byDefinition 21 119865[119909] is a principalC -ideal strong119867V-domain

(ii) Let 119891 isin 119865[119909] 0119865[119909] and 119869 = ⟨119891⟩ Then 119869 =H

119903(119891)

So for any ℎ isin 119869lowast there exist119860119894= [119891lowast119891

1198941lowast1198911198942lowastsdot sdot sdotlowast119891

119894119898119894]119903isin

H119903(119891) (119894 = 1 2 119896 119896119898

119894isin N) such that ℎ isin sum119896

119894=1119860119894

Thus ℎ = sum119896119894=1119905119894for some 119905

119894isin 119860119894 Since 119865[119909] is a strong119867V-

domain (by Proposition 13) with absorbing zero so 0119865[119909]isin

119860119894= [119891 lowast 119891

1198941lowast 1198911198942lowast sdot sdot sdot lowast 119891

119894119898119894]119903rArr [119891 lowast 119891

1198941lowast 1198911198942lowast

sdot sdot sdot lowast 119891119894119898119894]119903= 0119865[119909] Hence without any loss of generality

(since 0119865[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119865[119909])) we may

assume that 0119865[119909]notin 119860119894for any 119894 (since ℎ = 0

119865[119909]) Then

for any 119894 = 1 2 119896 and 119895 = 1 2 119898119894 119891119894119895= 0119865[119909]

So by Proposition 11(ii) ord 119905

119894= ord(119860

119894) ge ord119891 and so

ord ℎ = minord 119905119894 119894 ge ord119891 Thus ord119869lowast ge ord119891 So

by Proposition 17 119891 is a nonzero polynomial in the C-idealC(119869) such that ord119891 le ord ℎ for all ℎ isin C(119869)lowast So there existsan invertible element 119892 isin 119865[119909] and an integer119898 ge 0 such that119891 isin (119890119909

119898) lowast 119892 where 119890 is a hyperidentity in 119865 (see proof of

Proposition 22)(iii) For any integer119898 ge 1 since 119890119909119898 isin (119890119909) lowast (119890119909119898minus1) so

⟨119890119909119898⟩ sube ⟨119890119909⟩ Again for any119867V-ideal 119868 of the multiplicative

119867V-ring119865[119909] if119891 isin 119868lowast is such that ord119891 le ord119892 for all119892 isin 119868lowast

then 119868 sube ⟨119890119909119898⟩ where 119898 = ord119891 So ⟨119890119909⟩ is the uniquemaximal119867V-ideal in 119865[119909] Thus 119865[119909] is a local multiplicative119867V-ring Now 119890119909 isin C

1(as defined in Proposition 15) So

ord(119890119909) le ord119891 for any 119891 isin Clowast1 Thus C

1being a C-ideal

of 119865[119909] we have C1= ⟨119890119909⟩ (see proof of Proposition 22)

Hence 119865[119909] is aC-local multiplicative119867V-ring

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vougiouklis ldquoThe fundamental relation in hyperrings Thegeneral hyperfieldrdquo in Algebraic Hyperstructures and Appli-cations (Xanthi1990) pp 203ndash211 World Science PublisherTeaneck NJ USA 1991

[2] M R Darafsheh and B Davvaz ldquo119867V-ring of fractionsrdquo ItalianJournal of Pure and Applied Mathematics no 5 pp 25ndash34 1999

[3] B Davvaz and V Leoreanu-FoteaHyperring Theory and Appli-cations International Academic Press Palm Harbor Fla USA2007

[4] S Spartalis A Dramalides and T Vougiouklis ldquoOn 119867V-groupringsrdquoAlgebras Groups and Geometries vol 15 no 1 pp 47ndash541998

[5] T Vougiouklis ldquo119867V-groups defined on the same setrdquo DiscreteMathematics vol 155 no 1ndash3 pp 259ndash265 1996

[6] R Procesi and R Rota ldquoMultiplicative H119907-rings and com-

plementary hyperstructuresrdquo Journal of Discrete MathematicalSciences amp Cryptography vol 11 no 4 pp 447ndash456 2008

[7] S Spartalis ldquoOn the number of 119867Vminus119903119894119899119892119904 with 119875-hyperopera-tionsrdquoDiscrete Mathematics vol 155 no 1ndash3 pp 225ndash231 1996

[8] P Corsini Prolegomena of Hypergroup Theory Rivista di Mate-matica Pura ed Applicata Aviani Tricesimo Italy 1993

[9] M K Sen and U Dasgupta ldquoSome aspects of119866119867-ringsrdquoAnnals

of the Alexandru Ioan Cuza UniversitymdashMathematics vol 56no 2 pp 253ndash272 2010

[10] C Namnak N Triphop and Y Kemprasit ldquoHomomorphismsof somemultiplicative hyperringsrdquo Set-ValuedMathematics andApplications vol 1 no 2 pp 145ndash152 2008

[11] D M Olson and V K Ward ldquoA note on multiplicativehyperringrdquo Italian Journal of Pure andAppliedMathematics vol1 pp 77ndash84 1997

[12] R Procesi and R Rota ldquoComplementary multiplicative hyper-ringsrdquoDiscreteMathematics vol 308 no 2-3 pp 188ndash191 2008

[13] R Procesi and R Rota ldquoOn some classes of hyperstructuresrdquoDiscrete Mathematics vol 208-209 pp 485ndash497 1999

[14] R Rota ldquoStrongly distributive multiplicative hyperringsrdquo Jour-nal of Geometry vol 39 no 1-2 pp 130ndash138 1990

[15] R Rota ldquoSugli Iperanelli Moltiplicativirdquo Rendiconti di Matem-atica Series VII vol 2 no 4 pp 711ndash724 1982

[16] R Procesi Ciampi andR Rota ldquoPolynomials overmultiplicativehyperringsrdquo Journal of Discrete Mathematical Sciences amp Cryp-tography vol 6 no 2-3 pp 217ndash225 2003

[17] T Vougiouklis Hyperstructures and Their RepresentationsMonographs in Mathematics Hardonic Press 1994

[18] U Dasgupta ldquoOn prime and primary hyperideals of a multi-plicative hyperringrdquoAnalele Stiintifice ale Universitatii Al I Cuzadin IasimdashMatematica vol 58 no 1 pp 19ndash37 2012

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Function Spaces

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Algebra

119867V-ring wherever considered will always be assumed to bea semidistributive multiplicative 119867V-ring with the condition(R)

In 1982 the notion ofmultiplicative hyperring is inductedin the study on hyperring theory by Rota which is subse-quently investigated in [10ndash14] A commutative group (119877 +)endowed with an associative hyperoperation ∘ is called amultiplicative hyperring [15] if (i) ∘ is semidistributive acrossthe operation + on 119877 and (ii) ∘ satisfies the condition (R) forelements in 119877 An associative hyperoperation is eventuallyweakly associative and thus a multiplicative hyperring iseventually a semidistributive multiplicative 119867V-ring withcondition (R)

Procesi Ciampi and Rota define in [16] polynomialsover multiplicative hyperring as follows let (119877 + sdot) be amultiplicative hyperring with absorbing zero 0

119877and let 119909 be

any symbol out of 119877Then a polynomial in 119909 is an expressionof the form 119891(119909) = 119886

01199090+ 11988611199091+ 11988621199092+ sdot sdot sdot = sum 119886

119896119909119896

(119896 isin N 119886119896isin 119877) in which + is a connective and only a

finite number of the 119886119896rsquos (called the coefficients of 119909119896 in 119891(119909))

are different from zero (0119877) of 119877 The degree of a polynomial

119891(119909) = sum119886119896119909119896 (in short deg 119891(119909)) is a nonnegative integer 119899

such that 119886119899= 0119877and 119886119896= 0119877 for all 119896 gt 119899 A polynomial

119891(119909) = sum119886119896119909119896 over a multiplicative hyperring 119877 will be

written as 119891(119909) = sum119899119896=1119886119896119909119896 when and only when deg119891(119909) =

119899 For an integer 119898 isin N and any 119886 119886119896 119887 isin 119877 (119886 = 0

119877

119887 = 0119877 119896 = 1 2 119898 minus 1) we write the polynomials

119886(119909) = 01198771199090+sdot sdot sdot+0

119877119909119898minus1+119886119909119898 and119891(119909) = 1198861199090+119886

11199091+sdot sdot sdot+

119886119898minus1119909119898minus1+119887119909119898 simply as 119886119909119898 and 119886+119886

1119909+ sdot sdot sdot + 119886

119898minus1119909119898minus1+

119887119909119898 respectively Denote by 119877[119909] the set of all polynomials

in 119909 over 119877 and define on 119877[119909] a binary operation + and ahyperoperation lowast as follows for any two polynomials 119891(119909) =sum119886119896119909119896 and 119892(119909) = sum 119887

119896119909119896 from 119877[119909] 119891(119909) + 119892(119909) =

sum(119886119896+ 119887119896)119909119896 and 119891(119909) lowast 119892(119909) = sum 119888

119896119909119896 119888119896isin sum119894+119895=119896119886119894119887119895

where for any 119886 119887 isin 119877 the juxtaposition 119886119887 means theset 119886 sdot 119887 The purpose of the present paper is to study theproperties of the hyperstructure (119877[119909] + lowast) in connection tothose of a particular class of multiplicative hyperrings calledpolynomially structured multiplicative hyperrings which wedescribe formally in the following section

2 Polynomially StructuredMultiplicative Hyperring

It is asserted in [16] that for a multiplicative hyperring 119877 thehyperstructure (119877[119909] + lowast) is always a multiplicative hyper-ring But we note here that given a multiplicative hyperring119877 with absorbing zero the hyperoperation lowast (as is definedin Section 1) does not necessarily induce a multiplicativehyperring structure over the group of polynomials (119877[119909] +)In fact we have the following example

Example 1 Let (Z + sdot) be the ring of integers and (Z + ∘)the multiplicative hyperring where for any 119886 119887 isin Z 119886 ∘ 119887 =2119886119887 3119886119887 (denoting the product 119886 sdot 119887 of elements in thering (Z + sdot) simply by the juxtaposition 119886119887) Consider three

polynomials 119891(119909) = 1 + 2119909 119892(119909) = 2 + 1119909 and ℎ(119909) = 1 + 3119909over the multiplicative hyperring (Z + ∘)

Then the set of coefficients of 119909 in the polynomialsbelonging to (119891(119909) lowast 119892(119909)) lowast ℎ(119909) is 119872 = 4 6 ∘ 3 +

10 11 14 15 ∘ 1 Again the set of coefficients of 119909 in thepolynomials belonging to 119891(119909) lowast (119892(119909) lowast ℎ(119909)) is 119873 = 1 ∘14 15 20 21+2∘4 6 By a tedious but routine calculationone can see that 57 74 82 sube 119872 whereas 57 74 82 cap 119873 =120601 So there are some polynomials in (119891(119909) lowast 119892(119909)) lowast ℎ(119909)the coefficient of 119909 in each of which is an element of the set57 74 82These polynomials do not belong to119891(119909)lowast(119892(119909)lowastℎ(119909)) (since 57 74 82cap119873 = 120601)Thus (119891(119909)lowast119892(119909))lowastℎ(119909) sube119891(119909) lowast (119892(119909) lowast ℎ(119909)) So there is no question of claiming(Z[119909] + lowast) to be a multiplicative hyperring

However we observe that it is possible to constructa multiplicative hyperring 119877 corresponding to which thehyperstructure (119877[119909] + lowast) turns out to be a multiplicative119867V-ring if not a multiplicative hyperring at all

Let us consider a commutative group (119866 +) Suppose that120601 = 119860 sube Hom(119866Hom(119866)) is such that for any 119886 119887 isin 119860and 119909 119910 isin 119866 (119909

119886(119910))119887= 119909119886119910119887 where 119909

119886denotes the image

of 119909 isin 119866 under 119886 isin 119860 and 119909119886119910119887is simply the mapping

composition Define a hyperoperation ∘119860on119866 by stating that

119909∘119860119910 = 119909

119886(119910) 119886 isin 119860 Then we have the following

Lemma2 (119866 + ∘119860) is amultiplicative hyperring with absorb-

ing zero

Proof Let 119909 119910 119911 isin 119866 Then 119901 isin 119909∘119860(119910∘119860119911) rArr 119901 isin 119909∘

119860119904 (for

some 119904 isin 119910∘119860119911) rArr 119901 = 119909

119886(119904) and 119904 = 119910

119887(119911) (for some

119886 119887 isin 119860) rArr 119901 = 119909119886(119910119887(119911)) = (119909

119886119910119887)(119911) = (119909

119886(119910))119887(119911) rArr

119901 isin (119909119886(119910))∘119860119911 rArr 119901 isin (119909∘

119860119910)∘119860119911 (since 119909

119886(119910) isin 119909∘

119860119910)

So 119909∘119860(119910∘119860119911) sube (119909∘

119860119910)∘119860119911 The reverse inclusion can also

be shown to be true by adopting similar arguments Hence119909∘119860(119910∘119860119911) = (119909∘

119860119910)∘119860119911 Now119901 isin 119909∘

119860(119910+119911) rArr 119901 isin 119909

119886(119910+119911)

(for some 119886 isin 119860) = 119909119886(119910)+119909

119886(119911) isin 119909∘

119860119910+119909119886∘119860119911 So 119909∘

119860(119910+

119911) sube 119909∘119860119910 + 119909119886∘119860119911 Again 119901 isin (119910 + 119911)∘

119860119909 rArr 119901 isin (119910 + 119911)

119886119909

(for some 119886 isin 119860)= (119910119886+119911119886)119909 (since 119886 isin Hom(119866Hom(119866))) =

119910119886(119909) + 119911

119886(119909) sube 119910∘

119860119909 + 119911∘

119860119909 So (119910 + 119911)∘

119860119909 sube 119910∘

119860119909 + 119911∘

119860119909

Moreover 119909∘119860(minus119910) = 119909

119886(minus119910) 119886 isin 119860 = minus(119909

119886(119910)) 119886 isin 119860

(since 119909119886isin Hom(119866)) = minus119909

119886(119910) 119886 isin 119860 = minus(119909∘

119860119910) and

(minus119909)∘119860119910 = (minus119909)

119886(119910) 119886 isin 119860 = (minus(119909

119886))(119910) 119886 isin 119860

(since 119886 isin Hom(119866Hom(119866))) = 119909119886(minus119910) 119886 isin 119860 = 119909∘

119860(minus119910)

Thus (119866 + ∘119860) is a multiplicative hyperring Finally if 0

119866

denotes the identity element of the group (119866 +) then 119909∘0119866=

119909119886(0119866) 119886 isin 119860 = 0

119866 (since for 119886 isin 119860 119909

119886isin Hom(119866)) and

also 0119866∘ 119909 = (0

119866)119886(119909) 119886 isin 119860 = 0

119866 (since for 119886 isin

Hom(119866Hom(119866)) (0119866)119886is the zero homomorphism from 119866

to 119866) Thus 0119866is absorbing in the multiplicative hyperring

(119866 + ∘119860)

The multiplicative hyperring (119866 + ∘119860) defined in

Lemma 2 is called a multiplicative 119860-hyperring (of course ifsuch a set 119860 exists for the group (119866 +)) That a multiplicative119860-hyperring exists is evident in the following example

Example 3 Let (119877 + sdot) be a ringThen as is shown in [9 13](119877 +

lowast

119875) is a multiplicative hyperring with absorbing zero

Algebra 3

where 119875 isin 119875(119877) with |119875| ge 2 andlowast

119875 is the 119875-hyperoperation[8 17] on the semigroup (119877 sdot) that is119909

lowast

119875 119910 = 119909sdot119886sdot119910 119886 isin 119875

for all 119909 119910 isin 119877 Now for each 119886 isin 119875 and 119909 isin 119877 we define amapping 119909

119886 119877 rarr 119877 by stating that 119909

119886(119910) = 119909 sdot 119886 sdot 119910 for all

119910 isin 119877 Then 119909119886isin Hom(119877 +) Thus corresponding to each

119886 isin 119875 we have a mapping 119886 (119877 +) rarr Hom(119877 +) givenby 119886(119909) = 119909

119886 for any 119909 isin 119877 Then 119875 sube Hom(119877Hom(119877))

Moreover for any 119909 119910 119911 isin 119877 and 119886 119887 isin 119875 (119909119886(119910))119887(119911) = (119909 sdot

119886sdot119910)sdot119887sdot119911 = 119909sdot119886sdot(119910sdot119887sdot119911) = 119909sdot119886sdot(119910119887(119911)) = 119909

119886(119910119887(119911)) = (119909

119886119910119887)(119911)

that is (119909119886(119910))119887= 119909119886119910119887 Thus the hyperoperation ∘

119860(for

119860 = 119875) is defined on the group (119877 +) Note that for any119909 119910 isin 119877 119909∘

119860119910 = 119909

119886(119910) 119886 isin 119860 = 119909 sdot 119886 sdot 119910 119886 isin 119875 = 119886

lowast

119875 119910Thus (119877 +

lowast

119875) is a multiplicative 119860-hyperring for 119860 = 119875

Proposition4 For amultiplicative119860-hyperring (119877 + ∘119860) the

hyperoperation lowast induces a multiplicative119867V-ring structure onthe group (119877[119909] +) of polynomials over 119877

Proof Let 119891(119909) = sum119886119896119909119896 119892(119909) = sum 119887

119896119909119896 and ℎ(119909) = sum119889

119896119909119896

be three polynomials in 119877[119909] Then (119891(119909) lowast 119892(119909)) lowast ℎ(119909) =sum 119905119896119909119896 119905119896isin sum119906+V=119896(sum119894+119895=119906 119886119894∘119860119887119895)∘119860119889V and 119891(119909) lowast (119892(119909) lowast

ℎ(119909)) = sum 119904119896119909119896 119904119896isin sum119906+V=119896 119886119906∘119860(sum119894+119895=V 119887119894∘119860119889119895) Now we

choose and fix an element 120572 isin 119860 Then for each 119896 isin N0

119901119896= sum

119906+V=119896( sum

119894+119895=119906

(119886119894)120572(119887119895))

120572

(119889V)

isin sum

119906+V=119896( sum

119894+119895=119906

119886119894∘119860119887119895)∘119860119889V

119902119896= sum

119906+V=119896(119886119906)120572( sum

119894+119895=V(119887119894)120572(119889119895))

isin sum

119906+V=119896119886119906∘119860( sum

119894+119895=V119887119894∘119860119889119895)

(1)

Again 119901119896

= sum119906+V=119896(sum119894+119895=119906(119886119894)120572(119887119895))120572(119889V) =

sum119906+V=119896sum119894+119895=119906((119886119894)120572(119887119895))120572(119889V) (since120572 is a homomorphism) =sum119906+V=119896sum119894+119895=119906((119886119894)120572(119887119895)120572)(119889V) (since (119909120572(119910))120572 = 119909120572119910120572) 119902119896 =sum119906+V=119896(119886119906)120572(sum119894+119895=V(119887119894)120572(119889119895)) = sum119906+V=119896sum119894+119895=V(119886119906)120572((119887119894)120572(119889119895))

(since (119886119906)120572

is a homomorphism) =sum119906+V=119896sum119894+119895=V((119886119906)120572(119887119894)120572)(119889119895) Clearly then for each 119896 isin N0

119901119896= 119902119896isin ( sum

119906+V=119896( sum

119894+119895=119906

119886119894∘119860119887119895)∘119860119889V)

⋂( sum

119906+V=119896119886119906∘119860( sum

119894+119895=V119887119894∘119860119889119895))

(2)

Hence (119891(119909) lowast 119892(119909)) lowast ℎ(119909) cap 119891(119909) lowast (119892(119909) lowast ℎ(119909)) = 120601It is shown in [16] that for any multiplicative hyperring 119877the hyperoperationlowast defined on119877[119909] is semidistributive overthe operation + on 119877[119909] and also satisfies the condition (R)for any two polynomials in 119877[119909] Thus for the multiplicative119860-hyperring (119877 + ∘

119860) the hyperstructure (119877[119909] + lowast) is a

multiplicative 119867V-ring (with absorbing zero 0119877[119909]= 01198771199090+

01198771199091+ 01198771199092+ sdot sdot sdot )

We call a multiplicative hyperring (119877 + sdot)with absorbingzero polynomially structured if (119877[119909] + lowast) is a multiplicative119867V-ringThe class ofmultiplicative119860-hyperrings is a subclassof the class of polynomially structured multiplicative hyper-rings (by Proposition 4) Throughout the rest of the paper119877 will stand for a polynomially structured multiplicativehyperring

3 Polynomials over Integral Hyperrings

An element 119886 ( = 0119877) of a multiplicative 119867V-ring (119878 + ∘) is

a left (resp right) divisor of zero in 119878 if there exists 119887 isin 119878lowast(resp 119888 isin 119878lowast = 119878 0

119878) such that 0

119878isin 119886 ∘ 119887 (resp 0

119904isin 119888 ∘ 119886)

and a divisor of zero in 119878 if it is either a left or a right divisorof zero in 119878 An element 119886 ( = 0

119878) of 119878 is a left (resp right)

strong divisor of zero in 119878 if there exists 119887 isin 119878lowast = 119878 0119878

(resp 119888 isin 119878lowast) such that 119886 ∘ 119887 = 0119878 (resp 119888 ∘ 119886 = 0

119878) and

a strong divisor of zero in 119878 if it is either a left or a right strongdivisor of zero in 119878

Definition 5 A multiplicative 119867V-ring is called an integral119867V-ring if there is no strong divisor of zero in it A commuta-tive integral119867V-ring is an119867V-domain A strong integral119867V-ring is a multiplicative119867V-ring in which there is no divisor ofzero A strong 119867V-domain is a commutative strong integral119867V-ring We call an integral119867V-ring (resp an119867V-domain) 119878simply an integral hyperring (resp a hyperdomain) [13]when the119867V-ring 119878 is a multiplicative hyperring

Before entering into the study of the multiplicative 119867V-ring 119877[119909] of polynomials over integral hyperring and hyper-domain let us go through the following useful observations

Remark 6 (a) For any polynomially structuredmultiplicativehyperring119877 the identity element 0

119877[119909]= 01198771199090+01198771199091+01198771199092+

sdot sdot sdot of the group (119877[119909] +) is absorbing in the multiplicative119867V-ring (119877[119909] + lowast)

(b) 119877[119909] is commutative if 119877 is a commutative multiplica-tive hyperring

(c)Themapping120595 119877 rarr 119877[119909] defined by for all 119903 isin 119877lowast120595(119903) = 119903119909

0 and 120595(0119877) = 0119877[119909]

is a strong monomorphism Infact for any 119903 119904 isin 119877120595(119903119904) = 120595(119888) 119888 isin 119903119904 = 1198881199090 119888 isin 119903119904 =(1199031199090) lowast (119904119909

0) = 120595(119903) lowast 120595(119904) Thus 119877 can be identified with its

isomorphic image in 119877[119909] and for any 119903 isin 119877lowast we can writethe polynomial 1199031199090 simply as 119903 and the zero polynomial 0

119877[119909]

as 0119877

(d) If E = 1198901 1198902 119890

119899 is an 119894-set of 119877 then the set

E119909sube 119877[119909] is an 119894-set in the multiplicative 119867V-ring 119877[119909]

where for any119860 isin 119875lowast(119877)119860119909denotes the set 1198861199090 119886 isin 119860In

4 Algebra

fact for any 119891(119909) = sum119886119896119909119896isin 119877[119909] we have that sum119899

119894=1119890119894lowast

119891(119909) = sum119899

119894=1(119890119894lowast sum119886

119896119909119896) = sum

119899

119894=1sum 119886119894119896119909119896 119886119894119896isin 119890119894119886119896 =

sum(sum119899

119894=1119886119894119896)119909119896 119886119894119896isin 119890119894119886119896 = sum 119887

119896119909119896 119887119896isin sum119899

119894=1119890119894119886119896

Then since 119886119896isin sum119899

119894=1119890119894119886119896 we have that119891(119909) isin sum119899

119894=1119890119894lowast119891(119909)

identifying 1198901198941199090 with 119890

119894 Similarly one can see that 119891(119909) isin

sum119899

119894=1119891(119909) lowast 119890

119894

On the other hand if for some 119899 isin N E119877[119909]= 120598119894(119909) =

sum 119890119894119896119909119896isin 119877[119909] 119894 = 1 2 119899 is an 119894-set in 119877[119909] then

E = 1198901198940 119894 = 1 2 119899 is an 119894-set in 119877 In fact for any

119886 isin 119877 1198861199090 isin sum119899119894=1(1198861199090) lowast 120598119894(119909) = sum

119899

119894=1sum 119886119894119896119909119896isin 119877[119909] 119886

119894119896isin

119886119890119894119896 = sum(sum

119899

119894=1119886119894119896)119909119896isin 119877[119909] 119886

119894119896isin 119886119890119894119896 = sum 119886

119896119909119896isin

119877[119909] 119886119896isin sum119899

119894=1119886119890119894119896 rArr 119886 isin sum

119899

119894=11198861198901198940 Similarly from

1198861199090isin sum119899

119894=1120598119894(119909) lowast (119886119909

0) one may arrive at 119886 isin sum119899

119894=11198901198940119886

This is clear fromRemark 6 (d) that for any hyperidentity119890 of 119877 the polynomial 1198901199090 isin 119877[119909] is a hyperidentity in themultiplicative119867V-ring 119877[119909] Is every hyperidentity of 119877[119909] ofthe form 1198901199090 for some hyperidentity 119890 of 119877 Following is anexample of a multiplicative hyperring 119877 such that 119877[119909] has ahyperidentity 120598(119909) = 1198901199090 for any hyperidentity 119890 of 119877

Example 7 Let (Z + sdot) be the ring of integers and 119875 = 0 1Then (Z +

lowast

119875) is a commutative polynomially structuredmultiplicative hyperring (as is shown in Example 3) Denotethe multiplicative hyperring (Z +

lowast

119875) by Z119875 Consider a

polynomial 1 + 1119909 isin Z119875[119909] Then for any 119891(119909) = sum119886

119896119909119896isin

Z119875[119909] we see that (1+1119909)lowast119891(119909) = sum 119887

119896119909119896 1198870isin 1lowast

119875 1198860 and119887119896isin 1lowast

119875 119886119896 + 1lowast

119875 119886119896minus1 (for 119896 isin N) Clearly 1198860 isin 1lowast

119875 1198860 andfor any 119896 isin N 119886

119896isin 1lowast

119875 119886119896 + 1lowast

119875 119886119896minus1 (since 0 isin 1lowast

119875 119886119896minus1)Thus 119891(119909) isin (1 + 1119909) lowast 119891(119909) and so 1 + 1119909 is a hyperidentityin Z119875[119909] which is not in the form 1198901199090 for any hyperidentity

119890 of Z119875

Remark 8 Let 120598(119909) = sum 119890119896119909119896isin 119877[119909] be a hyperidentity in the

multiplicative119867V-ring 119877[119909] Then from Remark 6 (d) 1198900is a

hyperidentity in the multiplicative hyperring 119877

Proposition 9 Let 119877 be a strong integral hyperring Thenevery hyperidentity in the multiplicative119867V-ring 119877[119909] is of theform 1198901199090 for some hyperidentity 119890 of 119877

Proof Suppose that 120598(119909) = sum 119890119896119909119896isin 119877[119909] is a hyperidentity

in the multiplicative 119867V-ring 119877[119909] Then by the Remark 8119890 = 1198900is a hyperidentity in119877 Now let 119886 isin 1198770

119877 be arbitrary

and 119891(119909) = sum119886119896119909119896isin 119877[119909] where 119886

0= 119886 and 119886

119896= 0119877 for all

119896 isin N Then 119891(119909) isin 120598(119909) lowast 119891(119909) cap 119891(119909) lowast 120598(119909) whereby0119877isin 119890119896119886 cap 119886119890

119896 for all 119896 isin N whence 119890

119896= 0119877for all 119896 isin N

(since 119877 is a strong integral hyperring and 119886 = 0119877) Thus

120598(119909) = 1198901199090+01198771199091+01198771199092+sdot sdot sdot where 119890 = 119890

0is a hyperidentity

in 119877

Definition 10 If 0119877[119909]

= 119891(119909) = sum119886119896119909119896isin 119877[119909] then the

smallest integer 119899 such that 119886119899= 0119877is called the order of119891(119909)

and is denoted by ord119891(119909) The order of 0119877[119909]

is defined to bezero For a nonempty set119860 isin 119875(119877[119909]) the smallest element inthe set ord119891(119909) 119891(119909) isin 119860 does exist and is called the orderof 119860 being denoted by ord119860 that is ord119860 = minord119891(119909) 119891(119909) isin 119860

In the next proposition we will find some propertiesof ord119860 for some 119860 isin 119875lowast(119877[119909])(= 119875(119877[119909]) 120601) Forthat it is necessary at this point to frame some suitablenotations corresponding to different types of hyperproductsof elements in the multiplicative119867V-ring 119877[119909] Indeed in anymultiplicative 119867V-ring (119878 + ∘) (which is not a multiplicativehyperring) the expression like 119904

1∘1199042∘1199043∘sdot sdot sdot∘119904

119899(119904119894isin 119878 119899 ge 3)

bears no connotation in 119878 unless the parentheses ldquo(rdquo andldquo)rdquo are meaningfully inserted in Note that the following twoexpressions

1199041∘ (11199042∘ (21199043∘ (31199044∘ sdot sdot sdot ∘ (

119899minus3119904119899minus2∘ (119899minus2119904119899minus1∘ 119904119899)119899minus2)119899minus3sdot sdot sdot )3)2)1

(1(2(3sdot sdot sdot (119899minus3(119899minus21199041∘ 1199042)119899minus2∘ 1199043)119899minus3∘ sdot sdot sdot ∘ 119904

119899minus3)3∘ 119904119899minus2)2∘ 119904119899minus1)1∘ 119904119899

(3)

are meaningful called the finite hyperproducts of type 119897 andtype 119903 andwritten in notations respectively as [119904

1∘1199042∘sdot sdot sdot∘119904

119899]119897

and [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903

Proposition 11 (i) For any 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]

119890119894119905ℎ119890119903 0119877[119909]isin 119891 (119909) lowast 119892 (119909)

119900119903 ord (119891 (119909) lowast 119892 (119909)) ge ord 119891 (119909) + ord 119892 (119909) (4)

(ii) If 119877 is a strong hyperdomain then the implication that

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119903

) = ord 119892 (119909) (5)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119903holds true for any

119899 isin N with 119899 ge 2

Proof (i) Let ord119891(119909) = 119899 and ord119892(119909) = 119898 Then 119891(119909) =sum119886119896119909119896 where 119886

119899= 0119877 119886119896= 0119877for 119896 lt 119899 and 119892(119909) = sum 119887

119896119909119896

where 119887119898= 0119877 119887119896= 0119877for 119896 lt 119898 So for anysum119888

119896119909119896isin 119891(119909)lowast

119892(119909) we see that whenever 119896 lt 119899 + 119898 119888119896isin sum119894+119895=119896119886119894119887119895=

0119877 (since 0

119877is absorbing in 119877 and 119886

119896= 0119877(119896 lt 119899) 119887

119896=

0119877(119896 lt 119898)) Thus if 0

119877[119909]notin 119891(119909) lowast 119892(119909) then for any ℎ(119909) =

sum 119888119896119909119896isin 119891(119909) lowast 119892(119909) we have that ord ℎ(119909) ge 119899 + 119898 So

ord(119891(119909) lowast 119892(119909)) ge 119899 + 119898 = ord119891(119909) + ord119892(119909)(ii) Now suppose that 119877 is a strong hyperdomain with

absorbing zero Consider two polynomials 119891(119909) 119892(119909) isin119877[119909] 0

119877[119909] Let ord119891(119909) = 119901 and ord119892(119909) = 119898 Then

119891(119909) = sum119886119896119909119896 where 119886

119901= 0119877 119886119896= 0119877for 119896 lt 119901 and

Algebra 5

119892(119909) = sum 119887119896119909119896 where 119887

119898= 0119877 119887119896= 0119877for 119896 lt 119898

So 0119877notin 119886119901119887119898(since 119877 is a strong hyperdomain) and also

119886119901119887119898= sum119894+119895=119901+119898

119886119894119887119895(since 119886

119896= 0119877for 119896 lt 119901 and 119887

119896= 0119877

for 119896 lt 119898) Thus for any ℎ(119909) = sum 119888119896119909119896isin 119891(119909) lowast 119892(119909)

119888119901+119898

= 0119877(since 119888

119901+119898isin sum119894+119895=119901+119898

119886119894119887119895= 119886119901119887119898) and 119888

119896= 0119877

for all 119896 with 0 le 119896 lt 119901 + 119898 Thus ordℎ(119909) = 119901 + 119898 forany ℎ(119909) isin 119891(119909) lowast 119892(119909) and so ord(119891(119909) lowast 119892(119909)) = 119901 + 119898Hence ord119891(119909)+ord119892(119909) = ord(119891(119909)lowast119892(119909)) = ord ℎ(119909) forall ℎ(119909) isin 119891(119909) lowast 119892(119909) So the implication is true for 119899 = 2(noting that [119891(119909) lowast 119892(119909)]

119903= 119891(119909) lowast 119892(119909)) Suppose that for

some integer 119896 ge 2 the implication holds true for each valueof 119899 ranging from 2 to 119896minus1 and take any119891

119894(119909) isin 119877[119909]0

119877[119909]

for 119894 = 1 2 119896 Thensum119896minus1119894=1

ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast

sdot sdot sdotlowast119891119896minus1(119909)]119903) = ord ℎ(119909) for all ℎ(119909) isin [119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast

119891119896minus1(119909)]119903 Now let 119892(119909) isin [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903be

arbitraryThen 119892(119909) isin ℎ(119909) lowast119891119896(119909) for some ℎ(119909) isin [119891

1(119909) lowast

1198912(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903 So ord119892(119909) = ord(ℎ(119909) lowast 119891

119896(119909)) =

ord ℎ(119909) + ord119891119896(119909) = sum

119896minus1

119894=1ord119891119894(119909) + ord119891

119896(119909)(since ℎ(119909) isin

[1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903) = sum

119896

119894=1ord119891119894(119909) Thus

sum119896

119894=1ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119896(119909)]119903) = ord119892(119909)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903

Hence by strong induction the implication follows forany 119899 isin N with 119899 ge 2

Corollary 12 If 119877 is a strong hyperdomain then the implica-tion

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119897

) = ord 119892 (119909) (6)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897holds true for any

119899 isin N with 119899 ge 2

Proof Since themultiplicative hyperring119877 is a hyperdomainit is commutative and so 119877[119909] is also a commutative multi-plicative 119867V-ring Hence for any 119891119894(119909) isin 119877[119909] 0119877[119909] (119894 =1 2 119896) we have that [119891

1(119909)lowast119891

2(119909)lowast119891

3(119909)lowastsdot sdot sdotlowast119891

119896(119909)]119897=

[119891119896(119909) lowast 119891

119896minus1(119909) lowast sdot sdot sdot lowast 119891

2(119909) lowast 119891

1(119909)]119903 Hence the assertion

follows straight from Proposition 11

Proposition 13 If the multiplicative hyperring 119877 is a stronghyperdomain the multiplicative 119867V-ring 119877[119909] is a strong 119867V-domain

Proof 119877 being a strong hyperdomain is a commutativemultiplicative hyperring Thus 119877[119909] is a commutative mul-tiplicative 119867V-ring Again since 0119877 is absorbing in 119877 0

119877[119909]

is also absorbing in 119877[119909] Thus we take 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]Then by Proposition 11 ord(119891(119909)lowast119892(119909)) = ord119891(119909)+

ord119892(119909) gt 0 Hence 0119877[119909]notin 119891(119909) lowast 119892(119909) So 119877[119909] is a strong

119867V-domain

4 C-Ideals in 119877[119909]

A subgroup 119868 of the group (119878 +) is called a left (resp right)119867V-ideal of a multiplicative 119867V-ring (119878 + ∘) if for any 119904 isin 119878

and 119886 isin 119868 119904 ∘ 119886 sube 119868 (resp 119886 ∘ 119904 sube 119868) 119868 is an119867V-ideal of 119878 if itis both a left and a right119867V-ideal of 119878

We call an 119867V-ideal of a multiplicative 119867V-ring 119878 simplya hyperideal when 119878 is a multiplicative hyperringThe notionof a typical hyperideal in a multiplicative hyperring calledC-ideal is introduced in [18] to study prime and primaryhyperideals of multiplicative hyperrings A hyperideal 119868 of amultiplicative hyperring 119878 is a C-ideal if for any 119860 isin C 119860 cap119868 = 120601 rArr 119860 sube 119868 where C = 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899 119904119894isin 119878

119899 isin N sube 119875lowast(119878) Following is the definition of aC-ideal in anarbitrary multiplicative119867V-ring

Definition 14 A left (resp right) 119867V-ideal 119868 of a multiplica-tive 119867V-ring (119878 + ∘) is called a left (resp right) C-ideal iffor any type 119897 hyperproduct [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897(resp type 119903

hyperproduct [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903) of elements 119904

119894isin 119878 we have

that [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897cap 119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897sube 119868 (resp

[1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903cap119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903sube 119868) An119867V-ideal

119868 of a multiplicative119867V-ring 119878 is called a C-ideal if it is a leftas well as a rightC-ideal in 119878

We writeC-ideal(119878) (respC-ideallowast(119878)) to denote the setof all (resp proper)C-ideals of a multiplicative119867V-ring 119878 Ina commutative multiplicative119867V-ring 119878 every left C-ideal isa right C-ideal and vice versa since commutativity impliesthe equality [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903= [119904119899∘ 119904119899minus1∘ sdot sdot sdot ∘ 119904

1]119897 for any

119904119894isin 119877 Note that in amultiplicative hyperring (even if it is not

commutative) a hyperideal is a leftC-ideal (as an119867V-ideal) ifand only if it is a rightC-ideal

Proposition 15 If 119877 is a strong hyperdomain then the setC-119868119889119890119886119897lowast(119877[119909]) is nonempty

Proof For each 119899 isin N cup 0 let C119899= 119891(119909) isin 119877[119909]

ord119891(119909) ge 119899 cup 0119877[119909] Then C

119899is a subgroup of the group

(119877[119909] +) (since for any 119891(119909) 119892(119909) isin 119877[119909] ord(minus119891(119909)) =ord119891(119909) and ord(119891(119909) + 119892(119909)) = minord119891(119909) ord119892(119909)when 119891(119909)+119892(119909) = 0

119877[119909]) Let 119892(119909) isin 119877[119909] and 119891(119909) isin C

119899 If

119892(119909) = 0119877[119909]

or 119891(119909) = 0119877[119909]

then 119892(119909) lowast119891(119909) = 0119877[119909] sube C119899

(since 0119877[119909]

is absorbing in 119877[119909]) So let 119892(119909) = 0119877[119909]

and119891(119909) = 0

119877[119909] Then by Proposition 11(ii) for all ℎ(119909) isin 119892(119909)lowast

119891(119909) ord ℎ(119909) = ord(119892(119909) lowast 119891(119909)) = ord119891(119909) + ord119892(119909) gt119899 Thus 119892(119909) lowast 119891(119909) sube C

119899 Hence C

119899is an 119867V-ideal of

119877[119909] (since 119877[119909] is a commutative multiplicative 119867V-ring)Let [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897be a type-119897 hyperproduct of

elements of the119867V-ring 119877[119909] such that [1198911(119909) lowast 1198912(119909) lowast sdot sdot sdot lowast119891119899(119909)]119897sube C119899 Then 119891

119894(119909) = 0

119877[119909]for each 119894 (since 0

119877[119909]isin C119899

is absorbing in119877[119909])Thus ord([1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897) =

ord119892(119909) for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 Now

since [1198911(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119899(119909)]119897sube C119899 so there exists ℎ(119909) isin

[1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897such that ℎ(119909) notin C

119899Then for any

119892(119909) isin [1198911(119909) lowast119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 ord119892(119909) = ord ℎ(119909) lt 119899

Hence by definition ofC119899 [1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897capC119899=

120601 and soC119899is a leftC-ideal and thus aC-ideal of 119877[119909] (since

119877[119909] is commutative) Hence C119899isin C-Ideallowast(119877[119909]) for all

119899 gt 0Since the intersection of left C-ideals of a multiplicative

119867V-ring 119878 is also a left C-ideal of 119878 and 119878 is itself a left C-ideal so the smallest left C-ideal containing a subset 119860 of 119878

6 Algebra

being naturally called the left C-ideal generated by 119860 existsand is in fact the intersection of all left C-ideals containing119860 The left C-ideal generated by a left 119867V-ideal 119868 of 119878 iscalled the leftC-closure of 119868 and is denoted byC

119897(119868) Clearly

C119897(C119897(119868)) = C

119897(119868) for a left 119867V-ideal 119868 C119903(119868) and C(119868)

respectively denote the right C-closure of a right 119867V-ideal 119868and theC-closure of an119867V-ideal 119868 of 119878 For an119867V-ideal 119868 of acommutativemultiplicative119867V-ring 119878C119897(119868) = C119903(119868) = C(119868)The following lemma presents a description of the set C

119897(119868)

for a left119867V-ideal 119868 of amultiplicative119867V-ring 119878The setC119903(119868)

for a right119867V-ideal 119868 can be described dually

Lemma 16 Let 119878 be a multiplicative 119867V-ring with an 119894-set Eand let L denote the set of all left 119867V-ideals of 119878 119865 119875(119878) rarr119875(119878) is a mapping defined by

119865 (119860) = ⋃

119899

sum

119894=1

[1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897 119899 119898

119894isin N 119903

1198941isin 119878

119904119886119905119894119904119891119910119894119899119892 [1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897cap 119860 = 120601

(7)

for any 119860 isin 119875(119878) Then for any 119868 1198681 1198682isin L we have the

following (i) 119865(119868) isin L (ii) 119868 sube 119865(119868) (iii) 1198681sube 1198682rArr 119865(119868

1) sube

119865(1198682) (iv) 119865(119868) = 119868 if and only if 119868 is a left C-ideal of 119878 and

(v) left C-closure of 119868 is C119897(119868) = ⋃

infin

119896=1119865119896(119868) where for any

119896 isin N 119865119896 denotes the (119896 minus 1)-times mapping composition of 119865

Proof All the assertions made in this lemma can be estab-lished by adopting the arguments that are applied in provingwell-known analogous results on ldquocomplete closure of a setrdquoin semihypergroup theory (see [3 8])

Proposition 17 Let 119877 be a strong hyperdomain with an 119894-setThen for any 119867V-ideal 119868 of the multiplicative 119867V-ring 119877[119909]ord(C(I)lowast) ge ordIlowast where for any 119867V-ideal 119869 of 119877[119909] 119869lowast =119869 0119877[119909]

Proof Since 119877 is a (strong) hyperdomain the multiplicative119867V-ring is commutative So for any 119867V-ideal 119868 of 119877[119909]C(119868) = C

119897(119868) = ⋃

infin

119896=1119865119896(119868) (by Lemma 16(v)) For any

119899 isin N 119891(119909) isin 119865119899(119868)lowast rArr 119891(119909) isin sum119898

119894=1119860119894 for some

type-119897 hyperproducts 119860119894= [1198911198941(119909) lowast 119891

1198942(119909) lowast sdot sdot sdot lowast 119891

119894119898119894(119909)]119897

of elements of 119877[119909] satisfying 119860119894cap 119865119899minus1(119868) = 120601 Since

here 119877[119909] is a strong 119867V-domain (by Proposition 13) withabsorbing zero we may assume that 0

119877[119909]notin 119860119894for each 119894

(since 0119877[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119877[119909])) Then for

each 119894 (= 1 2 119898) and 119895119894(= 1 2 119898

119894) 119891119894119895119894

(119909) = 0119877[119909]

Thus by Corollary 12 ord119860

119894= ord119892

119894(119909) for any 119892

119894(119909) isin 119860

119894

Now since for each 119894 119860119894cap 119865119899minus1(119868) = 120601 so there exists

119892119894(119909) isin 119860

119894such that 119892

119894(119909) isin 119865

119899minus1(119868)lowast Then for any 119891(119909) isin

119865119899(119868)lowast ord119891(119909) ge sum119898

119894=1ord119860119894= minord119892

119894(119909) 119894 ge

ord(119865119899minus1(119868)lowast) Consequently ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast)for any 119899 isin N Now let 119891(119909) isin C(119868)

lowast be arbitrary Then119891(119909) isin 119865

119899(119868)lowast for some 119899 isin N So we have that ord119891(119909) ge

ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast) ge sdot sdot sdot ge ord(119865(119868)lowast) ge ord119868lowastThus ord(C(119868)lowast) ge ord119868lowast

5 Polynomials over Multiplicative Hyperfield

A nonzero element 119886 of a multiplicative119867V-ring (119878 + ∘) withan 119894-set E = 119890

1 1198902 119890

119899 is referred to be an E-invertible

element (or anE-unit) of 119878 if for each 119894 = 1 2 119899 (119899 isin N)there exist 119860

119894119895isin H119897(119886) = [119904

119901∘ 119904119901minus1∘ sdot sdot sdot ∘ 119904

1∘ 119886]119897 119904119894isin 119878

119901 isin N and 119861119894119896isin H119903(119886) = [119886 ∘ 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119902]119903 119904119894isin 119878

119902 isin N such that 119890119894isin (sum119898

119895=1119860119894119895) cap (sum

119897

119896=1119861119894119896) An element

119886 ( =0119878) of the multiplicative119867V-ring 119878with a hyperidentity 119890

is said to be 119890-hyperinvertible (or an 119890-hyperunit) in 119878 if thereexist 119860 isinH

119897(119886) and 119861 isinH

119903(119886) such that 119890 isin 119860 cap 119861

If 119886 isin 119878 is an E-unit (resp 119890-hyperunit) in a multiplica-tive 119867V-ring (119878 + ∘) with two 119894-sets E and E1015840 (resp withtwo hyperidentities 119890 and 1198901015840) then one can easily verify that119886 is also an E1015840-unit (resp an 1198901015840-hyperunit) in 119878 We thuscall an E-unit (resp an 119890-hyperunit) of a multiplicative 119867V-ring 119878 simply a unit (resp a hyperunit) Denote by 119880(119878)and 119880

ℎ(119878) respectively the sets of units and hyperunits of a

multiplicative119867V-ring 119878An 119867V-ideal 119868 ( = 119878) of a multiplicative 119867V-ring 119878 is

maximal in 119878 if for any 119867V-ideal 119869 of 119878 119868 ⊊ 119869 sube 119878 rArr 119869 = 119878For a commutative multiplicative119867V-ring 119878with an 119894-set thisis immediate to observe that 119886 isin 119880(119878) if and only if 119886 notin 119872for any maximal119867V-ideal119872 of 119878

Proposition 18 Let themultiplicative hyperring119877 be commu-tative and contain a hyperidentity 119890 Then for a polynomial119891(119909) = sum119886

119896119909119896isin 119877[119909] 119891(119909) isin 119880

ℎ(119877[119909]) if and only if

1198860isin 119880ℎ(119877)

Proof Since 119890 is a hyperidentity in the multiplicative hyper-ring 119877 1198901199090 isin 119877[119909] is a hyperidentity in the multiplicative119867V-ring 119877[119909] Now let 119891(119909) = sum119886

119896119909119896isin 119880ℎ(119877[119909]) Then there

exist 119892119894(119909) = sum 119887

119894119896119909119896isin 119877[119909] (119894 = 1 2 119899 119899 isin N) such that

1198901199090isin [119891(119909) lowast 119892

1(119909) lowast sdot sdot sdot lowast 119892

119899(119909)]119903 So 119890 isin 119886

0(1198871011988720 1198871198990)

and thus there exists 119887 isin 1198871011988720 1198871198990sube 119877 such that 119890 isin 119886

0119887

whence 1198860isin 119880ℎ(119877) (since 119877 is commutative)

Conversely let 1198860be hyperinvertible in 119877 with respect to

the hyperidentity 119890 Then there exists 1198870isin 119877 such that 119890 isin

11988601198870 We assert that there is a sequence 119887

119896119896isinN in 119877 whose

119899th term 119887119899(119899 isin N) is inductively defined so as to satisfy the

relation that

0119877isin 1198860119887119899+ 1198861119887119899minus1+ 1198862119887119899minus2+ sdot sdot sdot + 119886

1198991198870sdot sdot sdot () (8)

In fact we see that 0119877isin 01198771198870= (minus119886

1+ 1198861)1198870sube (119890(minus119886

1) +

1198861)1198870sube ((119886

01198870)(minus1198861) + 1198861)1198870sube (119886

01198870)(minus1198861)1198870+ 11988611198870=

1198860((minus1198870)11988611198870) + 11988611198870rArr there exist 119887

1isin (minus119887

0)11988611198870such that

0119877isin 11988601198871+ 11988611198870(the relation for 119899 = 1) Suppose for

some 119898 ge 1 the terms 119887119896(1 le 119896 le 119898 minus 1) are defined

in such a way that each 119887119896(1 le 119896 le 119898 minus 1) satisfies the

relation () for 119896 Then (1198861119887119898minus1+1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+1198861198981198870)

is defined to be a nonempty subset of 119877 Let 119905 isin 1198861119887119898minus1+

1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870 Then 0

119877= minus119905 + 119905 isin 119890(minus119905) + 119905 sube

(11988601198870)(minus119905) + 119905 sube 119886

0(minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+

1198861198981198870) + (1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) rArr there exists

119887119898isin (minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) such that

0119877isin 1198860119887119898+ 1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

1198981198870(the relation

for 119896 = 119898) Hence the assertion is true for all 119899 isin N Thus

Algebra 7

consider the polynomial 119892(119909) = sum 119887119896119909119896isin 119877[119909] Then from

the definition of lowast 119891(119909) lowast 119892(119909) = sum 119888119896119909119896 1198880isin 11988601198870and

119888119896isin 1198860119887119896+ 1198861119887119896minus1+ 1198862119887119896minus2+ sdot sdot sdot + 119886

1198961198870for 119896 isin N Thus

1198901199090isin 119891(119909) lowast 119892(119909) (due to relation and since 119890 isin 119886

01198870) So

119891(119909) is hyperinvertible in 119877[119909] that is 119891(119909) isin 119880ℎ(119877[119909])

The (left right) 119867V-ideal of a multiplicative 119867V-ring 119878generated by 119860 isin 119875(119878) is the smallest (resp left right) 119867V-ideal of 119878 containing119860which is denoted by (resp ⟨119860⟩

119897 ⟨119860⟩119903)

⟨119860⟩ The principal (left right) 119867V-ideal of the multiplicative119867V-ring 119878 generated by an element 119886 of 119878 denoted by (resp⟨119886⟩119897 ⟨119886⟩119903) ⟨119886⟩ is the (resp left right)119867V-ideal (resp ⟨119886⟩119897

⟨119886⟩119903) ⟨119886⟩ of the multiplicative119867V-ring 119878

If the multiplicative 119867V-ring 119878 has an 119894-set then for any119886 isin 119878 ⟨119886⟩

119897= H119897(119886) = ⋃sum

119899

119894=1119860119894 119860119894isin H119897(119886) 119899 isin N and

⟨119886⟩119903=H119903(119886) = ⋃sum

119899

119894=1119860119894 119860119894isinH119903(119886) 119899 isin N

Definition 19 A commutative multiplicative 119867V-ring 119878 withan 119894-setE is called amultiplicative119867V-field (resp an inversivemultiplicative 119867V-field) if 119878 ⟨0

119878⟩ sube 119880(119878) (resp 119878

⟨0119878⟩ sube 119880

ℎ(119878)) If a multiplicative119867V-field (resp an inversive

multiplicative 119867V-field) is a multiplicative hyperring thenwe call it a multiplicative hyperfield (resp an inversivemultiplicative hyperfield)

Proposition 20 Let 119865 be a polynomially structured inversivemultiplicative hyperfield Then a polynomial 119891(119909) = sum119886

119896119909119896isin

119865[119909]0119865[119909] is hyperinvertible in119865[119909] if and only if ord 119891 = 0

Proof If 119865 is an inversive multiplicative hyperfield then byProposition 18 any polynomial119891(119909) = sum119886

119896119909119896isin 119865[119909]0

119865[119909]

is hyperinvertible in 119865[119909] if and only if 1198860= 0119865 Hence the

result follows

Definition 21 A commutative multiplicative 119867V-ring 119878 iscalled a principal C-ideal 119867V-ring if every C-ideal of119878 is a principal 119867V-ideal A principal C-ideal 119867V-ringwhich is a (strong) 119867V-domain is called a principal C-ideal(strong) 119867V-domain

Proposition 22 Let 119865 be a polynomially structured inversivemultiplicative hyperfieldThen themultiplicative119867V-ring119865[119909]of polynomials over 119865 is a principalC-ideal119867V-ring

Proof Let 119890 be a hyperidentity of the inversive multiplicativehyperfield 119865 Then the polynomial 1198901199090 is a hyperidentity in119865[119909] and thus 119865[119909] = ⟨1198901199090⟩ (since 119865 is commutative) Solet 119868 be any proper C-ideal of 119865[119909] If 119868 = 0

119865[119909] then 119868 is

the principal hyperideal ⟨0119865[119909]⟩ (since for119865 having absorbing

zero 0119865[119909]

is absorbing in 119865[119909]) Suppose that 119868 = 0119865[119909]

Then take a nonzero polynomial 119891(119909) = sum119886119896119909119896isin 119868 such

that ord119891(119909) le ord 119905(119909) for any 119905(119909) isin 119868lowast Let us writeord119891(119909) = 119898 Then 119886

119898= 0119865and 119886119896= 0119865for any 0 le 119896 lt 119898

Consider then the polynomial 119892(119909) = sumlowast119896isinN0119887119896119909119896isin 119865[119909]

where 119887119896= 119886119898+119896

Then clearly 119891(119909) isin (119890119909119898) lowast 119892(119909) Alsoby Proposition 18 119892(119909) is hyperinvertible in 119865[119909] Thus thereexists ℎ(119909) isin 119865[119909] such that 1198901199090 isin 119892(119909) lowast ℎ(119909) Now119891(119909) isin (119890119909

119898) lowast 119892(119909) rArr 119891(119909) lowast ℎ(119909) sube [119890119909

119898lowast 119892(119909) lowast ℎ(119909)]

119903

Again 119891(119909) lowast ℎ(119909) sube 119868 (since 119891(119909) isin 119868) So [119890119909119898 lowast 119892(119909) lowastℎ(119909)]119903cap 119868 = 120601 Hence [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868 (since

119868 is a C-ideal and every C-ideal is a right C-ideal) Now[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap [119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119903= 120601 whence

[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap 119868 = 120601 (since [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868)

Consequently [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]119897sube 119868 (since 119868 is a left C-

ideal) Then 119890119909119898 isin (119890119909119898) lowast (1198901199090) sube 119890119909119898 lowast (119892(119909) lowast ℎ(119909)) =[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897sube 119868 Thus ⟨119890119909119898⟩ sube 119868

Now let ℎ(119909) = sum 119887119896119909119896isin 119868lowast be arbitrary Suppose that

ord ℎ = 119899 Then 119887119899= 0119865and 119887119896= 0119865for any 0 le 119896 lt 119899

By choice of 119891(119909) from 119868 here 119898 le 119899 So one can definea polynomial 119892(119909) = sum 119888

119896119909119896isin 119865[119909] where 119888

119896= 0119865for all

0 le 119896 le 119899 minus 119898 minus 1 and 119888119896= 119887119898+119896

for all 119896 ge 119899 minus 119898 Clearlythen ℎ(119909) isin 119890119909119898 lowast 119892(119909) sube ⟨119890119909119898⟩ whence 119868 sube ⟨119890119909119898⟩ Thus119868 = ⟨119890119909

119898⟩

Remark 23 In a ring an invertible element can never be adivisor of zero This not true in general for a multiplicativehyperring In fact on the commutative group of integers(Z +) if we define a hyperoperation ∘ by stating that 119909 ∘ 119910 =0 119909119910 for all 119909 119910 isin Z then (Z + ∘) is a commutative mul-tiplicative hyperring with a hyperidentity 1 Every nonzeroelement of (Z + ∘) is a zero divisor and 1 minus1 isin Z are inparticular hyperunits of (Z + ∘) To get a parity with thering theory in this regard we perceive the notion of stronghyperinvertibility of an element of amultiplicative hyperring

Definition 24 A hyperinvertible element of a multiplicativehyperring with a hyperidentity is said to be strongly hyperin-vertible (or a strong hyperunit) if it is not a zero divisor inthat multiplicative hyperring A commutative multiplicativehyperring with absorbing zero and a hyperidentity 119890 is said tobe a strongly inversive multiplicative hyperfield if each of itsnonzero elements is a strong hyperunit

Example 25 Let 119877 = 119886radic2 + 119887radic3 119886 119887 isin Q and 119860 =radic2radic3 Then with respect to usual addition + of reals(119877 +) is a commutative group with identity 0 On 119877 ∘ is ahyperoperation defined by

(119886radic2 + 119887radic3) ∘ (119888radic2 + 119889radic3)

= (119886radic2 + 119887radic3) sdot 119905 sdot (119888radic2 + 119889radic3) 119905 isin 119860

= 119901radic2 + 2119902radic3 3119902radic2 + 119901radic3

(9)

where 119901 = 2119886119888 + 3119887119889 and 119902 = 119887119888 + 119886119889Then (119877 + ∘) is a strongly inversivemultiplicative hyper-

field which is polynomially structured

Definition 26 A local (C-local) multiplicative 119867V-ring is acommutative multiplicative119867V-ring with an 119894-set which hasa unique maximal119867V-ideal (respC-ideal)

Proposition 27 Let 119865 be a polynomially structured stronglyinversive multiplicative hyperfieldThen (i) 119865[119909] is a principalC-ideal strong 119867V-domain (ii) for any 119891 isin 119865[119909] 0

119865[119909]

there exist a hyperinvertible element 119892 isin 119865[119909] and 119898 isin

N cup 0 such that 119891 isin (119890119909119898) lowast 119892 where 119890 is a hyperidentity

8 Algebra

in 119865 and (iii) 119865[119909] is a local as well as aC-local multiplicative119867V-ring

Proof (i) Here 119865 is a strongly inversive multiplicative hyper-field So 119865 is a strong hyperdomain Thus by Proposition 13the multiplicative 119867V-ring 119865[119909] is a strong 119867V-domainAgain by Proposition 22 119865[119909] is a principal C-ideal 119867V-ring (since 119865 is an inversive multiplicative hyperfield) So byDefinition 21 119865[119909] is a principalC -ideal strong119867V-domain

(ii) Let 119891 isin 119865[119909] 0119865[119909] and 119869 = ⟨119891⟩ Then 119869 =H

119903(119891)

So for any ℎ isin 119869lowast there exist119860119894= [119891lowast119891

1198941lowast1198911198942lowastsdot sdot sdotlowast119891

119894119898119894]119903isin

H119903(119891) (119894 = 1 2 119896 119896119898

119894isin N) such that ℎ isin sum119896

119894=1119860119894

Thus ℎ = sum119896119894=1119905119894for some 119905

119894isin 119860119894 Since 119865[119909] is a strong119867V-

domain (by Proposition 13) with absorbing zero so 0119865[119909]isin

119860119894= [119891 lowast 119891

1198941lowast 1198911198942lowast sdot sdot sdot lowast 119891

119894119898119894]119903rArr [119891 lowast 119891

1198941lowast 1198911198942lowast

sdot sdot sdot lowast 119891119894119898119894]119903= 0119865[119909] Hence without any loss of generality

(since 0119865[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119865[119909])) we may

assume that 0119865[119909]notin 119860119894for any 119894 (since ℎ = 0

119865[119909]) Then

for any 119894 = 1 2 119896 and 119895 = 1 2 119898119894 119891119894119895= 0119865[119909]

So by Proposition 11(ii) ord 119905

119894= ord(119860

119894) ge ord119891 and so

ord ℎ = minord 119905119894 119894 ge ord119891 Thus ord119869lowast ge ord119891 So

by Proposition 17 119891 is a nonzero polynomial in the C-idealC(119869) such that ord119891 le ord ℎ for all ℎ isin C(119869)lowast So there existsan invertible element 119892 isin 119865[119909] and an integer119898 ge 0 such that119891 isin (119890119909

119898) lowast 119892 where 119890 is a hyperidentity in 119865 (see proof of

Proposition 22)(iii) For any integer119898 ge 1 since 119890119909119898 isin (119890119909) lowast (119890119909119898minus1) so

⟨119890119909119898⟩ sube ⟨119890119909⟩ Again for any119867V-ideal 119868 of the multiplicative

119867V-ring119865[119909] if119891 isin 119868lowast is such that ord119891 le ord119892 for all119892 isin 119868lowast

then 119868 sube ⟨119890119909119898⟩ where 119898 = ord119891 So ⟨119890119909⟩ is the uniquemaximal119867V-ideal in 119865[119909] Thus 119865[119909] is a local multiplicative119867V-ring Now 119890119909 isin C

1(as defined in Proposition 15) So

ord(119890119909) le ord119891 for any 119891 isin Clowast1 Thus C

1being a C-ideal

of 119865[119909] we have C1= ⟨119890119909⟩ (see proof of Proposition 22)

Hence 119865[119909] is aC-local multiplicative119867V-ring

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vougiouklis ldquoThe fundamental relation in hyperrings Thegeneral hyperfieldrdquo in Algebraic Hyperstructures and Appli-cations (Xanthi1990) pp 203ndash211 World Science PublisherTeaneck NJ USA 1991

[2] M R Darafsheh and B Davvaz ldquo119867V-ring of fractionsrdquo ItalianJournal of Pure and Applied Mathematics no 5 pp 25ndash34 1999

[3] B Davvaz and V Leoreanu-FoteaHyperring Theory and Appli-cations International Academic Press Palm Harbor Fla USA2007

[4] S Spartalis A Dramalides and T Vougiouklis ldquoOn 119867V-groupringsrdquoAlgebras Groups and Geometries vol 15 no 1 pp 47ndash541998

[5] T Vougiouklis ldquo119867V-groups defined on the same setrdquo DiscreteMathematics vol 155 no 1ndash3 pp 259ndash265 1996

[6] R Procesi and R Rota ldquoMultiplicative H119907-rings and com-

plementary hyperstructuresrdquo Journal of Discrete MathematicalSciences amp Cryptography vol 11 no 4 pp 447ndash456 2008

[7] S Spartalis ldquoOn the number of 119867Vminus119903119894119899119892119904 with 119875-hyperopera-tionsrdquoDiscrete Mathematics vol 155 no 1ndash3 pp 225ndash231 1996

[8] P Corsini Prolegomena of Hypergroup Theory Rivista di Mate-matica Pura ed Applicata Aviani Tricesimo Italy 1993

[9] M K Sen and U Dasgupta ldquoSome aspects of119866119867-ringsrdquoAnnals

of the Alexandru Ioan Cuza UniversitymdashMathematics vol 56no 2 pp 253ndash272 2010

[10] C Namnak N Triphop and Y Kemprasit ldquoHomomorphismsof somemultiplicative hyperringsrdquo Set-ValuedMathematics andApplications vol 1 no 2 pp 145ndash152 2008

[11] D M Olson and V K Ward ldquoA note on multiplicativehyperringrdquo Italian Journal of Pure andAppliedMathematics vol1 pp 77ndash84 1997

[12] R Procesi and R Rota ldquoComplementary multiplicative hyper-ringsrdquoDiscreteMathematics vol 308 no 2-3 pp 188ndash191 2008

[13] R Procesi and R Rota ldquoOn some classes of hyperstructuresrdquoDiscrete Mathematics vol 208-209 pp 485ndash497 1999

[14] R Rota ldquoStrongly distributive multiplicative hyperringsrdquo Jour-nal of Geometry vol 39 no 1-2 pp 130ndash138 1990

[15] R Rota ldquoSugli Iperanelli Moltiplicativirdquo Rendiconti di Matem-atica Series VII vol 2 no 4 pp 711ndash724 1982

[16] R Procesi Ciampi andR Rota ldquoPolynomials overmultiplicativehyperringsrdquo Journal of Discrete Mathematical Sciences amp Cryp-tography vol 6 no 2-3 pp 217ndash225 2003

[17] T Vougiouklis Hyperstructures and Their RepresentationsMonographs in Mathematics Hardonic Press 1994

[18] U Dasgupta ldquoOn prime and primary hyperideals of a multi-plicative hyperringrdquoAnalele Stiintifice ale Universitatii Al I Cuzadin IasimdashMatematica vol 58 no 1 pp 19ndash37 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Algebra 3

where 119875 isin 119875(119877) with |119875| ge 2 andlowast

119875 is the 119875-hyperoperation[8 17] on the semigroup (119877 sdot) that is119909

lowast

119875 119910 = 119909sdot119886sdot119910 119886 isin 119875

for all 119909 119910 isin 119877 Now for each 119886 isin 119875 and 119909 isin 119877 we define amapping 119909

119886 119877 rarr 119877 by stating that 119909

119886(119910) = 119909 sdot 119886 sdot 119910 for all

119910 isin 119877 Then 119909119886isin Hom(119877 +) Thus corresponding to each

119886 isin 119875 we have a mapping 119886 (119877 +) rarr Hom(119877 +) givenby 119886(119909) = 119909

119886 for any 119909 isin 119877 Then 119875 sube Hom(119877Hom(119877))

Moreover for any 119909 119910 119911 isin 119877 and 119886 119887 isin 119875 (119909119886(119910))119887(119911) = (119909 sdot

119886sdot119910)sdot119887sdot119911 = 119909sdot119886sdot(119910sdot119887sdot119911) = 119909sdot119886sdot(119910119887(119911)) = 119909

119886(119910119887(119911)) = (119909

119886119910119887)(119911)

that is (119909119886(119910))119887= 119909119886119910119887 Thus the hyperoperation ∘

119860(for

119860 = 119875) is defined on the group (119877 +) Note that for any119909 119910 isin 119877 119909∘

119860119910 = 119909

119886(119910) 119886 isin 119860 = 119909 sdot 119886 sdot 119910 119886 isin 119875 = 119886

lowast

119875 119910Thus (119877 +

lowast

119875) is a multiplicative 119860-hyperring for 119860 = 119875

Proposition4 For amultiplicative119860-hyperring (119877 + ∘119860) the

hyperoperation lowast induces a multiplicative119867V-ring structure onthe group (119877[119909] +) of polynomials over 119877

Proof Let 119891(119909) = sum119886119896119909119896 119892(119909) = sum 119887

119896119909119896 and ℎ(119909) = sum119889

119896119909119896

be three polynomials in 119877[119909] Then (119891(119909) lowast 119892(119909)) lowast ℎ(119909) =sum 119905119896119909119896 119905119896isin sum119906+V=119896(sum119894+119895=119906 119886119894∘119860119887119895)∘119860119889V and 119891(119909) lowast (119892(119909) lowast

ℎ(119909)) = sum 119904119896119909119896 119904119896isin sum119906+V=119896 119886119906∘119860(sum119894+119895=V 119887119894∘119860119889119895) Now we

choose and fix an element 120572 isin 119860 Then for each 119896 isin N0

119901119896= sum

119906+V=119896( sum

119894+119895=119906

(119886119894)120572(119887119895))

120572

(119889V)

isin sum

119906+V=119896( sum

119894+119895=119906

119886119894∘119860119887119895)∘119860119889V

119902119896= sum

119906+V=119896(119886119906)120572( sum

119894+119895=V(119887119894)120572(119889119895))

isin sum

119906+V=119896119886119906∘119860( sum

119894+119895=V119887119894∘119860119889119895)

(1)

Again 119901119896

= sum119906+V=119896(sum119894+119895=119906(119886119894)120572(119887119895))120572(119889V) =

sum119906+V=119896sum119894+119895=119906((119886119894)120572(119887119895))120572(119889V) (since120572 is a homomorphism) =sum119906+V=119896sum119894+119895=119906((119886119894)120572(119887119895)120572)(119889V) (since (119909120572(119910))120572 = 119909120572119910120572) 119902119896 =sum119906+V=119896(119886119906)120572(sum119894+119895=V(119887119894)120572(119889119895)) = sum119906+V=119896sum119894+119895=V(119886119906)120572((119887119894)120572(119889119895))

(since (119886119906)120572

is a homomorphism) =sum119906+V=119896sum119894+119895=V((119886119906)120572(119887119894)120572)(119889119895) Clearly then for each 119896 isin N0

119901119896= 119902119896isin ( sum

119906+V=119896( sum

119894+119895=119906

119886119894∘119860119887119895)∘119860119889V)

⋂( sum

119906+V=119896119886119906∘119860( sum

119894+119895=V119887119894∘119860119889119895))

(2)

Hence (119891(119909) lowast 119892(119909)) lowast ℎ(119909) cap 119891(119909) lowast (119892(119909) lowast ℎ(119909)) = 120601It is shown in [16] that for any multiplicative hyperring 119877the hyperoperationlowast defined on119877[119909] is semidistributive overthe operation + on 119877[119909] and also satisfies the condition (R)for any two polynomials in 119877[119909] Thus for the multiplicative119860-hyperring (119877 + ∘

119860) the hyperstructure (119877[119909] + lowast) is a

multiplicative 119867V-ring (with absorbing zero 0119877[119909]= 01198771199090+

01198771199091+ 01198771199092+ sdot sdot sdot )

We call a multiplicative hyperring (119877 + sdot)with absorbingzero polynomially structured if (119877[119909] + lowast) is a multiplicative119867V-ringThe class ofmultiplicative119860-hyperrings is a subclassof the class of polynomially structured multiplicative hyper-rings (by Proposition 4) Throughout the rest of the paper119877 will stand for a polynomially structured multiplicativehyperring

3 Polynomials over Integral Hyperrings

An element 119886 ( = 0119877) of a multiplicative 119867V-ring (119878 + ∘) is

a left (resp right) divisor of zero in 119878 if there exists 119887 isin 119878lowast(resp 119888 isin 119878lowast = 119878 0

119878) such that 0

119878isin 119886 ∘ 119887 (resp 0

119904isin 119888 ∘ 119886)

and a divisor of zero in 119878 if it is either a left or a right divisorof zero in 119878 An element 119886 ( = 0

119878) of 119878 is a left (resp right)

strong divisor of zero in 119878 if there exists 119887 isin 119878lowast = 119878 0119878

(resp 119888 isin 119878lowast) such that 119886 ∘ 119887 = 0119878 (resp 119888 ∘ 119886 = 0

119878) and

a strong divisor of zero in 119878 if it is either a left or a right strongdivisor of zero in 119878

Definition 5 A multiplicative 119867V-ring is called an integral119867V-ring if there is no strong divisor of zero in it A commuta-tive integral119867V-ring is an119867V-domain A strong integral119867V-ring is a multiplicative119867V-ring in which there is no divisor ofzero A strong 119867V-domain is a commutative strong integral119867V-ring We call an integral119867V-ring (resp an119867V-domain) 119878simply an integral hyperring (resp a hyperdomain) [13]when the119867V-ring 119878 is a multiplicative hyperring

Before entering into the study of the multiplicative 119867V-ring 119877[119909] of polynomials over integral hyperring and hyper-domain let us go through the following useful observations

Remark 6 (a) For any polynomially structuredmultiplicativehyperring119877 the identity element 0

119877[119909]= 01198771199090+01198771199091+01198771199092+

sdot sdot sdot of the group (119877[119909] +) is absorbing in the multiplicative119867V-ring (119877[119909] + lowast)

(b) 119877[119909] is commutative if 119877 is a commutative multiplica-tive hyperring

(c)Themapping120595 119877 rarr 119877[119909] defined by for all 119903 isin 119877lowast120595(119903) = 119903119909

0 and 120595(0119877) = 0119877[119909]

is a strong monomorphism Infact for any 119903 119904 isin 119877120595(119903119904) = 120595(119888) 119888 isin 119903119904 = 1198881199090 119888 isin 119903119904 =(1199031199090) lowast (119904119909

0) = 120595(119903) lowast 120595(119904) Thus 119877 can be identified with its

isomorphic image in 119877[119909] and for any 119903 isin 119877lowast we can writethe polynomial 1199031199090 simply as 119903 and the zero polynomial 0

119877[119909]

as 0119877

(d) If E = 1198901 1198902 119890

119899 is an 119894-set of 119877 then the set

E119909sube 119877[119909] is an 119894-set in the multiplicative 119867V-ring 119877[119909]

where for any119860 isin 119875lowast(119877)119860119909denotes the set 1198861199090 119886 isin 119860In

4 Algebra

fact for any 119891(119909) = sum119886119896119909119896isin 119877[119909] we have that sum119899

119894=1119890119894lowast

119891(119909) = sum119899

119894=1(119890119894lowast sum119886

119896119909119896) = sum

119899

119894=1sum 119886119894119896119909119896 119886119894119896isin 119890119894119886119896 =

sum(sum119899

119894=1119886119894119896)119909119896 119886119894119896isin 119890119894119886119896 = sum 119887

119896119909119896 119887119896isin sum119899

119894=1119890119894119886119896

Then since 119886119896isin sum119899

119894=1119890119894119886119896 we have that119891(119909) isin sum119899

119894=1119890119894lowast119891(119909)

identifying 1198901198941199090 with 119890

119894 Similarly one can see that 119891(119909) isin

sum119899

119894=1119891(119909) lowast 119890

119894

On the other hand if for some 119899 isin N E119877[119909]= 120598119894(119909) =

sum 119890119894119896119909119896isin 119877[119909] 119894 = 1 2 119899 is an 119894-set in 119877[119909] then

E = 1198901198940 119894 = 1 2 119899 is an 119894-set in 119877 In fact for any

119886 isin 119877 1198861199090 isin sum119899119894=1(1198861199090) lowast 120598119894(119909) = sum

119899

119894=1sum 119886119894119896119909119896isin 119877[119909] 119886

119894119896isin

119886119890119894119896 = sum(sum

119899

119894=1119886119894119896)119909119896isin 119877[119909] 119886

119894119896isin 119886119890119894119896 = sum 119886

119896119909119896isin

119877[119909] 119886119896isin sum119899

119894=1119886119890119894119896 rArr 119886 isin sum

119899

119894=11198861198901198940 Similarly from

1198861199090isin sum119899

119894=1120598119894(119909) lowast (119886119909

0) one may arrive at 119886 isin sum119899

119894=11198901198940119886

This is clear fromRemark 6 (d) that for any hyperidentity119890 of 119877 the polynomial 1198901199090 isin 119877[119909] is a hyperidentity in themultiplicative119867V-ring 119877[119909] Is every hyperidentity of 119877[119909] ofthe form 1198901199090 for some hyperidentity 119890 of 119877 Following is anexample of a multiplicative hyperring 119877 such that 119877[119909] has ahyperidentity 120598(119909) = 1198901199090 for any hyperidentity 119890 of 119877

Example 7 Let (Z + sdot) be the ring of integers and 119875 = 0 1Then (Z +

lowast

119875) is a commutative polynomially structuredmultiplicative hyperring (as is shown in Example 3) Denotethe multiplicative hyperring (Z +

lowast

119875) by Z119875 Consider a

polynomial 1 + 1119909 isin Z119875[119909] Then for any 119891(119909) = sum119886

119896119909119896isin

Z119875[119909] we see that (1+1119909)lowast119891(119909) = sum 119887

119896119909119896 1198870isin 1lowast

119875 1198860 and119887119896isin 1lowast

119875 119886119896 + 1lowast

119875 119886119896minus1 (for 119896 isin N) Clearly 1198860 isin 1lowast

119875 1198860 andfor any 119896 isin N 119886

119896isin 1lowast

119875 119886119896 + 1lowast

119875 119886119896minus1 (since 0 isin 1lowast

119875 119886119896minus1)Thus 119891(119909) isin (1 + 1119909) lowast 119891(119909) and so 1 + 1119909 is a hyperidentityin Z119875[119909] which is not in the form 1198901199090 for any hyperidentity

119890 of Z119875

Remark 8 Let 120598(119909) = sum 119890119896119909119896isin 119877[119909] be a hyperidentity in the

multiplicative119867V-ring 119877[119909] Then from Remark 6 (d) 1198900is a

hyperidentity in the multiplicative hyperring 119877

Proposition 9 Let 119877 be a strong integral hyperring Thenevery hyperidentity in the multiplicative119867V-ring 119877[119909] is of theform 1198901199090 for some hyperidentity 119890 of 119877

Proof Suppose that 120598(119909) = sum 119890119896119909119896isin 119877[119909] is a hyperidentity

in the multiplicative 119867V-ring 119877[119909] Then by the Remark 8119890 = 1198900is a hyperidentity in119877 Now let 119886 isin 1198770

119877 be arbitrary

and 119891(119909) = sum119886119896119909119896isin 119877[119909] where 119886

0= 119886 and 119886

119896= 0119877 for all

119896 isin N Then 119891(119909) isin 120598(119909) lowast 119891(119909) cap 119891(119909) lowast 120598(119909) whereby0119877isin 119890119896119886 cap 119886119890

119896 for all 119896 isin N whence 119890

119896= 0119877for all 119896 isin N

(since 119877 is a strong integral hyperring and 119886 = 0119877) Thus

120598(119909) = 1198901199090+01198771199091+01198771199092+sdot sdot sdot where 119890 = 119890

0is a hyperidentity

in 119877

Definition 10 If 0119877[119909]

= 119891(119909) = sum119886119896119909119896isin 119877[119909] then the

smallest integer 119899 such that 119886119899= 0119877is called the order of119891(119909)

and is denoted by ord119891(119909) The order of 0119877[119909]

is defined to bezero For a nonempty set119860 isin 119875(119877[119909]) the smallest element inthe set ord119891(119909) 119891(119909) isin 119860 does exist and is called the orderof 119860 being denoted by ord119860 that is ord119860 = minord119891(119909) 119891(119909) isin 119860

In the next proposition we will find some propertiesof ord119860 for some 119860 isin 119875lowast(119877[119909])(= 119875(119877[119909]) 120601) Forthat it is necessary at this point to frame some suitablenotations corresponding to different types of hyperproductsof elements in the multiplicative119867V-ring 119877[119909] Indeed in anymultiplicative 119867V-ring (119878 + ∘) (which is not a multiplicativehyperring) the expression like 119904

1∘1199042∘1199043∘sdot sdot sdot∘119904

119899(119904119894isin 119878 119899 ge 3)

bears no connotation in 119878 unless the parentheses ldquo(rdquo andldquo)rdquo are meaningfully inserted in Note that the following twoexpressions

1199041∘ (11199042∘ (21199043∘ (31199044∘ sdot sdot sdot ∘ (

119899minus3119904119899minus2∘ (119899minus2119904119899minus1∘ 119904119899)119899minus2)119899minus3sdot sdot sdot )3)2)1

(1(2(3sdot sdot sdot (119899minus3(119899minus21199041∘ 1199042)119899minus2∘ 1199043)119899minus3∘ sdot sdot sdot ∘ 119904

119899minus3)3∘ 119904119899minus2)2∘ 119904119899minus1)1∘ 119904119899

(3)

are meaningful called the finite hyperproducts of type 119897 andtype 119903 andwritten in notations respectively as [119904

1∘1199042∘sdot sdot sdot∘119904

119899]119897

and [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903

Proposition 11 (i) For any 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]

119890119894119905ℎ119890119903 0119877[119909]isin 119891 (119909) lowast 119892 (119909)

119900119903 ord (119891 (119909) lowast 119892 (119909)) ge ord 119891 (119909) + ord 119892 (119909) (4)

(ii) If 119877 is a strong hyperdomain then the implication that

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119903

) = ord 119892 (119909) (5)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119903holds true for any

119899 isin N with 119899 ge 2

Proof (i) Let ord119891(119909) = 119899 and ord119892(119909) = 119898 Then 119891(119909) =sum119886119896119909119896 where 119886

119899= 0119877 119886119896= 0119877for 119896 lt 119899 and 119892(119909) = sum 119887

119896119909119896

where 119887119898= 0119877 119887119896= 0119877for 119896 lt 119898 So for anysum119888

119896119909119896isin 119891(119909)lowast

119892(119909) we see that whenever 119896 lt 119899 + 119898 119888119896isin sum119894+119895=119896119886119894119887119895=

0119877 (since 0

119877is absorbing in 119877 and 119886

119896= 0119877(119896 lt 119899) 119887

119896=

0119877(119896 lt 119898)) Thus if 0

119877[119909]notin 119891(119909) lowast 119892(119909) then for any ℎ(119909) =

sum 119888119896119909119896isin 119891(119909) lowast 119892(119909) we have that ord ℎ(119909) ge 119899 + 119898 So

ord(119891(119909) lowast 119892(119909)) ge 119899 + 119898 = ord119891(119909) + ord119892(119909)(ii) Now suppose that 119877 is a strong hyperdomain with

absorbing zero Consider two polynomials 119891(119909) 119892(119909) isin119877[119909] 0

119877[119909] Let ord119891(119909) = 119901 and ord119892(119909) = 119898 Then

119891(119909) = sum119886119896119909119896 where 119886

119901= 0119877 119886119896= 0119877for 119896 lt 119901 and

Algebra 5

119892(119909) = sum 119887119896119909119896 where 119887

119898= 0119877 119887119896= 0119877for 119896 lt 119898

So 0119877notin 119886119901119887119898(since 119877 is a strong hyperdomain) and also

119886119901119887119898= sum119894+119895=119901+119898

119886119894119887119895(since 119886

119896= 0119877for 119896 lt 119901 and 119887

119896= 0119877

for 119896 lt 119898) Thus for any ℎ(119909) = sum 119888119896119909119896isin 119891(119909) lowast 119892(119909)

119888119901+119898

= 0119877(since 119888

119901+119898isin sum119894+119895=119901+119898

119886119894119887119895= 119886119901119887119898) and 119888

119896= 0119877

for all 119896 with 0 le 119896 lt 119901 + 119898 Thus ordℎ(119909) = 119901 + 119898 forany ℎ(119909) isin 119891(119909) lowast 119892(119909) and so ord(119891(119909) lowast 119892(119909)) = 119901 + 119898Hence ord119891(119909)+ord119892(119909) = ord(119891(119909)lowast119892(119909)) = ord ℎ(119909) forall ℎ(119909) isin 119891(119909) lowast 119892(119909) So the implication is true for 119899 = 2(noting that [119891(119909) lowast 119892(119909)]

119903= 119891(119909) lowast 119892(119909)) Suppose that for

some integer 119896 ge 2 the implication holds true for each valueof 119899 ranging from 2 to 119896minus1 and take any119891

119894(119909) isin 119877[119909]0

119877[119909]

for 119894 = 1 2 119896 Thensum119896minus1119894=1

ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast

sdot sdot sdotlowast119891119896minus1(119909)]119903) = ord ℎ(119909) for all ℎ(119909) isin [119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast

119891119896minus1(119909)]119903 Now let 119892(119909) isin [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903be

arbitraryThen 119892(119909) isin ℎ(119909) lowast119891119896(119909) for some ℎ(119909) isin [119891

1(119909) lowast

1198912(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903 So ord119892(119909) = ord(ℎ(119909) lowast 119891

119896(119909)) =

ord ℎ(119909) + ord119891119896(119909) = sum

119896minus1

119894=1ord119891119894(119909) + ord119891

119896(119909)(since ℎ(119909) isin

[1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903) = sum

119896

119894=1ord119891119894(119909) Thus

sum119896

119894=1ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119896(119909)]119903) = ord119892(119909)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903

Hence by strong induction the implication follows forany 119899 isin N with 119899 ge 2

Corollary 12 If 119877 is a strong hyperdomain then the implica-tion

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119897

) = ord 119892 (119909) (6)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897holds true for any

119899 isin N with 119899 ge 2

Proof Since themultiplicative hyperring119877 is a hyperdomainit is commutative and so 119877[119909] is also a commutative multi-plicative 119867V-ring Hence for any 119891119894(119909) isin 119877[119909] 0119877[119909] (119894 =1 2 119896) we have that [119891

1(119909)lowast119891

2(119909)lowast119891

3(119909)lowastsdot sdot sdotlowast119891

119896(119909)]119897=

[119891119896(119909) lowast 119891

119896minus1(119909) lowast sdot sdot sdot lowast 119891

2(119909) lowast 119891

1(119909)]119903 Hence the assertion

follows straight from Proposition 11

Proposition 13 If the multiplicative hyperring 119877 is a stronghyperdomain the multiplicative 119867V-ring 119877[119909] is a strong 119867V-domain

Proof 119877 being a strong hyperdomain is a commutativemultiplicative hyperring Thus 119877[119909] is a commutative mul-tiplicative 119867V-ring Again since 0119877 is absorbing in 119877 0

119877[119909]

is also absorbing in 119877[119909] Thus we take 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]Then by Proposition 11 ord(119891(119909)lowast119892(119909)) = ord119891(119909)+

ord119892(119909) gt 0 Hence 0119877[119909]notin 119891(119909) lowast 119892(119909) So 119877[119909] is a strong

119867V-domain

4 C-Ideals in 119877[119909]

A subgroup 119868 of the group (119878 +) is called a left (resp right)119867V-ideal of a multiplicative 119867V-ring (119878 + ∘) if for any 119904 isin 119878

and 119886 isin 119868 119904 ∘ 119886 sube 119868 (resp 119886 ∘ 119904 sube 119868) 119868 is an119867V-ideal of 119878 if itis both a left and a right119867V-ideal of 119878

We call an 119867V-ideal of a multiplicative 119867V-ring 119878 simplya hyperideal when 119878 is a multiplicative hyperringThe notionof a typical hyperideal in a multiplicative hyperring calledC-ideal is introduced in [18] to study prime and primaryhyperideals of multiplicative hyperrings A hyperideal 119868 of amultiplicative hyperring 119878 is a C-ideal if for any 119860 isin C 119860 cap119868 = 120601 rArr 119860 sube 119868 where C = 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899 119904119894isin 119878

119899 isin N sube 119875lowast(119878) Following is the definition of aC-ideal in anarbitrary multiplicative119867V-ring

Definition 14 A left (resp right) 119867V-ideal 119868 of a multiplica-tive 119867V-ring (119878 + ∘) is called a left (resp right) C-ideal iffor any type 119897 hyperproduct [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897(resp type 119903

hyperproduct [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903) of elements 119904

119894isin 119878 we have

that [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897cap 119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897sube 119868 (resp

[1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903cap119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903sube 119868) An119867V-ideal

119868 of a multiplicative119867V-ring 119878 is called a C-ideal if it is a leftas well as a rightC-ideal in 119878

We writeC-ideal(119878) (respC-ideallowast(119878)) to denote the setof all (resp proper)C-ideals of a multiplicative119867V-ring 119878 Ina commutative multiplicative119867V-ring 119878 every left C-ideal isa right C-ideal and vice versa since commutativity impliesthe equality [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903= [119904119899∘ 119904119899minus1∘ sdot sdot sdot ∘ 119904

1]119897 for any

119904119894isin 119877 Note that in amultiplicative hyperring (even if it is not

commutative) a hyperideal is a leftC-ideal (as an119867V-ideal) ifand only if it is a rightC-ideal

Proposition 15 If 119877 is a strong hyperdomain then the setC-119868119889119890119886119897lowast(119877[119909]) is nonempty

Proof For each 119899 isin N cup 0 let C119899= 119891(119909) isin 119877[119909]

ord119891(119909) ge 119899 cup 0119877[119909] Then C

119899is a subgroup of the group

(119877[119909] +) (since for any 119891(119909) 119892(119909) isin 119877[119909] ord(minus119891(119909)) =ord119891(119909) and ord(119891(119909) + 119892(119909)) = minord119891(119909) ord119892(119909)when 119891(119909)+119892(119909) = 0

119877[119909]) Let 119892(119909) isin 119877[119909] and 119891(119909) isin C

119899 If

119892(119909) = 0119877[119909]

or 119891(119909) = 0119877[119909]

then 119892(119909) lowast119891(119909) = 0119877[119909] sube C119899

(since 0119877[119909]

is absorbing in 119877[119909]) So let 119892(119909) = 0119877[119909]

and119891(119909) = 0

119877[119909] Then by Proposition 11(ii) for all ℎ(119909) isin 119892(119909)lowast

119891(119909) ord ℎ(119909) = ord(119892(119909) lowast 119891(119909)) = ord119891(119909) + ord119892(119909) gt119899 Thus 119892(119909) lowast 119891(119909) sube C

119899 Hence C

119899is an 119867V-ideal of

119877[119909] (since 119877[119909] is a commutative multiplicative 119867V-ring)Let [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897be a type-119897 hyperproduct of

elements of the119867V-ring 119877[119909] such that [1198911(119909) lowast 1198912(119909) lowast sdot sdot sdot lowast119891119899(119909)]119897sube C119899 Then 119891

119894(119909) = 0

119877[119909]for each 119894 (since 0

119877[119909]isin C119899

is absorbing in119877[119909])Thus ord([1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897) =

ord119892(119909) for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 Now

since [1198911(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119899(119909)]119897sube C119899 so there exists ℎ(119909) isin

[1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897such that ℎ(119909) notin C

119899Then for any

119892(119909) isin [1198911(119909) lowast119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 ord119892(119909) = ord ℎ(119909) lt 119899

Hence by definition ofC119899 [1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897capC119899=

120601 and soC119899is a leftC-ideal and thus aC-ideal of 119877[119909] (since

119877[119909] is commutative) Hence C119899isin C-Ideallowast(119877[119909]) for all

119899 gt 0Since the intersection of left C-ideals of a multiplicative

119867V-ring 119878 is also a left C-ideal of 119878 and 119878 is itself a left C-ideal so the smallest left C-ideal containing a subset 119860 of 119878

6 Algebra

being naturally called the left C-ideal generated by 119860 existsand is in fact the intersection of all left C-ideals containing119860 The left C-ideal generated by a left 119867V-ideal 119868 of 119878 iscalled the leftC-closure of 119868 and is denoted byC

119897(119868) Clearly

C119897(C119897(119868)) = C

119897(119868) for a left 119867V-ideal 119868 C119903(119868) and C(119868)

respectively denote the right C-closure of a right 119867V-ideal 119868and theC-closure of an119867V-ideal 119868 of 119878 For an119867V-ideal 119868 of acommutativemultiplicative119867V-ring 119878C119897(119868) = C119903(119868) = C(119868)The following lemma presents a description of the set C

119897(119868)

for a left119867V-ideal 119868 of amultiplicative119867V-ring 119878The setC119903(119868)

for a right119867V-ideal 119868 can be described dually

Lemma 16 Let 119878 be a multiplicative 119867V-ring with an 119894-set Eand let L denote the set of all left 119867V-ideals of 119878 119865 119875(119878) rarr119875(119878) is a mapping defined by

119865 (119860) = ⋃

119899

sum

119894=1

[1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897 119899 119898

119894isin N 119903

1198941isin 119878

119904119886119905119894119904119891119910119894119899119892 [1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897cap 119860 = 120601

(7)

for any 119860 isin 119875(119878) Then for any 119868 1198681 1198682isin L we have the

following (i) 119865(119868) isin L (ii) 119868 sube 119865(119868) (iii) 1198681sube 1198682rArr 119865(119868

1) sube

119865(1198682) (iv) 119865(119868) = 119868 if and only if 119868 is a left C-ideal of 119878 and

(v) left C-closure of 119868 is C119897(119868) = ⋃

infin

119896=1119865119896(119868) where for any

119896 isin N 119865119896 denotes the (119896 minus 1)-times mapping composition of 119865

Proof All the assertions made in this lemma can be estab-lished by adopting the arguments that are applied in provingwell-known analogous results on ldquocomplete closure of a setrdquoin semihypergroup theory (see [3 8])

Proposition 17 Let 119877 be a strong hyperdomain with an 119894-setThen for any 119867V-ideal 119868 of the multiplicative 119867V-ring 119877[119909]ord(C(I)lowast) ge ordIlowast where for any 119867V-ideal 119869 of 119877[119909] 119869lowast =119869 0119877[119909]

Proof Since 119877 is a (strong) hyperdomain the multiplicative119867V-ring is commutative So for any 119867V-ideal 119868 of 119877[119909]C(119868) = C

119897(119868) = ⋃

infin

119896=1119865119896(119868) (by Lemma 16(v)) For any

119899 isin N 119891(119909) isin 119865119899(119868)lowast rArr 119891(119909) isin sum119898

119894=1119860119894 for some

type-119897 hyperproducts 119860119894= [1198911198941(119909) lowast 119891

1198942(119909) lowast sdot sdot sdot lowast 119891

119894119898119894(119909)]119897

of elements of 119877[119909] satisfying 119860119894cap 119865119899minus1(119868) = 120601 Since

here 119877[119909] is a strong 119867V-domain (by Proposition 13) withabsorbing zero we may assume that 0

119877[119909]notin 119860119894for each 119894

(since 0119877[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119877[119909])) Then for

each 119894 (= 1 2 119898) and 119895119894(= 1 2 119898

119894) 119891119894119895119894

(119909) = 0119877[119909]

Thus by Corollary 12 ord119860

119894= ord119892

119894(119909) for any 119892

119894(119909) isin 119860

119894

Now since for each 119894 119860119894cap 119865119899minus1(119868) = 120601 so there exists

119892119894(119909) isin 119860

119894such that 119892

119894(119909) isin 119865

119899minus1(119868)lowast Then for any 119891(119909) isin

119865119899(119868)lowast ord119891(119909) ge sum119898

119894=1ord119860119894= minord119892

119894(119909) 119894 ge

ord(119865119899minus1(119868)lowast) Consequently ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast)for any 119899 isin N Now let 119891(119909) isin C(119868)

lowast be arbitrary Then119891(119909) isin 119865

119899(119868)lowast for some 119899 isin N So we have that ord119891(119909) ge

ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast) ge sdot sdot sdot ge ord(119865(119868)lowast) ge ord119868lowastThus ord(C(119868)lowast) ge ord119868lowast

5 Polynomials over Multiplicative Hyperfield

A nonzero element 119886 of a multiplicative119867V-ring (119878 + ∘) withan 119894-set E = 119890

1 1198902 119890

119899 is referred to be an E-invertible

element (or anE-unit) of 119878 if for each 119894 = 1 2 119899 (119899 isin N)there exist 119860

119894119895isin H119897(119886) = [119904

119901∘ 119904119901minus1∘ sdot sdot sdot ∘ 119904

1∘ 119886]119897 119904119894isin 119878

119901 isin N and 119861119894119896isin H119903(119886) = [119886 ∘ 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119902]119903 119904119894isin 119878

119902 isin N such that 119890119894isin (sum119898

119895=1119860119894119895) cap (sum

119897

119896=1119861119894119896) An element

119886 ( =0119878) of the multiplicative119867V-ring 119878with a hyperidentity 119890

is said to be 119890-hyperinvertible (or an 119890-hyperunit) in 119878 if thereexist 119860 isinH

119897(119886) and 119861 isinH

119903(119886) such that 119890 isin 119860 cap 119861

If 119886 isin 119878 is an E-unit (resp 119890-hyperunit) in a multiplica-tive 119867V-ring (119878 + ∘) with two 119894-sets E and E1015840 (resp withtwo hyperidentities 119890 and 1198901015840) then one can easily verify that119886 is also an E1015840-unit (resp an 1198901015840-hyperunit) in 119878 We thuscall an E-unit (resp an 119890-hyperunit) of a multiplicative 119867V-ring 119878 simply a unit (resp a hyperunit) Denote by 119880(119878)and 119880

ℎ(119878) respectively the sets of units and hyperunits of a

multiplicative119867V-ring 119878An 119867V-ideal 119868 ( = 119878) of a multiplicative 119867V-ring 119878 is

maximal in 119878 if for any 119867V-ideal 119869 of 119878 119868 ⊊ 119869 sube 119878 rArr 119869 = 119878For a commutative multiplicative119867V-ring 119878with an 119894-set thisis immediate to observe that 119886 isin 119880(119878) if and only if 119886 notin 119872for any maximal119867V-ideal119872 of 119878

Proposition 18 Let themultiplicative hyperring119877 be commu-tative and contain a hyperidentity 119890 Then for a polynomial119891(119909) = sum119886

119896119909119896isin 119877[119909] 119891(119909) isin 119880

ℎ(119877[119909]) if and only if

1198860isin 119880ℎ(119877)

Proof Since 119890 is a hyperidentity in the multiplicative hyper-ring 119877 1198901199090 isin 119877[119909] is a hyperidentity in the multiplicative119867V-ring 119877[119909] Now let 119891(119909) = sum119886

119896119909119896isin 119880ℎ(119877[119909]) Then there

exist 119892119894(119909) = sum 119887

119894119896119909119896isin 119877[119909] (119894 = 1 2 119899 119899 isin N) such that

1198901199090isin [119891(119909) lowast 119892

1(119909) lowast sdot sdot sdot lowast 119892

119899(119909)]119903 So 119890 isin 119886

0(1198871011988720 1198871198990)

and thus there exists 119887 isin 1198871011988720 1198871198990sube 119877 such that 119890 isin 119886

0119887

whence 1198860isin 119880ℎ(119877) (since 119877 is commutative)

Conversely let 1198860be hyperinvertible in 119877 with respect to

the hyperidentity 119890 Then there exists 1198870isin 119877 such that 119890 isin

11988601198870 We assert that there is a sequence 119887

119896119896isinN in 119877 whose

119899th term 119887119899(119899 isin N) is inductively defined so as to satisfy the

relation that

0119877isin 1198860119887119899+ 1198861119887119899minus1+ 1198862119887119899minus2+ sdot sdot sdot + 119886

1198991198870sdot sdot sdot () (8)

In fact we see that 0119877isin 01198771198870= (minus119886

1+ 1198861)1198870sube (119890(minus119886

1) +

1198861)1198870sube ((119886

01198870)(minus1198861) + 1198861)1198870sube (119886

01198870)(minus1198861)1198870+ 11988611198870=

1198860((minus1198870)11988611198870) + 11988611198870rArr there exist 119887

1isin (minus119887

0)11988611198870such that

0119877isin 11988601198871+ 11988611198870(the relation for 119899 = 1) Suppose for

some 119898 ge 1 the terms 119887119896(1 le 119896 le 119898 minus 1) are defined

in such a way that each 119887119896(1 le 119896 le 119898 minus 1) satisfies the

relation () for 119896 Then (1198861119887119898minus1+1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+1198861198981198870)

is defined to be a nonempty subset of 119877 Let 119905 isin 1198861119887119898minus1+

1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870 Then 0

119877= minus119905 + 119905 isin 119890(minus119905) + 119905 sube

(11988601198870)(minus119905) + 119905 sube 119886

0(minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+

1198861198981198870) + (1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) rArr there exists

119887119898isin (minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) such that

0119877isin 1198860119887119898+ 1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

1198981198870(the relation

for 119896 = 119898) Hence the assertion is true for all 119899 isin N Thus

Algebra 7

consider the polynomial 119892(119909) = sum 119887119896119909119896isin 119877[119909] Then from

the definition of lowast 119891(119909) lowast 119892(119909) = sum 119888119896119909119896 1198880isin 11988601198870and

119888119896isin 1198860119887119896+ 1198861119887119896minus1+ 1198862119887119896minus2+ sdot sdot sdot + 119886

1198961198870for 119896 isin N Thus

1198901199090isin 119891(119909) lowast 119892(119909) (due to relation and since 119890 isin 119886

01198870) So

119891(119909) is hyperinvertible in 119877[119909] that is 119891(119909) isin 119880ℎ(119877[119909])

The (left right) 119867V-ideal of a multiplicative 119867V-ring 119878generated by 119860 isin 119875(119878) is the smallest (resp left right) 119867V-ideal of 119878 containing119860which is denoted by (resp ⟨119860⟩

119897 ⟨119860⟩119903)

⟨119860⟩ The principal (left right) 119867V-ideal of the multiplicative119867V-ring 119878 generated by an element 119886 of 119878 denoted by (resp⟨119886⟩119897 ⟨119886⟩119903) ⟨119886⟩ is the (resp left right)119867V-ideal (resp ⟨119886⟩119897

⟨119886⟩119903) ⟨119886⟩ of the multiplicative119867V-ring 119878

If the multiplicative 119867V-ring 119878 has an 119894-set then for any119886 isin 119878 ⟨119886⟩

119897= H119897(119886) = ⋃sum

119899

119894=1119860119894 119860119894isin H119897(119886) 119899 isin N and

⟨119886⟩119903=H119903(119886) = ⋃sum

119899

119894=1119860119894 119860119894isinH119903(119886) 119899 isin N

Definition 19 A commutative multiplicative 119867V-ring 119878 withan 119894-setE is called amultiplicative119867V-field (resp an inversivemultiplicative 119867V-field) if 119878 ⟨0

119878⟩ sube 119880(119878) (resp 119878

⟨0119878⟩ sube 119880

ℎ(119878)) If a multiplicative119867V-field (resp an inversive

multiplicative 119867V-field) is a multiplicative hyperring thenwe call it a multiplicative hyperfield (resp an inversivemultiplicative hyperfield)

Proposition 20 Let 119865 be a polynomially structured inversivemultiplicative hyperfield Then a polynomial 119891(119909) = sum119886

119896119909119896isin

119865[119909]0119865[119909] is hyperinvertible in119865[119909] if and only if ord 119891 = 0

Proof If 119865 is an inversive multiplicative hyperfield then byProposition 18 any polynomial119891(119909) = sum119886

119896119909119896isin 119865[119909]0

119865[119909]

is hyperinvertible in 119865[119909] if and only if 1198860= 0119865 Hence the

result follows

Definition 21 A commutative multiplicative 119867V-ring 119878 iscalled a principal C-ideal 119867V-ring if every C-ideal of119878 is a principal 119867V-ideal A principal C-ideal 119867V-ringwhich is a (strong) 119867V-domain is called a principal C-ideal(strong) 119867V-domain

Proposition 22 Let 119865 be a polynomially structured inversivemultiplicative hyperfieldThen themultiplicative119867V-ring119865[119909]of polynomials over 119865 is a principalC-ideal119867V-ring

Proof Let 119890 be a hyperidentity of the inversive multiplicativehyperfield 119865 Then the polynomial 1198901199090 is a hyperidentity in119865[119909] and thus 119865[119909] = ⟨1198901199090⟩ (since 119865 is commutative) Solet 119868 be any proper C-ideal of 119865[119909] If 119868 = 0

119865[119909] then 119868 is

the principal hyperideal ⟨0119865[119909]⟩ (since for119865 having absorbing

zero 0119865[119909]

is absorbing in 119865[119909]) Suppose that 119868 = 0119865[119909]

Then take a nonzero polynomial 119891(119909) = sum119886119896119909119896isin 119868 such

that ord119891(119909) le ord 119905(119909) for any 119905(119909) isin 119868lowast Let us writeord119891(119909) = 119898 Then 119886

119898= 0119865and 119886119896= 0119865for any 0 le 119896 lt 119898

Consider then the polynomial 119892(119909) = sumlowast119896isinN0119887119896119909119896isin 119865[119909]

where 119887119896= 119886119898+119896

Then clearly 119891(119909) isin (119890119909119898) lowast 119892(119909) Alsoby Proposition 18 119892(119909) is hyperinvertible in 119865[119909] Thus thereexists ℎ(119909) isin 119865[119909] such that 1198901199090 isin 119892(119909) lowast ℎ(119909) Now119891(119909) isin (119890119909

119898) lowast 119892(119909) rArr 119891(119909) lowast ℎ(119909) sube [119890119909

119898lowast 119892(119909) lowast ℎ(119909)]

119903

Again 119891(119909) lowast ℎ(119909) sube 119868 (since 119891(119909) isin 119868) So [119890119909119898 lowast 119892(119909) lowastℎ(119909)]119903cap 119868 = 120601 Hence [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868 (since

119868 is a C-ideal and every C-ideal is a right C-ideal) Now[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap [119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119903= 120601 whence

[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap 119868 = 120601 (since [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868)

Consequently [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]119897sube 119868 (since 119868 is a left C-

ideal) Then 119890119909119898 isin (119890119909119898) lowast (1198901199090) sube 119890119909119898 lowast (119892(119909) lowast ℎ(119909)) =[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897sube 119868 Thus ⟨119890119909119898⟩ sube 119868

Now let ℎ(119909) = sum 119887119896119909119896isin 119868lowast be arbitrary Suppose that

ord ℎ = 119899 Then 119887119899= 0119865and 119887119896= 0119865for any 0 le 119896 lt 119899

By choice of 119891(119909) from 119868 here 119898 le 119899 So one can definea polynomial 119892(119909) = sum 119888

119896119909119896isin 119865[119909] where 119888

119896= 0119865for all

0 le 119896 le 119899 minus 119898 minus 1 and 119888119896= 119887119898+119896

for all 119896 ge 119899 minus 119898 Clearlythen ℎ(119909) isin 119890119909119898 lowast 119892(119909) sube ⟨119890119909119898⟩ whence 119868 sube ⟨119890119909119898⟩ Thus119868 = ⟨119890119909

119898⟩

Remark 23 In a ring an invertible element can never be adivisor of zero This not true in general for a multiplicativehyperring In fact on the commutative group of integers(Z +) if we define a hyperoperation ∘ by stating that 119909 ∘ 119910 =0 119909119910 for all 119909 119910 isin Z then (Z + ∘) is a commutative mul-tiplicative hyperring with a hyperidentity 1 Every nonzeroelement of (Z + ∘) is a zero divisor and 1 minus1 isin Z are inparticular hyperunits of (Z + ∘) To get a parity with thering theory in this regard we perceive the notion of stronghyperinvertibility of an element of amultiplicative hyperring

Definition 24 A hyperinvertible element of a multiplicativehyperring with a hyperidentity is said to be strongly hyperin-vertible (or a strong hyperunit) if it is not a zero divisor inthat multiplicative hyperring A commutative multiplicativehyperring with absorbing zero and a hyperidentity 119890 is said tobe a strongly inversive multiplicative hyperfield if each of itsnonzero elements is a strong hyperunit

Example 25 Let 119877 = 119886radic2 + 119887radic3 119886 119887 isin Q and 119860 =radic2radic3 Then with respect to usual addition + of reals(119877 +) is a commutative group with identity 0 On 119877 ∘ is ahyperoperation defined by

(119886radic2 + 119887radic3) ∘ (119888radic2 + 119889radic3)

= (119886radic2 + 119887radic3) sdot 119905 sdot (119888radic2 + 119889radic3) 119905 isin 119860

= 119901radic2 + 2119902radic3 3119902radic2 + 119901radic3

(9)

where 119901 = 2119886119888 + 3119887119889 and 119902 = 119887119888 + 119886119889Then (119877 + ∘) is a strongly inversivemultiplicative hyper-

field which is polynomially structured

Definition 26 A local (C-local) multiplicative 119867V-ring is acommutative multiplicative119867V-ring with an 119894-set which hasa unique maximal119867V-ideal (respC-ideal)

Proposition 27 Let 119865 be a polynomially structured stronglyinversive multiplicative hyperfieldThen (i) 119865[119909] is a principalC-ideal strong 119867V-domain (ii) for any 119891 isin 119865[119909] 0

119865[119909]

there exist a hyperinvertible element 119892 isin 119865[119909] and 119898 isin

N cup 0 such that 119891 isin (119890119909119898) lowast 119892 where 119890 is a hyperidentity

8 Algebra

in 119865 and (iii) 119865[119909] is a local as well as aC-local multiplicative119867V-ring

Proof (i) Here 119865 is a strongly inversive multiplicative hyper-field So 119865 is a strong hyperdomain Thus by Proposition 13the multiplicative 119867V-ring 119865[119909] is a strong 119867V-domainAgain by Proposition 22 119865[119909] is a principal C-ideal 119867V-ring (since 119865 is an inversive multiplicative hyperfield) So byDefinition 21 119865[119909] is a principalC -ideal strong119867V-domain

(ii) Let 119891 isin 119865[119909] 0119865[119909] and 119869 = ⟨119891⟩ Then 119869 =H

119903(119891)

So for any ℎ isin 119869lowast there exist119860119894= [119891lowast119891

1198941lowast1198911198942lowastsdot sdot sdotlowast119891

119894119898119894]119903isin

H119903(119891) (119894 = 1 2 119896 119896119898

119894isin N) such that ℎ isin sum119896

119894=1119860119894

Thus ℎ = sum119896119894=1119905119894for some 119905

119894isin 119860119894 Since 119865[119909] is a strong119867V-

domain (by Proposition 13) with absorbing zero so 0119865[119909]isin

119860119894= [119891 lowast 119891

1198941lowast 1198911198942lowast sdot sdot sdot lowast 119891

119894119898119894]119903rArr [119891 lowast 119891

1198941lowast 1198911198942lowast

sdot sdot sdot lowast 119891119894119898119894]119903= 0119865[119909] Hence without any loss of generality

(since 0119865[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119865[119909])) we may

assume that 0119865[119909]notin 119860119894for any 119894 (since ℎ = 0

119865[119909]) Then

for any 119894 = 1 2 119896 and 119895 = 1 2 119898119894 119891119894119895= 0119865[119909]

So by Proposition 11(ii) ord 119905

119894= ord(119860

119894) ge ord119891 and so

ord ℎ = minord 119905119894 119894 ge ord119891 Thus ord119869lowast ge ord119891 So

by Proposition 17 119891 is a nonzero polynomial in the C-idealC(119869) such that ord119891 le ord ℎ for all ℎ isin C(119869)lowast So there existsan invertible element 119892 isin 119865[119909] and an integer119898 ge 0 such that119891 isin (119890119909

119898) lowast 119892 where 119890 is a hyperidentity in 119865 (see proof of

Proposition 22)(iii) For any integer119898 ge 1 since 119890119909119898 isin (119890119909) lowast (119890119909119898minus1) so

⟨119890119909119898⟩ sube ⟨119890119909⟩ Again for any119867V-ideal 119868 of the multiplicative

119867V-ring119865[119909] if119891 isin 119868lowast is such that ord119891 le ord119892 for all119892 isin 119868lowast

then 119868 sube ⟨119890119909119898⟩ where 119898 = ord119891 So ⟨119890119909⟩ is the uniquemaximal119867V-ideal in 119865[119909] Thus 119865[119909] is a local multiplicative119867V-ring Now 119890119909 isin C

1(as defined in Proposition 15) So

ord(119890119909) le ord119891 for any 119891 isin Clowast1 Thus C

1being a C-ideal

of 119865[119909] we have C1= ⟨119890119909⟩ (see proof of Proposition 22)

Hence 119865[119909] is aC-local multiplicative119867V-ring

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vougiouklis ldquoThe fundamental relation in hyperrings Thegeneral hyperfieldrdquo in Algebraic Hyperstructures and Appli-cations (Xanthi1990) pp 203ndash211 World Science PublisherTeaneck NJ USA 1991

[2] M R Darafsheh and B Davvaz ldquo119867V-ring of fractionsrdquo ItalianJournal of Pure and Applied Mathematics no 5 pp 25ndash34 1999

[3] B Davvaz and V Leoreanu-FoteaHyperring Theory and Appli-cations International Academic Press Palm Harbor Fla USA2007

[4] S Spartalis A Dramalides and T Vougiouklis ldquoOn 119867V-groupringsrdquoAlgebras Groups and Geometries vol 15 no 1 pp 47ndash541998

[5] T Vougiouklis ldquo119867V-groups defined on the same setrdquo DiscreteMathematics vol 155 no 1ndash3 pp 259ndash265 1996

[6] R Procesi and R Rota ldquoMultiplicative H119907-rings and com-

plementary hyperstructuresrdquo Journal of Discrete MathematicalSciences amp Cryptography vol 11 no 4 pp 447ndash456 2008

[7] S Spartalis ldquoOn the number of 119867Vminus119903119894119899119892119904 with 119875-hyperopera-tionsrdquoDiscrete Mathematics vol 155 no 1ndash3 pp 225ndash231 1996

[8] P Corsini Prolegomena of Hypergroup Theory Rivista di Mate-matica Pura ed Applicata Aviani Tricesimo Italy 1993

[9] M K Sen and U Dasgupta ldquoSome aspects of119866119867-ringsrdquoAnnals

of the Alexandru Ioan Cuza UniversitymdashMathematics vol 56no 2 pp 253ndash272 2010

[10] C Namnak N Triphop and Y Kemprasit ldquoHomomorphismsof somemultiplicative hyperringsrdquo Set-ValuedMathematics andApplications vol 1 no 2 pp 145ndash152 2008

[11] D M Olson and V K Ward ldquoA note on multiplicativehyperringrdquo Italian Journal of Pure andAppliedMathematics vol1 pp 77ndash84 1997

[12] R Procesi and R Rota ldquoComplementary multiplicative hyper-ringsrdquoDiscreteMathematics vol 308 no 2-3 pp 188ndash191 2008

[13] R Procesi and R Rota ldquoOn some classes of hyperstructuresrdquoDiscrete Mathematics vol 208-209 pp 485ndash497 1999

[14] R Rota ldquoStrongly distributive multiplicative hyperringsrdquo Jour-nal of Geometry vol 39 no 1-2 pp 130ndash138 1990

[15] R Rota ldquoSugli Iperanelli Moltiplicativirdquo Rendiconti di Matem-atica Series VII vol 2 no 4 pp 711ndash724 1982

[16] R Procesi Ciampi andR Rota ldquoPolynomials overmultiplicativehyperringsrdquo Journal of Discrete Mathematical Sciences amp Cryp-tography vol 6 no 2-3 pp 217ndash225 2003

[17] T Vougiouklis Hyperstructures and Their RepresentationsMonographs in Mathematics Hardonic Press 1994

[18] U Dasgupta ldquoOn prime and primary hyperideals of a multi-plicative hyperringrdquoAnalele Stiintifice ale Universitatii Al I Cuzadin IasimdashMatematica vol 58 no 1 pp 19ndash37 2012

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Algebra

fact for any 119891(119909) = sum119886119896119909119896isin 119877[119909] we have that sum119899

119894=1119890119894lowast

119891(119909) = sum119899

119894=1(119890119894lowast sum119886

119896119909119896) = sum

119899

119894=1sum 119886119894119896119909119896 119886119894119896isin 119890119894119886119896 =

sum(sum119899

119894=1119886119894119896)119909119896 119886119894119896isin 119890119894119886119896 = sum 119887

119896119909119896 119887119896isin sum119899

119894=1119890119894119886119896

Then since 119886119896isin sum119899

119894=1119890119894119886119896 we have that119891(119909) isin sum119899

119894=1119890119894lowast119891(119909)

identifying 1198901198941199090 with 119890

119894 Similarly one can see that 119891(119909) isin

sum119899

119894=1119891(119909) lowast 119890

119894

On the other hand if for some 119899 isin N E119877[119909]= 120598119894(119909) =

sum 119890119894119896119909119896isin 119877[119909] 119894 = 1 2 119899 is an 119894-set in 119877[119909] then

E = 1198901198940 119894 = 1 2 119899 is an 119894-set in 119877 In fact for any

119886 isin 119877 1198861199090 isin sum119899119894=1(1198861199090) lowast 120598119894(119909) = sum

119899

119894=1sum 119886119894119896119909119896isin 119877[119909] 119886

119894119896isin

119886119890119894119896 = sum(sum

119899

119894=1119886119894119896)119909119896isin 119877[119909] 119886

119894119896isin 119886119890119894119896 = sum 119886

119896119909119896isin

119877[119909] 119886119896isin sum119899

119894=1119886119890119894119896 rArr 119886 isin sum

119899

119894=11198861198901198940 Similarly from

1198861199090isin sum119899

119894=1120598119894(119909) lowast (119886119909

0) one may arrive at 119886 isin sum119899

119894=11198901198940119886

This is clear fromRemark 6 (d) that for any hyperidentity119890 of 119877 the polynomial 1198901199090 isin 119877[119909] is a hyperidentity in themultiplicative119867V-ring 119877[119909] Is every hyperidentity of 119877[119909] ofthe form 1198901199090 for some hyperidentity 119890 of 119877 Following is anexample of a multiplicative hyperring 119877 such that 119877[119909] has ahyperidentity 120598(119909) = 1198901199090 for any hyperidentity 119890 of 119877

Example 7 Let (Z + sdot) be the ring of integers and 119875 = 0 1Then (Z +

lowast

119875) is a commutative polynomially structuredmultiplicative hyperring (as is shown in Example 3) Denotethe multiplicative hyperring (Z +

lowast

119875) by Z119875 Consider a

polynomial 1 + 1119909 isin Z119875[119909] Then for any 119891(119909) = sum119886

119896119909119896isin

Z119875[119909] we see that (1+1119909)lowast119891(119909) = sum 119887

119896119909119896 1198870isin 1lowast

119875 1198860 and119887119896isin 1lowast

119875 119886119896 + 1lowast

119875 119886119896minus1 (for 119896 isin N) Clearly 1198860 isin 1lowast

119875 1198860 andfor any 119896 isin N 119886

119896isin 1lowast

119875 119886119896 + 1lowast

119875 119886119896minus1 (since 0 isin 1lowast

119875 119886119896minus1)Thus 119891(119909) isin (1 + 1119909) lowast 119891(119909) and so 1 + 1119909 is a hyperidentityin Z119875[119909] which is not in the form 1198901199090 for any hyperidentity

119890 of Z119875

Remark 8 Let 120598(119909) = sum 119890119896119909119896isin 119877[119909] be a hyperidentity in the

multiplicative119867V-ring 119877[119909] Then from Remark 6 (d) 1198900is a

hyperidentity in the multiplicative hyperring 119877

Proposition 9 Let 119877 be a strong integral hyperring Thenevery hyperidentity in the multiplicative119867V-ring 119877[119909] is of theform 1198901199090 for some hyperidentity 119890 of 119877

Proof Suppose that 120598(119909) = sum 119890119896119909119896isin 119877[119909] is a hyperidentity

in the multiplicative 119867V-ring 119877[119909] Then by the Remark 8119890 = 1198900is a hyperidentity in119877 Now let 119886 isin 1198770

119877 be arbitrary

and 119891(119909) = sum119886119896119909119896isin 119877[119909] where 119886

0= 119886 and 119886

119896= 0119877 for all

119896 isin N Then 119891(119909) isin 120598(119909) lowast 119891(119909) cap 119891(119909) lowast 120598(119909) whereby0119877isin 119890119896119886 cap 119886119890

119896 for all 119896 isin N whence 119890

119896= 0119877for all 119896 isin N

(since 119877 is a strong integral hyperring and 119886 = 0119877) Thus

120598(119909) = 1198901199090+01198771199091+01198771199092+sdot sdot sdot where 119890 = 119890

0is a hyperidentity

in 119877

Definition 10 If 0119877[119909]

= 119891(119909) = sum119886119896119909119896isin 119877[119909] then the

smallest integer 119899 such that 119886119899= 0119877is called the order of119891(119909)

and is denoted by ord119891(119909) The order of 0119877[119909]

is defined to bezero For a nonempty set119860 isin 119875(119877[119909]) the smallest element inthe set ord119891(119909) 119891(119909) isin 119860 does exist and is called the orderof 119860 being denoted by ord119860 that is ord119860 = minord119891(119909) 119891(119909) isin 119860

In the next proposition we will find some propertiesof ord119860 for some 119860 isin 119875lowast(119877[119909])(= 119875(119877[119909]) 120601) Forthat it is necessary at this point to frame some suitablenotations corresponding to different types of hyperproductsof elements in the multiplicative119867V-ring 119877[119909] Indeed in anymultiplicative 119867V-ring (119878 + ∘) (which is not a multiplicativehyperring) the expression like 119904

1∘1199042∘1199043∘sdot sdot sdot∘119904

119899(119904119894isin 119878 119899 ge 3)

bears no connotation in 119878 unless the parentheses ldquo(rdquo andldquo)rdquo are meaningfully inserted in Note that the following twoexpressions

1199041∘ (11199042∘ (21199043∘ (31199044∘ sdot sdot sdot ∘ (

119899minus3119904119899minus2∘ (119899minus2119904119899minus1∘ 119904119899)119899minus2)119899minus3sdot sdot sdot )3)2)1

(1(2(3sdot sdot sdot (119899minus3(119899minus21199041∘ 1199042)119899minus2∘ 1199043)119899minus3∘ sdot sdot sdot ∘ 119904

119899minus3)3∘ 119904119899minus2)2∘ 119904119899minus1)1∘ 119904119899

(3)

are meaningful called the finite hyperproducts of type 119897 andtype 119903 andwritten in notations respectively as [119904

1∘1199042∘sdot sdot sdot∘119904

119899]119897

and [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903

Proposition 11 (i) For any 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]

119890119894119905ℎ119890119903 0119877[119909]isin 119891 (119909) lowast 119892 (119909)

119900119903 ord (119891 (119909) lowast 119892 (119909)) ge ord 119891 (119909) + ord 119892 (119909) (4)

(ii) If 119877 is a strong hyperdomain then the implication that

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119903

) = ord 119892 (119909) (5)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119903holds true for any

119899 isin N with 119899 ge 2

Proof (i) Let ord119891(119909) = 119899 and ord119892(119909) = 119898 Then 119891(119909) =sum119886119896119909119896 where 119886

119899= 0119877 119886119896= 0119877for 119896 lt 119899 and 119892(119909) = sum 119887

119896119909119896

where 119887119898= 0119877 119887119896= 0119877for 119896 lt 119898 So for anysum119888

119896119909119896isin 119891(119909)lowast

119892(119909) we see that whenever 119896 lt 119899 + 119898 119888119896isin sum119894+119895=119896119886119894119887119895=

0119877 (since 0

119877is absorbing in 119877 and 119886

119896= 0119877(119896 lt 119899) 119887

119896=

0119877(119896 lt 119898)) Thus if 0

119877[119909]notin 119891(119909) lowast 119892(119909) then for any ℎ(119909) =

sum 119888119896119909119896isin 119891(119909) lowast 119892(119909) we have that ord ℎ(119909) ge 119899 + 119898 So

ord(119891(119909) lowast 119892(119909)) ge 119899 + 119898 = ord119891(119909) + ord119892(119909)(ii) Now suppose that 119877 is a strong hyperdomain with

absorbing zero Consider two polynomials 119891(119909) 119892(119909) isin119877[119909] 0

119877[119909] Let ord119891(119909) = 119901 and ord119892(119909) = 119898 Then

119891(119909) = sum119886119896119909119896 where 119886

119901= 0119877 119886119896= 0119877for 119896 lt 119901 and

Algebra 5

119892(119909) = sum 119887119896119909119896 where 119887

119898= 0119877 119887119896= 0119877for 119896 lt 119898

So 0119877notin 119886119901119887119898(since 119877 is a strong hyperdomain) and also

119886119901119887119898= sum119894+119895=119901+119898

119886119894119887119895(since 119886

119896= 0119877for 119896 lt 119901 and 119887

119896= 0119877

for 119896 lt 119898) Thus for any ℎ(119909) = sum 119888119896119909119896isin 119891(119909) lowast 119892(119909)

119888119901+119898

= 0119877(since 119888

119901+119898isin sum119894+119895=119901+119898

119886119894119887119895= 119886119901119887119898) and 119888

119896= 0119877

for all 119896 with 0 le 119896 lt 119901 + 119898 Thus ordℎ(119909) = 119901 + 119898 forany ℎ(119909) isin 119891(119909) lowast 119892(119909) and so ord(119891(119909) lowast 119892(119909)) = 119901 + 119898Hence ord119891(119909)+ord119892(119909) = ord(119891(119909)lowast119892(119909)) = ord ℎ(119909) forall ℎ(119909) isin 119891(119909) lowast 119892(119909) So the implication is true for 119899 = 2(noting that [119891(119909) lowast 119892(119909)]

119903= 119891(119909) lowast 119892(119909)) Suppose that for

some integer 119896 ge 2 the implication holds true for each valueof 119899 ranging from 2 to 119896minus1 and take any119891

119894(119909) isin 119877[119909]0

119877[119909]

for 119894 = 1 2 119896 Thensum119896minus1119894=1

ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast

sdot sdot sdotlowast119891119896minus1(119909)]119903) = ord ℎ(119909) for all ℎ(119909) isin [119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast

119891119896minus1(119909)]119903 Now let 119892(119909) isin [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903be

arbitraryThen 119892(119909) isin ℎ(119909) lowast119891119896(119909) for some ℎ(119909) isin [119891

1(119909) lowast

1198912(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903 So ord119892(119909) = ord(ℎ(119909) lowast 119891

119896(119909)) =

ord ℎ(119909) + ord119891119896(119909) = sum

119896minus1

119894=1ord119891119894(119909) + ord119891

119896(119909)(since ℎ(119909) isin

[1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903) = sum

119896

119894=1ord119891119894(119909) Thus

sum119896

119894=1ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119896(119909)]119903) = ord119892(119909)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903

Hence by strong induction the implication follows forany 119899 isin N with 119899 ge 2

Corollary 12 If 119877 is a strong hyperdomain then the implica-tion

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119897

) = ord 119892 (119909) (6)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897holds true for any

119899 isin N with 119899 ge 2

Proof Since themultiplicative hyperring119877 is a hyperdomainit is commutative and so 119877[119909] is also a commutative multi-plicative 119867V-ring Hence for any 119891119894(119909) isin 119877[119909] 0119877[119909] (119894 =1 2 119896) we have that [119891

1(119909)lowast119891

2(119909)lowast119891

3(119909)lowastsdot sdot sdotlowast119891

119896(119909)]119897=

[119891119896(119909) lowast 119891

119896minus1(119909) lowast sdot sdot sdot lowast 119891

2(119909) lowast 119891

1(119909)]119903 Hence the assertion

follows straight from Proposition 11

Proposition 13 If the multiplicative hyperring 119877 is a stronghyperdomain the multiplicative 119867V-ring 119877[119909] is a strong 119867V-domain

Proof 119877 being a strong hyperdomain is a commutativemultiplicative hyperring Thus 119877[119909] is a commutative mul-tiplicative 119867V-ring Again since 0119877 is absorbing in 119877 0

119877[119909]

is also absorbing in 119877[119909] Thus we take 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]Then by Proposition 11 ord(119891(119909)lowast119892(119909)) = ord119891(119909)+

ord119892(119909) gt 0 Hence 0119877[119909]notin 119891(119909) lowast 119892(119909) So 119877[119909] is a strong

119867V-domain

4 C-Ideals in 119877[119909]

A subgroup 119868 of the group (119878 +) is called a left (resp right)119867V-ideal of a multiplicative 119867V-ring (119878 + ∘) if for any 119904 isin 119878

and 119886 isin 119868 119904 ∘ 119886 sube 119868 (resp 119886 ∘ 119904 sube 119868) 119868 is an119867V-ideal of 119878 if itis both a left and a right119867V-ideal of 119878

We call an 119867V-ideal of a multiplicative 119867V-ring 119878 simplya hyperideal when 119878 is a multiplicative hyperringThe notionof a typical hyperideal in a multiplicative hyperring calledC-ideal is introduced in [18] to study prime and primaryhyperideals of multiplicative hyperrings A hyperideal 119868 of amultiplicative hyperring 119878 is a C-ideal if for any 119860 isin C 119860 cap119868 = 120601 rArr 119860 sube 119868 where C = 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899 119904119894isin 119878

119899 isin N sube 119875lowast(119878) Following is the definition of aC-ideal in anarbitrary multiplicative119867V-ring

Definition 14 A left (resp right) 119867V-ideal 119868 of a multiplica-tive 119867V-ring (119878 + ∘) is called a left (resp right) C-ideal iffor any type 119897 hyperproduct [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897(resp type 119903

hyperproduct [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903) of elements 119904

119894isin 119878 we have

that [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897cap 119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897sube 119868 (resp

[1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903cap119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903sube 119868) An119867V-ideal

119868 of a multiplicative119867V-ring 119878 is called a C-ideal if it is a leftas well as a rightC-ideal in 119878

We writeC-ideal(119878) (respC-ideallowast(119878)) to denote the setof all (resp proper)C-ideals of a multiplicative119867V-ring 119878 Ina commutative multiplicative119867V-ring 119878 every left C-ideal isa right C-ideal and vice versa since commutativity impliesthe equality [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903= [119904119899∘ 119904119899minus1∘ sdot sdot sdot ∘ 119904

1]119897 for any

119904119894isin 119877 Note that in amultiplicative hyperring (even if it is not

commutative) a hyperideal is a leftC-ideal (as an119867V-ideal) ifand only if it is a rightC-ideal

Proposition 15 If 119877 is a strong hyperdomain then the setC-119868119889119890119886119897lowast(119877[119909]) is nonempty

Proof For each 119899 isin N cup 0 let C119899= 119891(119909) isin 119877[119909]

ord119891(119909) ge 119899 cup 0119877[119909] Then C

119899is a subgroup of the group

(119877[119909] +) (since for any 119891(119909) 119892(119909) isin 119877[119909] ord(minus119891(119909)) =ord119891(119909) and ord(119891(119909) + 119892(119909)) = minord119891(119909) ord119892(119909)when 119891(119909)+119892(119909) = 0

119877[119909]) Let 119892(119909) isin 119877[119909] and 119891(119909) isin C

119899 If

119892(119909) = 0119877[119909]

or 119891(119909) = 0119877[119909]

then 119892(119909) lowast119891(119909) = 0119877[119909] sube C119899

(since 0119877[119909]

is absorbing in 119877[119909]) So let 119892(119909) = 0119877[119909]

and119891(119909) = 0

119877[119909] Then by Proposition 11(ii) for all ℎ(119909) isin 119892(119909)lowast

119891(119909) ord ℎ(119909) = ord(119892(119909) lowast 119891(119909)) = ord119891(119909) + ord119892(119909) gt119899 Thus 119892(119909) lowast 119891(119909) sube C

119899 Hence C

119899is an 119867V-ideal of

119877[119909] (since 119877[119909] is a commutative multiplicative 119867V-ring)Let [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897be a type-119897 hyperproduct of

elements of the119867V-ring 119877[119909] such that [1198911(119909) lowast 1198912(119909) lowast sdot sdot sdot lowast119891119899(119909)]119897sube C119899 Then 119891

119894(119909) = 0

119877[119909]for each 119894 (since 0

119877[119909]isin C119899

is absorbing in119877[119909])Thus ord([1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897) =

ord119892(119909) for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 Now

since [1198911(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119899(119909)]119897sube C119899 so there exists ℎ(119909) isin

[1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897such that ℎ(119909) notin C

119899Then for any

119892(119909) isin [1198911(119909) lowast119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 ord119892(119909) = ord ℎ(119909) lt 119899

Hence by definition ofC119899 [1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897capC119899=

120601 and soC119899is a leftC-ideal and thus aC-ideal of 119877[119909] (since

119877[119909] is commutative) Hence C119899isin C-Ideallowast(119877[119909]) for all

119899 gt 0Since the intersection of left C-ideals of a multiplicative

119867V-ring 119878 is also a left C-ideal of 119878 and 119878 is itself a left C-ideal so the smallest left C-ideal containing a subset 119860 of 119878

6 Algebra

being naturally called the left C-ideal generated by 119860 existsand is in fact the intersection of all left C-ideals containing119860 The left C-ideal generated by a left 119867V-ideal 119868 of 119878 iscalled the leftC-closure of 119868 and is denoted byC

119897(119868) Clearly

C119897(C119897(119868)) = C

119897(119868) for a left 119867V-ideal 119868 C119903(119868) and C(119868)

respectively denote the right C-closure of a right 119867V-ideal 119868and theC-closure of an119867V-ideal 119868 of 119878 For an119867V-ideal 119868 of acommutativemultiplicative119867V-ring 119878C119897(119868) = C119903(119868) = C(119868)The following lemma presents a description of the set C

119897(119868)

for a left119867V-ideal 119868 of amultiplicative119867V-ring 119878The setC119903(119868)

for a right119867V-ideal 119868 can be described dually

Lemma 16 Let 119878 be a multiplicative 119867V-ring with an 119894-set Eand let L denote the set of all left 119867V-ideals of 119878 119865 119875(119878) rarr119875(119878) is a mapping defined by

119865 (119860) = ⋃

119899

sum

119894=1

[1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897 119899 119898

119894isin N 119903

1198941isin 119878

119904119886119905119894119904119891119910119894119899119892 [1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897cap 119860 = 120601

(7)

for any 119860 isin 119875(119878) Then for any 119868 1198681 1198682isin L we have the

following (i) 119865(119868) isin L (ii) 119868 sube 119865(119868) (iii) 1198681sube 1198682rArr 119865(119868

1) sube

119865(1198682) (iv) 119865(119868) = 119868 if and only if 119868 is a left C-ideal of 119878 and

(v) left C-closure of 119868 is C119897(119868) = ⋃

infin

119896=1119865119896(119868) where for any

119896 isin N 119865119896 denotes the (119896 minus 1)-times mapping composition of 119865

Proof All the assertions made in this lemma can be estab-lished by adopting the arguments that are applied in provingwell-known analogous results on ldquocomplete closure of a setrdquoin semihypergroup theory (see [3 8])

Proposition 17 Let 119877 be a strong hyperdomain with an 119894-setThen for any 119867V-ideal 119868 of the multiplicative 119867V-ring 119877[119909]ord(C(I)lowast) ge ordIlowast where for any 119867V-ideal 119869 of 119877[119909] 119869lowast =119869 0119877[119909]

Proof Since 119877 is a (strong) hyperdomain the multiplicative119867V-ring is commutative So for any 119867V-ideal 119868 of 119877[119909]C(119868) = C

119897(119868) = ⋃

infin

119896=1119865119896(119868) (by Lemma 16(v)) For any

119899 isin N 119891(119909) isin 119865119899(119868)lowast rArr 119891(119909) isin sum119898

119894=1119860119894 for some

type-119897 hyperproducts 119860119894= [1198911198941(119909) lowast 119891

1198942(119909) lowast sdot sdot sdot lowast 119891

119894119898119894(119909)]119897

of elements of 119877[119909] satisfying 119860119894cap 119865119899minus1(119868) = 120601 Since

here 119877[119909] is a strong 119867V-domain (by Proposition 13) withabsorbing zero we may assume that 0

119877[119909]notin 119860119894for each 119894

(since 0119877[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119877[119909])) Then for

each 119894 (= 1 2 119898) and 119895119894(= 1 2 119898

119894) 119891119894119895119894

(119909) = 0119877[119909]

Thus by Corollary 12 ord119860

119894= ord119892

119894(119909) for any 119892

119894(119909) isin 119860

119894

Now since for each 119894 119860119894cap 119865119899minus1(119868) = 120601 so there exists

119892119894(119909) isin 119860

119894such that 119892

119894(119909) isin 119865

119899minus1(119868)lowast Then for any 119891(119909) isin

119865119899(119868)lowast ord119891(119909) ge sum119898

119894=1ord119860119894= minord119892

119894(119909) 119894 ge

ord(119865119899minus1(119868)lowast) Consequently ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast)for any 119899 isin N Now let 119891(119909) isin C(119868)

lowast be arbitrary Then119891(119909) isin 119865

119899(119868)lowast for some 119899 isin N So we have that ord119891(119909) ge

ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast) ge sdot sdot sdot ge ord(119865(119868)lowast) ge ord119868lowastThus ord(C(119868)lowast) ge ord119868lowast

5 Polynomials over Multiplicative Hyperfield

A nonzero element 119886 of a multiplicative119867V-ring (119878 + ∘) withan 119894-set E = 119890

1 1198902 119890

119899 is referred to be an E-invertible

element (or anE-unit) of 119878 if for each 119894 = 1 2 119899 (119899 isin N)there exist 119860

119894119895isin H119897(119886) = [119904

119901∘ 119904119901minus1∘ sdot sdot sdot ∘ 119904

1∘ 119886]119897 119904119894isin 119878

119901 isin N and 119861119894119896isin H119903(119886) = [119886 ∘ 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119902]119903 119904119894isin 119878

119902 isin N such that 119890119894isin (sum119898

119895=1119860119894119895) cap (sum

119897

119896=1119861119894119896) An element

119886 ( =0119878) of the multiplicative119867V-ring 119878with a hyperidentity 119890

is said to be 119890-hyperinvertible (or an 119890-hyperunit) in 119878 if thereexist 119860 isinH

119897(119886) and 119861 isinH

119903(119886) such that 119890 isin 119860 cap 119861

If 119886 isin 119878 is an E-unit (resp 119890-hyperunit) in a multiplica-tive 119867V-ring (119878 + ∘) with two 119894-sets E and E1015840 (resp withtwo hyperidentities 119890 and 1198901015840) then one can easily verify that119886 is also an E1015840-unit (resp an 1198901015840-hyperunit) in 119878 We thuscall an E-unit (resp an 119890-hyperunit) of a multiplicative 119867V-ring 119878 simply a unit (resp a hyperunit) Denote by 119880(119878)and 119880

ℎ(119878) respectively the sets of units and hyperunits of a

multiplicative119867V-ring 119878An 119867V-ideal 119868 ( = 119878) of a multiplicative 119867V-ring 119878 is

maximal in 119878 if for any 119867V-ideal 119869 of 119878 119868 ⊊ 119869 sube 119878 rArr 119869 = 119878For a commutative multiplicative119867V-ring 119878with an 119894-set thisis immediate to observe that 119886 isin 119880(119878) if and only if 119886 notin 119872for any maximal119867V-ideal119872 of 119878

Proposition 18 Let themultiplicative hyperring119877 be commu-tative and contain a hyperidentity 119890 Then for a polynomial119891(119909) = sum119886

119896119909119896isin 119877[119909] 119891(119909) isin 119880

ℎ(119877[119909]) if and only if

1198860isin 119880ℎ(119877)

Proof Since 119890 is a hyperidentity in the multiplicative hyper-ring 119877 1198901199090 isin 119877[119909] is a hyperidentity in the multiplicative119867V-ring 119877[119909] Now let 119891(119909) = sum119886

119896119909119896isin 119880ℎ(119877[119909]) Then there

exist 119892119894(119909) = sum 119887

119894119896119909119896isin 119877[119909] (119894 = 1 2 119899 119899 isin N) such that

1198901199090isin [119891(119909) lowast 119892

1(119909) lowast sdot sdot sdot lowast 119892

119899(119909)]119903 So 119890 isin 119886

0(1198871011988720 1198871198990)

and thus there exists 119887 isin 1198871011988720 1198871198990sube 119877 such that 119890 isin 119886

0119887

whence 1198860isin 119880ℎ(119877) (since 119877 is commutative)

Conversely let 1198860be hyperinvertible in 119877 with respect to

the hyperidentity 119890 Then there exists 1198870isin 119877 such that 119890 isin

11988601198870 We assert that there is a sequence 119887

119896119896isinN in 119877 whose

119899th term 119887119899(119899 isin N) is inductively defined so as to satisfy the

relation that

0119877isin 1198860119887119899+ 1198861119887119899minus1+ 1198862119887119899minus2+ sdot sdot sdot + 119886

1198991198870sdot sdot sdot () (8)

In fact we see that 0119877isin 01198771198870= (minus119886

1+ 1198861)1198870sube (119890(minus119886

1) +

1198861)1198870sube ((119886

01198870)(minus1198861) + 1198861)1198870sube (119886

01198870)(minus1198861)1198870+ 11988611198870=

1198860((minus1198870)11988611198870) + 11988611198870rArr there exist 119887

1isin (minus119887

0)11988611198870such that

0119877isin 11988601198871+ 11988611198870(the relation for 119899 = 1) Suppose for

some 119898 ge 1 the terms 119887119896(1 le 119896 le 119898 minus 1) are defined

in such a way that each 119887119896(1 le 119896 le 119898 minus 1) satisfies the

relation () for 119896 Then (1198861119887119898minus1+1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+1198861198981198870)

is defined to be a nonempty subset of 119877 Let 119905 isin 1198861119887119898minus1+

1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870 Then 0

119877= minus119905 + 119905 isin 119890(minus119905) + 119905 sube

(11988601198870)(minus119905) + 119905 sube 119886

0(minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+

1198861198981198870) + (1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) rArr there exists

119887119898isin (minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) such that

0119877isin 1198860119887119898+ 1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

1198981198870(the relation

for 119896 = 119898) Hence the assertion is true for all 119899 isin N Thus

Algebra 7

consider the polynomial 119892(119909) = sum 119887119896119909119896isin 119877[119909] Then from

the definition of lowast 119891(119909) lowast 119892(119909) = sum 119888119896119909119896 1198880isin 11988601198870and

119888119896isin 1198860119887119896+ 1198861119887119896minus1+ 1198862119887119896minus2+ sdot sdot sdot + 119886

1198961198870for 119896 isin N Thus

1198901199090isin 119891(119909) lowast 119892(119909) (due to relation and since 119890 isin 119886

01198870) So

119891(119909) is hyperinvertible in 119877[119909] that is 119891(119909) isin 119880ℎ(119877[119909])

The (left right) 119867V-ideal of a multiplicative 119867V-ring 119878generated by 119860 isin 119875(119878) is the smallest (resp left right) 119867V-ideal of 119878 containing119860which is denoted by (resp ⟨119860⟩

119897 ⟨119860⟩119903)

⟨119860⟩ The principal (left right) 119867V-ideal of the multiplicative119867V-ring 119878 generated by an element 119886 of 119878 denoted by (resp⟨119886⟩119897 ⟨119886⟩119903) ⟨119886⟩ is the (resp left right)119867V-ideal (resp ⟨119886⟩119897

⟨119886⟩119903) ⟨119886⟩ of the multiplicative119867V-ring 119878

If the multiplicative 119867V-ring 119878 has an 119894-set then for any119886 isin 119878 ⟨119886⟩

119897= H119897(119886) = ⋃sum

119899

119894=1119860119894 119860119894isin H119897(119886) 119899 isin N and

⟨119886⟩119903=H119903(119886) = ⋃sum

119899

119894=1119860119894 119860119894isinH119903(119886) 119899 isin N

Definition 19 A commutative multiplicative 119867V-ring 119878 withan 119894-setE is called amultiplicative119867V-field (resp an inversivemultiplicative 119867V-field) if 119878 ⟨0

119878⟩ sube 119880(119878) (resp 119878

⟨0119878⟩ sube 119880

ℎ(119878)) If a multiplicative119867V-field (resp an inversive

multiplicative 119867V-field) is a multiplicative hyperring thenwe call it a multiplicative hyperfield (resp an inversivemultiplicative hyperfield)

Proposition 20 Let 119865 be a polynomially structured inversivemultiplicative hyperfield Then a polynomial 119891(119909) = sum119886

119896119909119896isin

119865[119909]0119865[119909] is hyperinvertible in119865[119909] if and only if ord 119891 = 0

Proof If 119865 is an inversive multiplicative hyperfield then byProposition 18 any polynomial119891(119909) = sum119886

119896119909119896isin 119865[119909]0

119865[119909]

is hyperinvertible in 119865[119909] if and only if 1198860= 0119865 Hence the

result follows

Definition 21 A commutative multiplicative 119867V-ring 119878 iscalled a principal C-ideal 119867V-ring if every C-ideal of119878 is a principal 119867V-ideal A principal C-ideal 119867V-ringwhich is a (strong) 119867V-domain is called a principal C-ideal(strong) 119867V-domain

Proposition 22 Let 119865 be a polynomially structured inversivemultiplicative hyperfieldThen themultiplicative119867V-ring119865[119909]of polynomials over 119865 is a principalC-ideal119867V-ring

Proof Let 119890 be a hyperidentity of the inversive multiplicativehyperfield 119865 Then the polynomial 1198901199090 is a hyperidentity in119865[119909] and thus 119865[119909] = ⟨1198901199090⟩ (since 119865 is commutative) Solet 119868 be any proper C-ideal of 119865[119909] If 119868 = 0

119865[119909] then 119868 is

the principal hyperideal ⟨0119865[119909]⟩ (since for119865 having absorbing

zero 0119865[119909]

is absorbing in 119865[119909]) Suppose that 119868 = 0119865[119909]

Then take a nonzero polynomial 119891(119909) = sum119886119896119909119896isin 119868 such

that ord119891(119909) le ord 119905(119909) for any 119905(119909) isin 119868lowast Let us writeord119891(119909) = 119898 Then 119886

119898= 0119865and 119886119896= 0119865for any 0 le 119896 lt 119898

Consider then the polynomial 119892(119909) = sumlowast119896isinN0119887119896119909119896isin 119865[119909]

where 119887119896= 119886119898+119896

Then clearly 119891(119909) isin (119890119909119898) lowast 119892(119909) Alsoby Proposition 18 119892(119909) is hyperinvertible in 119865[119909] Thus thereexists ℎ(119909) isin 119865[119909] such that 1198901199090 isin 119892(119909) lowast ℎ(119909) Now119891(119909) isin (119890119909

119898) lowast 119892(119909) rArr 119891(119909) lowast ℎ(119909) sube [119890119909

119898lowast 119892(119909) lowast ℎ(119909)]

119903

Again 119891(119909) lowast ℎ(119909) sube 119868 (since 119891(119909) isin 119868) So [119890119909119898 lowast 119892(119909) lowastℎ(119909)]119903cap 119868 = 120601 Hence [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868 (since

119868 is a C-ideal and every C-ideal is a right C-ideal) Now[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap [119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119903= 120601 whence

[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap 119868 = 120601 (since [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868)

Consequently [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]119897sube 119868 (since 119868 is a left C-

ideal) Then 119890119909119898 isin (119890119909119898) lowast (1198901199090) sube 119890119909119898 lowast (119892(119909) lowast ℎ(119909)) =[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897sube 119868 Thus ⟨119890119909119898⟩ sube 119868

Now let ℎ(119909) = sum 119887119896119909119896isin 119868lowast be arbitrary Suppose that

ord ℎ = 119899 Then 119887119899= 0119865and 119887119896= 0119865for any 0 le 119896 lt 119899

By choice of 119891(119909) from 119868 here 119898 le 119899 So one can definea polynomial 119892(119909) = sum 119888

119896119909119896isin 119865[119909] where 119888

119896= 0119865for all

0 le 119896 le 119899 minus 119898 minus 1 and 119888119896= 119887119898+119896

for all 119896 ge 119899 minus 119898 Clearlythen ℎ(119909) isin 119890119909119898 lowast 119892(119909) sube ⟨119890119909119898⟩ whence 119868 sube ⟨119890119909119898⟩ Thus119868 = ⟨119890119909

119898⟩

Remark 23 In a ring an invertible element can never be adivisor of zero This not true in general for a multiplicativehyperring In fact on the commutative group of integers(Z +) if we define a hyperoperation ∘ by stating that 119909 ∘ 119910 =0 119909119910 for all 119909 119910 isin Z then (Z + ∘) is a commutative mul-tiplicative hyperring with a hyperidentity 1 Every nonzeroelement of (Z + ∘) is a zero divisor and 1 minus1 isin Z are inparticular hyperunits of (Z + ∘) To get a parity with thering theory in this regard we perceive the notion of stronghyperinvertibility of an element of amultiplicative hyperring

Definition 24 A hyperinvertible element of a multiplicativehyperring with a hyperidentity is said to be strongly hyperin-vertible (or a strong hyperunit) if it is not a zero divisor inthat multiplicative hyperring A commutative multiplicativehyperring with absorbing zero and a hyperidentity 119890 is said tobe a strongly inversive multiplicative hyperfield if each of itsnonzero elements is a strong hyperunit

Example 25 Let 119877 = 119886radic2 + 119887radic3 119886 119887 isin Q and 119860 =radic2radic3 Then with respect to usual addition + of reals(119877 +) is a commutative group with identity 0 On 119877 ∘ is ahyperoperation defined by

(119886radic2 + 119887radic3) ∘ (119888radic2 + 119889radic3)

= (119886radic2 + 119887radic3) sdot 119905 sdot (119888radic2 + 119889radic3) 119905 isin 119860

= 119901radic2 + 2119902radic3 3119902radic2 + 119901radic3

(9)

where 119901 = 2119886119888 + 3119887119889 and 119902 = 119887119888 + 119886119889Then (119877 + ∘) is a strongly inversivemultiplicative hyper-

field which is polynomially structured

Definition 26 A local (C-local) multiplicative 119867V-ring is acommutative multiplicative119867V-ring with an 119894-set which hasa unique maximal119867V-ideal (respC-ideal)

Proposition 27 Let 119865 be a polynomially structured stronglyinversive multiplicative hyperfieldThen (i) 119865[119909] is a principalC-ideal strong 119867V-domain (ii) for any 119891 isin 119865[119909] 0

119865[119909]

there exist a hyperinvertible element 119892 isin 119865[119909] and 119898 isin

N cup 0 such that 119891 isin (119890119909119898) lowast 119892 where 119890 is a hyperidentity

8 Algebra

in 119865 and (iii) 119865[119909] is a local as well as aC-local multiplicative119867V-ring

Proof (i) Here 119865 is a strongly inversive multiplicative hyper-field So 119865 is a strong hyperdomain Thus by Proposition 13the multiplicative 119867V-ring 119865[119909] is a strong 119867V-domainAgain by Proposition 22 119865[119909] is a principal C-ideal 119867V-ring (since 119865 is an inversive multiplicative hyperfield) So byDefinition 21 119865[119909] is a principalC -ideal strong119867V-domain

(ii) Let 119891 isin 119865[119909] 0119865[119909] and 119869 = ⟨119891⟩ Then 119869 =H

119903(119891)

So for any ℎ isin 119869lowast there exist119860119894= [119891lowast119891

1198941lowast1198911198942lowastsdot sdot sdotlowast119891

119894119898119894]119903isin

H119903(119891) (119894 = 1 2 119896 119896119898

119894isin N) such that ℎ isin sum119896

119894=1119860119894

Thus ℎ = sum119896119894=1119905119894for some 119905

119894isin 119860119894 Since 119865[119909] is a strong119867V-

domain (by Proposition 13) with absorbing zero so 0119865[119909]isin

119860119894= [119891 lowast 119891

1198941lowast 1198911198942lowast sdot sdot sdot lowast 119891

119894119898119894]119903rArr [119891 lowast 119891

1198941lowast 1198911198942lowast

sdot sdot sdot lowast 119891119894119898119894]119903= 0119865[119909] Hence without any loss of generality

(since 0119865[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119865[119909])) we may

assume that 0119865[119909]notin 119860119894for any 119894 (since ℎ = 0

119865[119909]) Then

for any 119894 = 1 2 119896 and 119895 = 1 2 119898119894 119891119894119895= 0119865[119909]

So by Proposition 11(ii) ord 119905

119894= ord(119860

119894) ge ord119891 and so

ord ℎ = minord 119905119894 119894 ge ord119891 Thus ord119869lowast ge ord119891 So

by Proposition 17 119891 is a nonzero polynomial in the C-idealC(119869) such that ord119891 le ord ℎ for all ℎ isin C(119869)lowast So there existsan invertible element 119892 isin 119865[119909] and an integer119898 ge 0 such that119891 isin (119890119909

119898) lowast 119892 where 119890 is a hyperidentity in 119865 (see proof of

Proposition 22)(iii) For any integer119898 ge 1 since 119890119909119898 isin (119890119909) lowast (119890119909119898minus1) so

⟨119890119909119898⟩ sube ⟨119890119909⟩ Again for any119867V-ideal 119868 of the multiplicative

119867V-ring119865[119909] if119891 isin 119868lowast is such that ord119891 le ord119892 for all119892 isin 119868lowast

then 119868 sube ⟨119890119909119898⟩ where 119898 = ord119891 So ⟨119890119909⟩ is the uniquemaximal119867V-ideal in 119865[119909] Thus 119865[119909] is a local multiplicative119867V-ring Now 119890119909 isin C

1(as defined in Proposition 15) So

ord(119890119909) le ord119891 for any 119891 isin Clowast1 Thus C

1being a C-ideal

of 119865[119909] we have C1= ⟨119890119909⟩ (see proof of Proposition 22)

Hence 119865[119909] is aC-local multiplicative119867V-ring

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vougiouklis ldquoThe fundamental relation in hyperrings Thegeneral hyperfieldrdquo in Algebraic Hyperstructures and Appli-cations (Xanthi1990) pp 203ndash211 World Science PublisherTeaneck NJ USA 1991

[2] M R Darafsheh and B Davvaz ldquo119867V-ring of fractionsrdquo ItalianJournal of Pure and Applied Mathematics no 5 pp 25ndash34 1999

[3] B Davvaz and V Leoreanu-FoteaHyperring Theory and Appli-cations International Academic Press Palm Harbor Fla USA2007

[4] S Spartalis A Dramalides and T Vougiouklis ldquoOn 119867V-groupringsrdquoAlgebras Groups and Geometries vol 15 no 1 pp 47ndash541998

[5] T Vougiouklis ldquo119867V-groups defined on the same setrdquo DiscreteMathematics vol 155 no 1ndash3 pp 259ndash265 1996

[6] R Procesi and R Rota ldquoMultiplicative H119907-rings and com-

plementary hyperstructuresrdquo Journal of Discrete MathematicalSciences amp Cryptography vol 11 no 4 pp 447ndash456 2008

[7] S Spartalis ldquoOn the number of 119867Vminus119903119894119899119892119904 with 119875-hyperopera-tionsrdquoDiscrete Mathematics vol 155 no 1ndash3 pp 225ndash231 1996

[8] P Corsini Prolegomena of Hypergroup Theory Rivista di Mate-matica Pura ed Applicata Aviani Tricesimo Italy 1993

[9] M K Sen and U Dasgupta ldquoSome aspects of119866119867-ringsrdquoAnnals

of the Alexandru Ioan Cuza UniversitymdashMathematics vol 56no 2 pp 253ndash272 2010

[10] C Namnak N Triphop and Y Kemprasit ldquoHomomorphismsof somemultiplicative hyperringsrdquo Set-ValuedMathematics andApplications vol 1 no 2 pp 145ndash152 2008

[11] D M Olson and V K Ward ldquoA note on multiplicativehyperringrdquo Italian Journal of Pure andAppliedMathematics vol1 pp 77ndash84 1997

[12] R Procesi and R Rota ldquoComplementary multiplicative hyper-ringsrdquoDiscreteMathematics vol 308 no 2-3 pp 188ndash191 2008

[13] R Procesi and R Rota ldquoOn some classes of hyperstructuresrdquoDiscrete Mathematics vol 208-209 pp 485ndash497 1999

[14] R Rota ldquoStrongly distributive multiplicative hyperringsrdquo Jour-nal of Geometry vol 39 no 1-2 pp 130ndash138 1990

[15] R Rota ldquoSugli Iperanelli Moltiplicativirdquo Rendiconti di Matem-atica Series VII vol 2 no 4 pp 711ndash724 1982

[16] R Procesi Ciampi andR Rota ldquoPolynomials overmultiplicativehyperringsrdquo Journal of Discrete Mathematical Sciences amp Cryp-tography vol 6 no 2-3 pp 217ndash225 2003

[17] T Vougiouklis Hyperstructures and Their RepresentationsMonographs in Mathematics Hardonic Press 1994

[18] U Dasgupta ldquoOn prime and primary hyperideals of a multi-plicative hyperringrdquoAnalele Stiintifice ale Universitatii Al I Cuzadin IasimdashMatematica vol 58 no 1 pp 19ndash37 2012

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Algebra 5

119892(119909) = sum 119887119896119909119896 where 119887

119898= 0119877 119887119896= 0119877for 119896 lt 119898

So 0119877notin 119886119901119887119898(since 119877 is a strong hyperdomain) and also

119886119901119887119898= sum119894+119895=119901+119898

119886119894119887119895(since 119886

119896= 0119877for 119896 lt 119901 and 119887

119896= 0119877

for 119896 lt 119898) Thus for any ℎ(119909) = sum 119888119896119909119896isin 119891(119909) lowast 119892(119909)

119888119901+119898

= 0119877(since 119888

119901+119898isin sum119894+119895=119901+119898

119886119894119887119895= 119886119901119887119898) and 119888

119896= 0119877

for all 119896 with 0 le 119896 lt 119901 + 119898 Thus ordℎ(119909) = 119901 + 119898 forany ℎ(119909) isin 119891(119909) lowast 119892(119909) and so ord(119891(119909) lowast 119892(119909)) = 119901 + 119898Hence ord119891(119909)+ord119892(119909) = ord(119891(119909)lowast119892(119909)) = ord ℎ(119909) forall ℎ(119909) isin 119891(119909) lowast 119892(119909) So the implication is true for 119899 = 2(noting that [119891(119909) lowast 119892(119909)]

119903= 119891(119909) lowast 119892(119909)) Suppose that for

some integer 119896 ge 2 the implication holds true for each valueof 119899 ranging from 2 to 119896minus1 and take any119891

119894(119909) isin 119877[119909]0

119877[119909]

for 119894 = 1 2 119896 Thensum119896minus1119894=1

ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast

sdot sdot sdotlowast119891119896minus1(119909)]119903) = ord ℎ(119909) for all ℎ(119909) isin [119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast

119891119896minus1(119909)]119903 Now let 119892(119909) isin [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903be

arbitraryThen 119892(119909) isin ℎ(119909) lowast119891119896(119909) for some ℎ(119909) isin [119891

1(119909) lowast

1198912(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903 So ord119892(119909) = ord(ℎ(119909) lowast 119891

119896(119909)) =

ord ℎ(119909) + ord119891119896(119909) = sum

119896minus1

119894=1ord119891119894(119909) + ord119891

119896(119909)(since ℎ(119909) isin

[1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896minus1(119909)]119903) = sum

119896

119894=1ord119891119894(119909) Thus

sum119896

119894=1ord119891119894(119909) = ord([119891

1(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119896(119909)]119903) = ord119892(119909)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119896(119909)]119903

Hence by strong induction the implication follows forany 119899 isin N with 119899 ge 2

Corollary 12 If 119877 is a strong hyperdomain then the implica-tion

119891119894 (119909) isin 119877 [119909] 0119877[119909] 997904rArr

119899

sum

119894=1

ord 119891119894 (119909)

= ord ([1198911 (119909) lowast 1198912 (119909) lowast sdot sdot sdot lowast 119891119899 (119909)]119897

) = ord 119892 (119909) (6)

for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897holds true for any

119899 isin N with 119899 ge 2

Proof Since themultiplicative hyperring119877 is a hyperdomainit is commutative and so 119877[119909] is also a commutative multi-plicative 119867V-ring Hence for any 119891119894(119909) isin 119877[119909] 0119877[119909] (119894 =1 2 119896) we have that [119891

1(119909)lowast119891

2(119909)lowast119891

3(119909)lowastsdot sdot sdotlowast119891

119896(119909)]119897=

[119891119896(119909) lowast 119891

119896minus1(119909) lowast sdot sdot sdot lowast 119891

2(119909) lowast 119891

1(119909)]119903 Hence the assertion

follows straight from Proposition 11

Proposition 13 If the multiplicative hyperring 119877 is a stronghyperdomain the multiplicative 119867V-ring 119877[119909] is a strong 119867V-domain

Proof 119877 being a strong hyperdomain is a commutativemultiplicative hyperring Thus 119877[119909] is a commutative mul-tiplicative 119867V-ring Again since 0119877 is absorbing in 119877 0

119877[119909]

is also absorbing in 119877[119909] Thus we take 119891(119909) 119892(119909) isin 119877[119909] 0119877[119909]Then by Proposition 11 ord(119891(119909)lowast119892(119909)) = ord119891(119909)+

ord119892(119909) gt 0 Hence 0119877[119909]notin 119891(119909) lowast 119892(119909) So 119877[119909] is a strong

119867V-domain

4 C-Ideals in 119877[119909]

A subgroup 119868 of the group (119878 +) is called a left (resp right)119867V-ideal of a multiplicative 119867V-ring (119878 + ∘) if for any 119904 isin 119878

and 119886 isin 119868 119904 ∘ 119886 sube 119868 (resp 119886 ∘ 119904 sube 119868) 119868 is an119867V-ideal of 119878 if itis both a left and a right119867V-ideal of 119878

We call an 119867V-ideal of a multiplicative 119867V-ring 119878 simplya hyperideal when 119878 is a multiplicative hyperringThe notionof a typical hyperideal in a multiplicative hyperring calledC-ideal is introduced in [18] to study prime and primaryhyperideals of multiplicative hyperrings A hyperideal 119868 of amultiplicative hyperring 119878 is a C-ideal if for any 119860 isin C 119860 cap119868 = 120601 rArr 119860 sube 119868 where C = 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899 119904119894isin 119878

119899 isin N sube 119875lowast(119878) Following is the definition of aC-ideal in anarbitrary multiplicative119867V-ring

Definition 14 A left (resp right) 119867V-ideal 119868 of a multiplica-tive 119867V-ring (119878 + ∘) is called a left (resp right) C-ideal iffor any type 119897 hyperproduct [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897(resp type 119903

hyperproduct [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903) of elements 119904

119894isin 119878 we have

that [1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897cap 119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119897sube 119868 (resp

[1199041∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903cap119868 = 120601 rArr [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903sube 119868) An119867V-ideal

119868 of a multiplicative119867V-ring 119878 is called a C-ideal if it is a leftas well as a rightC-ideal in 119878

We writeC-ideal(119878) (respC-ideallowast(119878)) to denote the setof all (resp proper)C-ideals of a multiplicative119867V-ring 119878 Ina commutative multiplicative119867V-ring 119878 every left C-ideal isa right C-ideal and vice versa since commutativity impliesthe equality [119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119899]119903= [119904119899∘ 119904119899minus1∘ sdot sdot sdot ∘ 119904

1]119897 for any

119904119894isin 119877 Note that in amultiplicative hyperring (even if it is not

commutative) a hyperideal is a leftC-ideal (as an119867V-ideal) ifand only if it is a rightC-ideal

Proposition 15 If 119877 is a strong hyperdomain then the setC-119868119889119890119886119897lowast(119877[119909]) is nonempty

Proof For each 119899 isin N cup 0 let C119899= 119891(119909) isin 119877[119909]

ord119891(119909) ge 119899 cup 0119877[119909] Then C

119899is a subgroup of the group

(119877[119909] +) (since for any 119891(119909) 119892(119909) isin 119877[119909] ord(minus119891(119909)) =ord119891(119909) and ord(119891(119909) + 119892(119909)) = minord119891(119909) ord119892(119909)when 119891(119909)+119892(119909) = 0

119877[119909]) Let 119892(119909) isin 119877[119909] and 119891(119909) isin C

119899 If

119892(119909) = 0119877[119909]

or 119891(119909) = 0119877[119909]

then 119892(119909) lowast119891(119909) = 0119877[119909] sube C119899

(since 0119877[119909]

is absorbing in 119877[119909]) So let 119892(119909) = 0119877[119909]

and119891(119909) = 0

119877[119909] Then by Proposition 11(ii) for all ℎ(119909) isin 119892(119909)lowast

119891(119909) ord ℎ(119909) = ord(119892(119909) lowast 119891(119909)) = ord119891(119909) + ord119892(119909) gt119899 Thus 119892(119909) lowast 119891(119909) sube C

119899 Hence C

119899is an 119867V-ideal of

119877[119909] (since 119877[119909] is a commutative multiplicative 119867V-ring)Let [119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897be a type-119897 hyperproduct of

elements of the119867V-ring 119877[119909] such that [1198911(119909) lowast 1198912(119909) lowast sdot sdot sdot lowast119891119899(119909)]119897sube C119899 Then 119891

119894(119909) = 0

119877[119909]for each 119894 (since 0

119877[119909]isin C119899

is absorbing in119877[119909])Thus ord([1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897) =

ord119892(119909) for all 119892(119909) isin [1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 Now

since [1198911(119909)lowast119891

2(119909)lowast sdot sdot sdotlowast119891

119899(119909)]119897sube C119899 so there exists ℎ(119909) isin

[1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897such that ℎ(119909) notin C

119899Then for any

119892(119909) isin [1198911(119909) lowast119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909)]119897 ord119892(119909) = ord ℎ(119909) lt 119899

Hence by definition ofC119899 [1198911(119909)lowast119891

2(119909)lowastsdot sdot sdotlowast119891

119899(119909)]119897capC119899=

120601 and soC119899is a leftC-ideal and thus aC-ideal of 119877[119909] (since

119877[119909] is commutative) Hence C119899isin C-Ideallowast(119877[119909]) for all

119899 gt 0Since the intersection of left C-ideals of a multiplicative

119867V-ring 119878 is also a left C-ideal of 119878 and 119878 is itself a left C-ideal so the smallest left C-ideal containing a subset 119860 of 119878

6 Algebra

being naturally called the left C-ideal generated by 119860 existsand is in fact the intersection of all left C-ideals containing119860 The left C-ideal generated by a left 119867V-ideal 119868 of 119878 iscalled the leftC-closure of 119868 and is denoted byC

119897(119868) Clearly

C119897(C119897(119868)) = C

119897(119868) for a left 119867V-ideal 119868 C119903(119868) and C(119868)

respectively denote the right C-closure of a right 119867V-ideal 119868and theC-closure of an119867V-ideal 119868 of 119878 For an119867V-ideal 119868 of acommutativemultiplicative119867V-ring 119878C119897(119868) = C119903(119868) = C(119868)The following lemma presents a description of the set C

119897(119868)

for a left119867V-ideal 119868 of amultiplicative119867V-ring 119878The setC119903(119868)

for a right119867V-ideal 119868 can be described dually

Lemma 16 Let 119878 be a multiplicative 119867V-ring with an 119894-set Eand let L denote the set of all left 119867V-ideals of 119878 119865 119875(119878) rarr119875(119878) is a mapping defined by

119865 (119860) = ⋃

119899

sum

119894=1

[1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897 119899 119898

119894isin N 119903

1198941isin 119878

119904119886119905119894119904119891119910119894119899119892 [1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897cap 119860 = 120601

(7)

for any 119860 isin 119875(119878) Then for any 119868 1198681 1198682isin L we have the

following (i) 119865(119868) isin L (ii) 119868 sube 119865(119868) (iii) 1198681sube 1198682rArr 119865(119868

1) sube

119865(1198682) (iv) 119865(119868) = 119868 if and only if 119868 is a left C-ideal of 119878 and

(v) left C-closure of 119868 is C119897(119868) = ⋃

infin

119896=1119865119896(119868) where for any

119896 isin N 119865119896 denotes the (119896 minus 1)-times mapping composition of 119865

Proof All the assertions made in this lemma can be estab-lished by adopting the arguments that are applied in provingwell-known analogous results on ldquocomplete closure of a setrdquoin semihypergroup theory (see [3 8])

Proposition 17 Let 119877 be a strong hyperdomain with an 119894-setThen for any 119867V-ideal 119868 of the multiplicative 119867V-ring 119877[119909]ord(C(I)lowast) ge ordIlowast where for any 119867V-ideal 119869 of 119877[119909] 119869lowast =119869 0119877[119909]

Proof Since 119877 is a (strong) hyperdomain the multiplicative119867V-ring is commutative So for any 119867V-ideal 119868 of 119877[119909]C(119868) = C

119897(119868) = ⋃

infin

119896=1119865119896(119868) (by Lemma 16(v)) For any

119899 isin N 119891(119909) isin 119865119899(119868)lowast rArr 119891(119909) isin sum119898

119894=1119860119894 for some

type-119897 hyperproducts 119860119894= [1198911198941(119909) lowast 119891

1198942(119909) lowast sdot sdot sdot lowast 119891

119894119898119894(119909)]119897

of elements of 119877[119909] satisfying 119860119894cap 119865119899minus1(119868) = 120601 Since

here 119877[119909] is a strong 119867V-domain (by Proposition 13) withabsorbing zero we may assume that 0

119877[119909]notin 119860119894for each 119894

(since 0119877[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119877[119909])) Then for

each 119894 (= 1 2 119898) and 119895119894(= 1 2 119898

119894) 119891119894119895119894

(119909) = 0119877[119909]

Thus by Corollary 12 ord119860

119894= ord119892

119894(119909) for any 119892

119894(119909) isin 119860

119894

Now since for each 119894 119860119894cap 119865119899minus1(119868) = 120601 so there exists

119892119894(119909) isin 119860

119894such that 119892

119894(119909) isin 119865

119899minus1(119868)lowast Then for any 119891(119909) isin

119865119899(119868)lowast ord119891(119909) ge sum119898

119894=1ord119860119894= minord119892

119894(119909) 119894 ge

ord(119865119899minus1(119868)lowast) Consequently ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast)for any 119899 isin N Now let 119891(119909) isin C(119868)

lowast be arbitrary Then119891(119909) isin 119865

119899(119868)lowast for some 119899 isin N So we have that ord119891(119909) ge

ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast) ge sdot sdot sdot ge ord(119865(119868)lowast) ge ord119868lowastThus ord(C(119868)lowast) ge ord119868lowast

5 Polynomials over Multiplicative Hyperfield

A nonzero element 119886 of a multiplicative119867V-ring (119878 + ∘) withan 119894-set E = 119890

1 1198902 119890

119899 is referred to be an E-invertible

element (or anE-unit) of 119878 if for each 119894 = 1 2 119899 (119899 isin N)there exist 119860

119894119895isin H119897(119886) = [119904

119901∘ 119904119901minus1∘ sdot sdot sdot ∘ 119904

1∘ 119886]119897 119904119894isin 119878

119901 isin N and 119861119894119896isin H119903(119886) = [119886 ∘ 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119902]119903 119904119894isin 119878

119902 isin N such that 119890119894isin (sum119898

119895=1119860119894119895) cap (sum

119897

119896=1119861119894119896) An element

119886 ( =0119878) of the multiplicative119867V-ring 119878with a hyperidentity 119890

is said to be 119890-hyperinvertible (or an 119890-hyperunit) in 119878 if thereexist 119860 isinH

119897(119886) and 119861 isinH

119903(119886) such that 119890 isin 119860 cap 119861

If 119886 isin 119878 is an E-unit (resp 119890-hyperunit) in a multiplica-tive 119867V-ring (119878 + ∘) with two 119894-sets E and E1015840 (resp withtwo hyperidentities 119890 and 1198901015840) then one can easily verify that119886 is also an E1015840-unit (resp an 1198901015840-hyperunit) in 119878 We thuscall an E-unit (resp an 119890-hyperunit) of a multiplicative 119867V-ring 119878 simply a unit (resp a hyperunit) Denote by 119880(119878)and 119880

ℎ(119878) respectively the sets of units and hyperunits of a

multiplicative119867V-ring 119878An 119867V-ideal 119868 ( = 119878) of a multiplicative 119867V-ring 119878 is

maximal in 119878 if for any 119867V-ideal 119869 of 119878 119868 ⊊ 119869 sube 119878 rArr 119869 = 119878For a commutative multiplicative119867V-ring 119878with an 119894-set thisis immediate to observe that 119886 isin 119880(119878) if and only if 119886 notin 119872for any maximal119867V-ideal119872 of 119878

Proposition 18 Let themultiplicative hyperring119877 be commu-tative and contain a hyperidentity 119890 Then for a polynomial119891(119909) = sum119886

119896119909119896isin 119877[119909] 119891(119909) isin 119880

ℎ(119877[119909]) if and only if

1198860isin 119880ℎ(119877)

Proof Since 119890 is a hyperidentity in the multiplicative hyper-ring 119877 1198901199090 isin 119877[119909] is a hyperidentity in the multiplicative119867V-ring 119877[119909] Now let 119891(119909) = sum119886

119896119909119896isin 119880ℎ(119877[119909]) Then there

exist 119892119894(119909) = sum 119887

119894119896119909119896isin 119877[119909] (119894 = 1 2 119899 119899 isin N) such that

1198901199090isin [119891(119909) lowast 119892

1(119909) lowast sdot sdot sdot lowast 119892

119899(119909)]119903 So 119890 isin 119886

0(1198871011988720 1198871198990)

and thus there exists 119887 isin 1198871011988720 1198871198990sube 119877 such that 119890 isin 119886

0119887

whence 1198860isin 119880ℎ(119877) (since 119877 is commutative)

Conversely let 1198860be hyperinvertible in 119877 with respect to

the hyperidentity 119890 Then there exists 1198870isin 119877 such that 119890 isin

11988601198870 We assert that there is a sequence 119887

119896119896isinN in 119877 whose

119899th term 119887119899(119899 isin N) is inductively defined so as to satisfy the

relation that

0119877isin 1198860119887119899+ 1198861119887119899minus1+ 1198862119887119899minus2+ sdot sdot sdot + 119886

1198991198870sdot sdot sdot () (8)

In fact we see that 0119877isin 01198771198870= (minus119886

1+ 1198861)1198870sube (119890(minus119886

1) +

1198861)1198870sube ((119886

01198870)(minus1198861) + 1198861)1198870sube (119886

01198870)(minus1198861)1198870+ 11988611198870=

1198860((minus1198870)11988611198870) + 11988611198870rArr there exist 119887

1isin (minus119887

0)11988611198870such that

0119877isin 11988601198871+ 11988611198870(the relation for 119899 = 1) Suppose for

some 119898 ge 1 the terms 119887119896(1 le 119896 le 119898 minus 1) are defined

in such a way that each 119887119896(1 le 119896 le 119898 minus 1) satisfies the

relation () for 119896 Then (1198861119887119898minus1+1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+1198861198981198870)

is defined to be a nonempty subset of 119877 Let 119905 isin 1198861119887119898minus1+

1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870 Then 0

119877= minus119905 + 119905 isin 119890(minus119905) + 119905 sube

(11988601198870)(minus119905) + 119905 sube 119886

0(minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+

1198861198981198870) + (1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) rArr there exists

119887119898isin (minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) such that

0119877isin 1198860119887119898+ 1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

1198981198870(the relation

for 119896 = 119898) Hence the assertion is true for all 119899 isin N Thus

Algebra 7

consider the polynomial 119892(119909) = sum 119887119896119909119896isin 119877[119909] Then from

the definition of lowast 119891(119909) lowast 119892(119909) = sum 119888119896119909119896 1198880isin 11988601198870and

119888119896isin 1198860119887119896+ 1198861119887119896minus1+ 1198862119887119896minus2+ sdot sdot sdot + 119886

1198961198870for 119896 isin N Thus

1198901199090isin 119891(119909) lowast 119892(119909) (due to relation and since 119890 isin 119886

01198870) So

119891(119909) is hyperinvertible in 119877[119909] that is 119891(119909) isin 119880ℎ(119877[119909])

The (left right) 119867V-ideal of a multiplicative 119867V-ring 119878generated by 119860 isin 119875(119878) is the smallest (resp left right) 119867V-ideal of 119878 containing119860which is denoted by (resp ⟨119860⟩

119897 ⟨119860⟩119903)

⟨119860⟩ The principal (left right) 119867V-ideal of the multiplicative119867V-ring 119878 generated by an element 119886 of 119878 denoted by (resp⟨119886⟩119897 ⟨119886⟩119903) ⟨119886⟩ is the (resp left right)119867V-ideal (resp ⟨119886⟩119897

⟨119886⟩119903) ⟨119886⟩ of the multiplicative119867V-ring 119878

If the multiplicative 119867V-ring 119878 has an 119894-set then for any119886 isin 119878 ⟨119886⟩

119897= H119897(119886) = ⋃sum

119899

119894=1119860119894 119860119894isin H119897(119886) 119899 isin N and

⟨119886⟩119903=H119903(119886) = ⋃sum

119899

119894=1119860119894 119860119894isinH119903(119886) 119899 isin N

Definition 19 A commutative multiplicative 119867V-ring 119878 withan 119894-setE is called amultiplicative119867V-field (resp an inversivemultiplicative 119867V-field) if 119878 ⟨0

119878⟩ sube 119880(119878) (resp 119878

⟨0119878⟩ sube 119880

ℎ(119878)) If a multiplicative119867V-field (resp an inversive

multiplicative 119867V-field) is a multiplicative hyperring thenwe call it a multiplicative hyperfield (resp an inversivemultiplicative hyperfield)

Proposition 20 Let 119865 be a polynomially structured inversivemultiplicative hyperfield Then a polynomial 119891(119909) = sum119886

119896119909119896isin

119865[119909]0119865[119909] is hyperinvertible in119865[119909] if and only if ord 119891 = 0

Proof If 119865 is an inversive multiplicative hyperfield then byProposition 18 any polynomial119891(119909) = sum119886

119896119909119896isin 119865[119909]0

119865[119909]

is hyperinvertible in 119865[119909] if and only if 1198860= 0119865 Hence the

result follows

Definition 21 A commutative multiplicative 119867V-ring 119878 iscalled a principal C-ideal 119867V-ring if every C-ideal of119878 is a principal 119867V-ideal A principal C-ideal 119867V-ringwhich is a (strong) 119867V-domain is called a principal C-ideal(strong) 119867V-domain

Proposition 22 Let 119865 be a polynomially structured inversivemultiplicative hyperfieldThen themultiplicative119867V-ring119865[119909]of polynomials over 119865 is a principalC-ideal119867V-ring

Proof Let 119890 be a hyperidentity of the inversive multiplicativehyperfield 119865 Then the polynomial 1198901199090 is a hyperidentity in119865[119909] and thus 119865[119909] = ⟨1198901199090⟩ (since 119865 is commutative) Solet 119868 be any proper C-ideal of 119865[119909] If 119868 = 0

119865[119909] then 119868 is

the principal hyperideal ⟨0119865[119909]⟩ (since for119865 having absorbing

zero 0119865[119909]

is absorbing in 119865[119909]) Suppose that 119868 = 0119865[119909]

Then take a nonzero polynomial 119891(119909) = sum119886119896119909119896isin 119868 such

that ord119891(119909) le ord 119905(119909) for any 119905(119909) isin 119868lowast Let us writeord119891(119909) = 119898 Then 119886

119898= 0119865and 119886119896= 0119865for any 0 le 119896 lt 119898

Consider then the polynomial 119892(119909) = sumlowast119896isinN0119887119896119909119896isin 119865[119909]

where 119887119896= 119886119898+119896

Then clearly 119891(119909) isin (119890119909119898) lowast 119892(119909) Alsoby Proposition 18 119892(119909) is hyperinvertible in 119865[119909] Thus thereexists ℎ(119909) isin 119865[119909] such that 1198901199090 isin 119892(119909) lowast ℎ(119909) Now119891(119909) isin (119890119909

119898) lowast 119892(119909) rArr 119891(119909) lowast ℎ(119909) sube [119890119909

119898lowast 119892(119909) lowast ℎ(119909)]

119903

Again 119891(119909) lowast ℎ(119909) sube 119868 (since 119891(119909) isin 119868) So [119890119909119898 lowast 119892(119909) lowastℎ(119909)]119903cap 119868 = 120601 Hence [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868 (since

119868 is a C-ideal and every C-ideal is a right C-ideal) Now[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap [119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119903= 120601 whence

[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap 119868 = 120601 (since [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868)

Consequently [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]119897sube 119868 (since 119868 is a left C-

ideal) Then 119890119909119898 isin (119890119909119898) lowast (1198901199090) sube 119890119909119898 lowast (119892(119909) lowast ℎ(119909)) =[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897sube 119868 Thus ⟨119890119909119898⟩ sube 119868

Now let ℎ(119909) = sum 119887119896119909119896isin 119868lowast be arbitrary Suppose that

ord ℎ = 119899 Then 119887119899= 0119865and 119887119896= 0119865for any 0 le 119896 lt 119899

By choice of 119891(119909) from 119868 here 119898 le 119899 So one can definea polynomial 119892(119909) = sum 119888

119896119909119896isin 119865[119909] where 119888

119896= 0119865for all

0 le 119896 le 119899 minus 119898 minus 1 and 119888119896= 119887119898+119896

for all 119896 ge 119899 minus 119898 Clearlythen ℎ(119909) isin 119890119909119898 lowast 119892(119909) sube ⟨119890119909119898⟩ whence 119868 sube ⟨119890119909119898⟩ Thus119868 = ⟨119890119909

119898⟩

Remark 23 In a ring an invertible element can never be adivisor of zero This not true in general for a multiplicativehyperring In fact on the commutative group of integers(Z +) if we define a hyperoperation ∘ by stating that 119909 ∘ 119910 =0 119909119910 for all 119909 119910 isin Z then (Z + ∘) is a commutative mul-tiplicative hyperring with a hyperidentity 1 Every nonzeroelement of (Z + ∘) is a zero divisor and 1 minus1 isin Z are inparticular hyperunits of (Z + ∘) To get a parity with thering theory in this regard we perceive the notion of stronghyperinvertibility of an element of amultiplicative hyperring

Definition 24 A hyperinvertible element of a multiplicativehyperring with a hyperidentity is said to be strongly hyperin-vertible (or a strong hyperunit) if it is not a zero divisor inthat multiplicative hyperring A commutative multiplicativehyperring with absorbing zero and a hyperidentity 119890 is said tobe a strongly inversive multiplicative hyperfield if each of itsnonzero elements is a strong hyperunit

Example 25 Let 119877 = 119886radic2 + 119887radic3 119886 119887 isin Q and 119860 =radic2radic3 Then with respect to usual addition + of reals(119877 +) is a commutative group with identity 0 On 119877 ∘ is ahyperoperation defined by

(119886radic2 + 119887radic3) ∘ (119888radic2 + 119889radic3)

= (119886radic2 + 119887radic3) sdot 119905 sdot (119888radic2 + 119889radic3) 119905 isin 119860

= 119901radic2 + 2119902radic3 3119902radic2 + 119901radic3

(9)

where 119901 = 2119886119888 + 3119887119889 and 119902 = 119887119888 + 119886119889Then (119877 + ∘) is a strongly inversivemultiplicative hyper-

field which is polynomially structured

Definition 26 A local (C-local) multiplicative 119867V-ring is acommutative multiplicative119867V-ring with an 119894-set which hasa unique maximal119867V-ideal (respC-ideal)

Proposition 27 Let 119865 be a polynomially structured stronglyinversive multiplicative hyperfieldThen (i) 119865[119909] is a principalC-ideal strong 119867V-domain (ii) for any 119891 isin 119865[119909] 0

119865[119909]

there exist a hyperinvertible element 119892 isin 119865[119909] and 119898 isin

N cup 0 such that 119891 isin (119890119909119898) lowast 119892 where 119890 is a hyperidentity

8 Algebra

in 119865 and (iii) 119865[119909] is a local as well as aC-local multiplicative119867V-ring

Proof (i) Here 119865 is a strongly inversive multiplicative hyper-field So 119865 is a strong hyperdomain Thus by Proposition 13the multiplicative 119867V-ring 119865[119909] is a strong 119867V-domainAgain by Proposition 22 119865[119909] is a principal C-ideal 119867V-ring (since 119865 is an inversive multiplicative hyperfield) So byDefinition 21 119865[119909] is a principalC -ideal strong119867V-domain

(ii) Let 119891 isin 119865[119909] 0119865[119909] and 119869 = ⟨119891⟩ Then 119869 =H

119903(119891)

So for any ℎ isin 119869lowast there exist119860119894= [119891lowast119891

1198941lowast1198911198942lowastsdot sdot sdotlowast119891

119894119898119894]119903isin

H119903(119891) (119894 = 1 2 119896 119896119898

119894isin N) such that ℎ isin sum119896

119894=1119860119894

Thus ℎ = sum119896119894=1119905119894for some 119905

119894isin 119860119894 Since 119865[119909] is a strong119867V-

domain (by Proposition 13) with absorbing zero so 0119865[119909]isin

119860119894= [119891 lowast 119891

1198941lowast 1198911198942lowast sdot sdot sdot lowast 119891

119894119898119894]119903rArr [119891 lowast 119891

1198941lowast 1198911198942lowast

sdot sdot sdot lowast 119891119894119898119894]119903= 0119865[119909] Hence without any loss of generality

(since 0119865[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119865[119909])) we may

assume that 0119865[119909]notin 119860119894for any 119894 (since ℎ = 0

119865[119909]) Then

for any 119894 = 1 2 119896 and 119895 = 1 2 119898119894 119891119894119895= 0119865[119909]

So by Proposition 11(ii) ord 119905

119894= ord(119860

119894) ge ord119891 and so

ord ℎ = minord 119905119894 119894 ge ord119891 Thus ord119869lowast ge ord119891 So

by Proposition 17 119891 is a nonzero polynomial in the C-idealC(119869) such that ord119891 le ord ℎ for all ℎ isin C(119869)lowast So there existsan invertible element 119892 isin 119865[119909] and an integer119898 ge 0 such that119891 isin (119890119909

119898) lowast 119892 where 119890 is a hyperidentity in 119865 (see proof of

Proposition 22)(iii) For any integer119898 ge 1 since 119890119909119898 isin (119890119909) lowast (119890119909119898minus1) so

⟨119890119909119898⟩ sube ⟨119890119909⟩ Again for any119867V-ideal 119868 of the multiplicative

119867V-ring119865[119909] if119891 isin 119868lowast is such that ord119891 le ord119892 for all119892 isin 119868lowast

then 119868 sube ⟨119890119909119898⟩ where 119898 = ord119891 So ⟨119890119909⟩ is the uniquemaximal119867V-ideal in 119865[119909] Thus 119865[119909] is a local multiplicative119867V-ring Now 119890119909 isin C

1(as defined in Proposition 15) So

ord(119890119909) le ord119891 for any 119891 isin Clowast1 Thus C

1being a C-ideal

of 119865[119909] we have C1= ⟨119890119909⟩ (see proof of Proposition 22)

Hence 119865[119909] is aC-local multiplicative119867V-ring

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vougiouklis ldquoThe fundamental relation in hyperrings Thegeneral hyperfieldrdquo in Algebraic Hyperstructures and Appli-cations (Xanthi1990) pp 203ndash211 World Science PublisherTeaneck NJ USA 1991

[2] M R Darafsheh and B Davvaz ldquo119867V-ring of fractionsrdquo ItalianJournal of Pure and Applied Mathematics no 5 pp 25ndash34 1999

[3] B Davvaz and V Leoreanu-FoteaHyperring Theory and Appli-cations International Academic Press Palm Harbor Fla USA2007

[4] S Spartalis A Dramalides and T Vougiouklis ldquoOn 119867V-groupringsrdquoAlgebras Groups and Geometries vol 15 no 1 pp 47ndash541998

[5] T Vougiouklis ldquo119867V-groups defined on the same setrdquo DiscreteMathematics vol 155 no 1ndash3 pp 259ndash265 1996

[6] R Procesi and R Rota ldquoMultiplicative H119907-rings and com-

plementary hyperstructuresrdquo Journal of Discrete MathematicalSciences amp Cryptography vol 11 no 4 pp 447ndash456 2008

[7] S Spartalis ldquoOn the number of 119867Vminus119903119894119899119892119904 with 119875-hyperopera-tionsrdquoDiscrete Mathematics vol 155 no 1ndash3 pp 225ndash231 1996

[8] P Corsini Prolegomena of Hypergroup Theory Rivista di Mate-matica Pura ed Applicata Aviani Tricesimo Italy 1993

[9] M K Sen and U Dasgupta ldquoSome aspects of119866119867-ringsrdquoAnnals

of the Alexandru Ioan Cuza UniversitymdashMathematics vol 56no 2 pp 253ndash272 2010

[10] C Namnak N Triphop and Y Kemprasit ldquoHomomorphismsof somemultiplicative hyperringsrdquo Set-ValuedMathematics andApplications vol 1 no 2 pp 145ndash152 2008

[11] D M Olson and V K Ward ldquoA note on multiplicativehyperringrdquo Italian Journal of Pure andAppliedMathematics vol1 pp 77ndash84 1997

[12] R Procesi and R Rota ldquoComplementary multiplicative hyper-ringsrdquoDiscreteMathematics vol 308 no 2-3 pp 188ndash191 2008

[13] R Procesi and R Rota ldquoOn some classes of hyperstructuresrdquoDiscrete Mathematics vol 208-209 pp 485ndash497 1999

[14] R Rota ldquoStrongly distributive multiplicative hyperringsrdquo Jour-nal of Geometry vol 39 no 1-2 pp 130ndash138 1990

[15] R Rota ldquoSugli Iperanelli Moltiplicativirdquo Rendiconti di Matem-atica Series VII vol 2 no 4 pp 711ndash724 1982

[16] R Procesi Ciampi andR Rota ldquoPolynomials overmultiplicativehyperringsrdquo Journal of Discrete Mathematical Sciences amp Cryp-tography vol 6 no 2-3 pp 217ndash225 2003

[17] T Vougiouklis Hyperstructures and Their RepresentationsMonographs in Mathematics Hardonic Press 1994

[18] U Dasgupta ldquoOn prime and primary hyperideals of a multi-plicative hyperringrdquoAnalele Stiintifice ale Universitatii Al I Cuzadin IasimdashMatematica vol 58 no 1 pp 19ndash37 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Algebra

being naturally called the left C-ideal generated by 119860 existsand is in fact the intersection of all left C-ideals containing119860 The left C-ideal generated by a left 119867V-ideal 119868 of 119878 iscalled the leftC-closure of 119868 and is denoted byC

119897(119868) Clearly

C119897(C119897(119868)) = C

119897(119868) for a left 119867V-ideal 119868 C119903(119868) and C(119868)

respectively denote the right C-closure of a right 119867V-ideal 119868and theC-closure of an119867V-ideal 119868 of 119878 For an119867V-ideal 119868 of acommutativemultiplicative119867V-ring 119878C119897(119868) = C119903(119868) = C(119868)The following lemma presents a description of the set C

119897(119868)

for a left119867V-ideal 119868 of amultiplicative119867V-ring 119878The setC119903(119868)

for a right119867V-ideal 119868 can be described dually

Lemma 16 Let 119878 be a multiplicative 119867V-ring with an 119894-set Eand let L denote the set of all left 119867V-ideals of 119878 119865 119875(119878) rarr119875(119878) is a mapping defined by

119865 (119860) = ⋃

119899

sum

119894=1

[1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897 119899 119898

119894isin N 119903

1198941isin 119878

119904119886119905119894119904119891119910119894119899119892 [1199031198941∘ 1199031198942∘ sdot sdot sdot ∘ 119903

119894119898119894]119897cap 119860 = 120601

(7)

for any 119860 isin 119875(119878) Then for any 119868 1198681 1198682isin L we have the

following (i) 119865(119868) isin L (ii) 119868 sube 119865(119868) (iii) 1198681sube 1198682rArr 119865(119868

1) sube

119865(1198682) (iv) 119865(119868) = 119868 if and only if 119868 is a left C-ideal of 119878 and

(v) left C-closure of 119868 is C119897(119868) = ⋃

infin

119896=1119865119896(119868) where for any

119896 isin N 119865119896 denotes the (119896 minus 1)-times mapping composition of 119865

Proof All the assertions made in this lemma can be estab-lished by adopting the arguments that are applied in provingwell-known analogous results on ldquocomplete closure of a setrdquoin semihypergroup theory (see [3 8])

Proposition 17 Let 119877 be a strong hyperdomain with an 119894-setThen for any 119867V-ideal 119868 of the multiplicative 119867V-ring 119877[119909]ord(C(I)lowast) ge ordIlowast where for any 119867V-ideal 119869 of 119877[119909] 119869lowast =119869 0119877[119909]

Proof Since 119877 is a (strong) hyperdomain the multiplicative119867V-ring is commutative So for any 119867V-ideal 119868 of 119877[119909]C(119868) = C

119897(119868) = ⋃

infin

119896=1119865119896(119868) (by Lemma 16(v)) For any

119899 isin N 119891(119909) isin 119865119899(119868)lowast rArr 119891(119909) isin sum119898

119894=1119860119894 for some

type-119897 hyperproducts 119860119894= [1198911198941(119909) lowast 119891

1198942(119909) lowast sdot sdot sdot lowast 119891

119894119898119894(119909)]119897

of elements of 119877[119909] satisfying 119860119894cap 119865119899minus1(119868) = 120601 Since

here 119877[119909] is a strong 119867V-domain (by Proposition 13) withabsorbing zero we may assume that 0

119877[119909]notin 119860119894for each 119894

(since 0119877[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119877[119909])) Then for

each 119894 (= 1 2 119898) and 119895119894(= 1 2 119898

119894) 119891119894119895119894

(119909) = 0119877[119909]

Thus by Corollary 12 ord119860

119894= ord119892

119894(119909) for any 119892

119894(119909) isin 119860

119894

Now since for each 119894 119860119894cap 119865119899minus1(119868) = 120601 so there exists

119892119894(119909) isin 119860

119894such that 119892

119894(119909) isin 119865

119899minus1(119868)lowast Then for any 119891(119909) isin

119865119899(119868)lowast ord119891(119909) ge sum119898

119894=1ord119860119894= minord119892

119894(119909) 119894 ge

ord(119865119899minus1(119868)lowast) Consequently ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast)for any 119899 isin N Now let 119891(119909) isin C(119868)

lowast be arbitrary Then119891(119909) isin 119865

119899(119868)lowast for some 119899 isin N So we have that ord119891(119909) ge

ord(119865119899(119868)lowast) ge ord(119865119899minus1(119868)lowast) ge sdot sdot sdot ge ord(119865(119868)lowast) ge ord119868lowastThus ord(C(119868)lowast) ge ord119868lowast

5 Polynomials over Multiplicative Hyperfield

A nonzero element 119886 of a multiplicative119867V-ring (119878 + ∘) withan 119894-set E = 119890

1 1198902 119890

119899 is referred to be an E-invertible

element (or anE-unit) of 119878 if for each 119894 = 1 2 119899 (119899 isin N)there exist 119860

119894119895isin H119897(119886) = [119904

119901∘ 119904119901minus1∘ sdot sdot sdot ∘ 119904

1∘ 119886]119897 119904119894isin 119878

119901 isin N and 119861119894119896isin H119903(119886) = [119886 ∘ 119904

1∘ 1199042∘ sdot sdot sdot ∘ 119904

119902]119903 119904119894isin 119878

119902 isin N such that 119890119894isin (sum119898

119895=1119860119894119895) cap (sum

119897

119896=1119861119894119896) An element

119886 ( =0119878) of the multiplicative119867V-ring 119878with a hyperidentity 119890

is said to be 119890-hyperinvertible (or an 119890-hyperunit) in 119878 if thereexist 119860 isinH

119897(119886) and 119861 isinH

119903(119886) such that 119890 isin 119860 cap 119861

If 119886 isin 119878 is an E-unit (resp 119890-hyperunit) in a multiplica-tive 119867V-ring (119878 + ∘) with two 119894-sets E and E1015840 (resp withtwo hyperidentities 119890 and 1198901015840) then one can easily verify that119886 is also an E1015840-unit (resp an 1198901015840-hyperunit) in 119878 We thuscall an E-unit (resp an 119890-hyperunit) of a multiplicative 119867V-ring 119878 simply a unit (resp a hyperunit) Denote by 119880(119878)and 119880

ℎ(119878) respectively the sets of units and hyperunits of a

multiplicative119867V-ring 119878An 119867V-ideal 119868 ( = 119878) of a multiplicative 119867V-ring 119878 is

maximal in 119878 if for any 119867V-ideal 119869 of 119878 119868 ⊊ 119869 sube 119878 rArr 119869 = 119878For a commutative multiplicative119867V-ring 119878with an 119894-set thisis immediate to observe that 119886 isin 119880(119878) if and only if 119886 notin 119872for any maximal119867V-ideal119872 of 119878

Proposition 18 Let themultiplicative hyperring119877 be commu-tative and contain a hyperidentity 119890 Then for a polynomial119891(119909) = sum119886

119896119909119896isin 119877[119909] 119891(119909) isin 119880

ℎ(119877[119909]) if and only if

1198860isin 119880ℎ(119877)

Proof Since 119890 is a hyperidentity in the multiplicative hyper-ring 119877 1198901199090 isin 119877[119909] is a hyperidentity in the multiplicative119867V-ring 119877[119909] Now let 119891(119909) = sum119886

119896119909119896isin 119880ℎ(119877[119909]) Then there

exist 119892119894(119909) = sum 119887

119894119896119909119896isin 119877[119909] (119894 = 1 2 119899 119899 isin N) such that

1198901199090isin [119891(119909) lowast 119892

1(119909) lowast sdot sdot sdot lowast 119892

119899(119909)]119903 So 119890 isin 119886

0(1198871011988720 1198871198990)

and thus there exists 119887 isin 1198871011988720 1198871198990sube 119877 such that 119890 isin 119886

0119887

whence 1198860isin 119880ℎ(119877) (since 119877 is commutative)

Conversely let 1198860be hyperinvertible in 119877 with respect to

the hyperidentity 119890 Then there exists 1198870isin 119877 such that 119890 isin

11988601198870 We assert that there is a sequence 119887

119896119896isinN in 119877 whose

119899th term 119887119899(119899 isin N) is inductively defined so as to satisfy the

relation that

0119877isin 1198860119887119899+ 1198861119887119899minus1+ 1198862119887119899minus2+ sdot sdot sdot + 119886

1198991198870sdot sdot sdot () (8)

In fact we see that 0119877isin 01198771198870= (minus119886

1+ 1198861)1198870sube (119890(minus119886

1) +

1198861)1198870sube ((119886

01198870)(minus1198861) + 1198861)1198870sube (119886

01198870)(minus1198861)1198870+ 11988611198870=

1198860((minus1198870)11988611198870) + 11988611198870rArr there exist 119887

1isin (minus119887

0)11988611198870such that

0119877isin 11988601198871+ 11988611198870(the relation for 119899 = 1) Suppose for

some 119898 ge 1 the terms 119887119896(1 le 119896 le 119898 minus 1) are defined

in such a way that each 119887119896(1 le 119896 le 119898 minus 1) satisfies the

relation () for 119896 Then (1198861119887119898minus1+1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+1198861198981198870)

is defined to be a nonempty subset of 119877 Let 119905 isin 1198861119887119898minus1+

1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870 Then 0

119877= minus119905 + 119905 isin 119890(minus119905) + 119905 sube

(11988601198870)(minus119905) + 119905 sube 119886

0(minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+

1198861198981198870) + (1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) rArr there exists

119887119898isin (minus1198870)(1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

119898minus11198871+ 1198861198981198870) such that

0119877isin 1198860119887119898+ 1198861119887119898minus1+ 1198862119887119898minus2+ sdot sdot sdot + 119886

1198981198870(the relation

for 119896 = 119898) Hence the assertion is true for all 119899 isin N Thus

Algebra 7

consider the polynomial 119892(119909) = sum 119887119896119909119896isin 119877[119909] Then from

the definition of lowast 119891(119909) lowast 119892(119909) = sum 119888119896119909119896 1198880isin 11988601198870and

119888119896isin 1198860119887119896+ 1198861119887119896minus1+ 1198862119887119896minus2+ sdot sdot sdot + 119886

1198961198870for 119896 isin N Thus

1198901199090isin 119891(119909) lowast 119892(119909) (due to relation and since 119890 isin 119886

01198870) So

119891(119909) is hyperinvertible in 119877[119909] that is 119891(119909) isin 119880ℎ(119877[119909])

The (left right) 119867V-ideal of a multiplicative 119867V-ring 119878generated by 119860 isin 119875(119878) is the smallest (resp left right) 119867V-ideal of 119878 containing119860which is denoted by (resp ⟨119860⟩

119897 ⟨119860⟩119903)

⟨119860⟩ The principal (left right) 119867V-ideal of the multiplicative119867V-ring 119878 generated by an element 119886 of 119878 denoted by (resp⟨119886⟩119897 ⟨119886⟩119903) ⟨119886⟩ is the (resp left right)119867V-ideal (resp ⟨119886⟩119897

⟨119886⟩119903) ⟨119886⟩ of the multiplicative119867V-ring 119878

If the multiplicative 119867V-ring 119878 has an 119894-set then for any119886 isin 119878 ⟨119886⟩

119897= H119897(119886) = ⋃sum

119899

119894=1119860119894 119860119894isin H119897(119886) 119899 isin N and

⟨119886⟩119903=H119903(119886) = ⋃sum

119899

119894=1119860119894 119860119894isinH119903(119886) 119899 isin N

Definition 19 A commutative multiplicative 119867V-ring 119878 withan 119894-setE is called amultiplicative119867V-field (resp an inversivemultiplicative 119867V-field) if 119878 ⟨0

119878⟩ sube 119880(119878) (resp 119878

⟨0119878⟩ sube 119880

ℎ(119878)) If a multiplicative119867V-field (resp an inversive

multiplicative 119867V-field) is a multiplicative hyperring thenwe call it a multiplicative hyperfield (resp an inversivemultiplicative hyperfield)

Proposition 20 Let 119865 be a polynomially structured inversivemultiplicative hyperfield Then a polynomial 119891(119909) = sum119886

119896119909119896isin

119865[119909]0119865[119909] is hyperinvertible in119865[119909] if and only if ord 119891 = 0

Proof If 119865 is an inversive multiplicative hyperfield then byProposition 18 any polynomial119891(119909) = sum119886

119896119909119896isin 119865[119909]0

119865[119909]

is hyperinvertible in 119865[119909] if and only if 1198860= 0119865 Hence the

result follows

Definition 21 A commutative multiplicative 119867V-ring 119878 iscalled a principal C-ideal 119867V-ring if every C-ideal of119878 is a principal 119867V-ideal A principal C-ideal 119867V-ringwhich is a (strong) 119867V-domain is called a principal C-ideal(strong) 119867V-domain

Proposition 22 Let 119865 be a polynomially structured inversivemultiplicative hyperfieldThen themultiplicative119867V-ring119865[119909]of polynomials over 119865 is a principalC-ideal119867V-ring

Proof Let 119890 be a hyperidentity of the inversive multiplicativehyperfield 119865 Then the polynomial 1198901199090 is a hyperidentity in119865[119909] and thus 119865[119909] = ⟨1198901199090⟩ (since 119865 is commutative) Solet 119868 be any proper C-ideal of 119865[119909] If 119868 = 0

119865[119909] then 119868 is

the principal hyperideal ⟨0119865[119909]⟩ (since for119865 having absorbing

zero 0119865[119909]

is absorbing in 119865[119909]) Suppose that 119868 = 0119865[119909]

Then take a nonzero polynomial 119891(119909) = sum119886119896119909119896isin 119868 such

that ord119891(119909) le ord 119905(119909) for any 119905(119909) isin 119868lowast Let us writeord119891(119909) = 119898 Then 119886

119898= 0119865and 119886119896= 0119865for any 0 le 119896 lt 119898

Consider then the polynomial 119892(119909) = sumlowast119896isinN0119887119896119909119896isin 119865[119909]

where 119887119896= 119886119898+119896

Then clearly 119891(119909) isin (119890119909119898) lowast 119892(119909) Alsoby Proposition 18 119892(119909) is hyperinvertible in 119865[119909] Thus thereexists ℎ(119909) isin 119865[119909] such that 1198901199090 isin 119892(119909) lowast ℎ(119909) Now119891(119909) isin (119890119909

119898) lowast 119892(119909) rArr 119891(119909) lowast ℎ(119909) sube [119890119909

119898lowast 119892(119909) lowast ℎ(119909)]

119903

Again 119891(119909) lowast ℎ(119909) sube 119868 (since 119891(119909) isin 119868) So [119890119909119898 lowast 119892(119909) lowastℎ(119909)]119903cap 119868 = 120601 Hence [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868 (since

119868 is a C-ideal and every C-ideal is a right C-ideal) Now[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap [119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119903= 120601 whence

[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap 119868 = 120601 (since [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868)

Consequently [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]119897sube 119868 (since 119868 is a left C-

ideal) Then 119890119909119898 isin (119890119909119898) lowast (1198901199090) sube 119890119909119898 lowast (119892(119909) lowast ℎ(119909)) =[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897sube 119868 Thus ⟨119890119909119898⟩ sube 119868

Now let ℎ(119909) = sum 119887119896119909119896isin 119868lowast be arbitrary Suppose that

ord ℎ = 119899 Then 119887119899= 0119865and 119887119896= 0119865for any 0 le 119896 lt 119899

By choice of 119891(119909) from 119868 here 119898 le 119899 So one can definea polynomial 119892(119909) = sum 119888

119896119909119896isin 119865[119909] where 119888

119896= 0119865for all

0 le 119896 le 119899 minus 119898 minus 1 and 119888119896= 119887119898+119896

for all 119896 ge 119899 minus 119898 Clearlythen ℎ(119909) isin 119890119909119898 lowast 119892(119909) sube ⟨119890119909119898⟩ whence 119868 sube ⟨119890119909119898⟩ Thus119868 = ⟨119890119909

119898⟩

Remark 23 In a ring an invertible element can never be adivisor of zero This not true in general for a multiplicativehyperring In fact on the commutative group of integers(Z +) if we define a hyperoperation ∘ by stating that 119909 ∘ 119910 =0 119909119910 for all 119909 119910 isin Z then (Z + ∘) is a commutative mul-tiplicative hyperring with a hyperidentity 1 Every nonzeroelement of (Z + ∘) is a zero divisor and 1 minus1 isin Z are inparticular hyperunits of (Z + ∘) To get a parity with thering theory in this regard we perceive the notion of stronghyperinvertibility of an element of amultiplicative hyperring

Definition 24 A hyperinvertible element of a multiplicativehyperring with a hyperidentity is said to be strongly hyperin-vertible (or a strong hyperunit) if it is not a zero divisor inthat multiplicative hyperring A commutative multiplicativehyperring with absorbing zero and a hyperidentity 119890 is said tobe a strongly inversive multiplicative hyperfield if each of itsnonzero elements is a strong hyperunit

Example 25 Let 119877 = 119886radic2 + 119887radic3 119886 119887 isin Q and 119860 =radic2radic3 Then with respect to usual addition + of reals(119877 +) is a commutative group with identity 0 On 119877 ∘ is ahyperoperation defined by

(119886radic2 + 119887radic3) ∘ (119888radic2 + 119889radic3)

= (119886radic2 + 119887radic3) sdot 119905 sdot (119888radic2 + 119889radic3) 119905 isin 119860

= 119901radic2 + 2119902radic3 3119902radic2 + 119901radic3

(9)

where 119901 = 2119886119888 + 3119887119889 and 119902 = 119887119888 + 119886119889Then (119877 + ∘) is a strongly inversivemultiplicative hyper-

field which is polynomially structured

Definition 26 A local (C-local) multiplicative 119867V-ring is acommutative multiplicative119867V-ring with an 119894-set which hasa unique maximal119867V-ideal (respC-ideal)

Proposition 27 Let 119865 be a polynomially structured stronglyinversive multiplicative hyperfieldThen (i) 119865[119909] is a principalC-ideal strong 119867V-domain (ii) for any 119891 isin 119865[119909] 0

119865[119909]

there exist a hyperinvertible element 119892 isin 119865[119909] and 119898 isin

N cup 0 such that 119891 isin (119890119909119898) lowast 119892 where 119890 is a hyperidentity

8 Algebra

in 119865 and (iii) 119865[119909] is a local as well as aC-local multiplicative119867V-ring

Proof (i) Here 119865 is a strongly inversive multiplicative hyper-field So 119865 is a strong hyperdomain Thus by Proposition 13the multiplicative 119867V-ring 119865[119909] is a strong 119867V-domainAgain by Proposition 22 119865[119909] is a principal C-ideal 119867V-ring (since 119865 is an inversive multiplicative hyperfield) So byDefinition 21 119865[119909] is a principalC -ideal strong119867V-domain

(ii) Let 119891 isin 119865[119909] 0119865[119909] and 119869 = ⟨119891⟩ Then 119869 =H

119903(119891)

So for any ℎ isin 119869lowast there exist119860119894= [119891lowast119891

1198941lowast1198911198942lowastsdot sdot sdotlowast119891

119894119898119894]119903isin

H119903(119891) (119894 = 1 2 119896 119896119898

119894isin N) such that ℎ isin sum119896

119894=1119860119894

Thus ℎ = sum119896119894=1119905119894for some 119905

119894isin 119860119894 Since 119865[119909] is a strong119867V-

domain (by Proposition 13) with absorbing zero so 0119865[119909]isin

119860119894= [119891 lowast 119891

1198941lowast 1198911198942lowast sdot sdot sdot lowast 119891

119894119898119894]119903rArr [119891 lowast 119891

1198941lowast 1198911198942lowast

sdot sdot sdot lowast 119891119894119898119894]119903= 0119865[119909] Hence without any loss of generality

(since 0119865[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119865[119909])) we may

assume that 0119865[119909]notin 119860119894for any 119894 (since ℎ = 0

119865[119909]) Then

for any 119894 = 1 2 119896 and 119895 = 1 2 119898119894 119891119894119895= 0119865[119909]

So by Proposition 11(ii) ord 119905

119894= ord(119860

119894) ge ord119891 and so

ord ℎ = minord 119905119894 119894 ge ord119891 Thus ord119869lowast ge ord119891 So

by Proposition 17 119891 is a nonzero polynomial in the C-idealC(119869) such that ord119891 le ord ℎ for all ℎ isin C(119869)lowast So there existsan invertible element 119892 isin 119865[119909] and an integer119898 ge 0 such that119891 isin (119890119909

119898) lowast 119892 where 119890 is a hyperidentity in 119865 (see proof of

Proposition 22)(iii) For any integer119898 ge 1 since 119890119909119898 isin (119890119909) lowast (119890119909119898minus1) so

⟨119890119909119898⟩ sube ⟨119890119909⟩ Again for any119867V-ideal 119868 of the multiplicative

119867V-ring119865[119909] if119891 isin 119868lowast is such that ord119891 le ord119892 for all119892 isin 119868lowast

then 119868 sube ⟨119890119909119898⟩ where 119898 = ord119891 So ⟨119890119909⟩ is the uniquemaximal119867V-ideal in 119865[119909] Thus 119865[119909] is a local multiplicative119867V-ring Now 119890119909 isin C

1(as defined in Proposition 15) So

ord(119890119909) le ord119891 for any 119891 isin Clowast1 Thus C

1being a C-ideal

of 119865[119909] we have C1= ⟨119890119909⟩ (see proof of Proposition 22)

Hence 119865[119909] is aC-local multiplicative119867V-ring

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vougiouklis ldquoThe fundamental relation in hyperrings Thegeneral hyperfieldrdquo in Algebraic Hyperstructures and Appli-cations (Xanthi1990) pp 203ndash211 World Science PublisherTeaneck NJ USA 1991

[2] M R Darafsheh and B Davvaz ldquo119867V-ring of fractionsrdquo ItalianJournal of Pure and Applied Mathematics no 5 pp 25ndash34 1999

[3] B Davvaz and V Leoreanu-FoteaHyperring Theory and Appli-cations International Academic Press Palm Harbor Fla USA2007

[4] S Spartalis A Dramalides and T Vougiouklis ldquoOn 119867V-groupringsrdquoAlgebras Groups and Geometries vol 15 no 1 pp 47ndash541998

[5] T Vougiouklis ldquo119867V-groups defined on the same setrdquo DiscreteMathematics vol 155 no 1ndash3 pp 259ndash265 1996

[6] R Procesi and R Rota ldquoMultiplicative H119907-rings and com-

plementary hyperstructuresrdquo Journal of Discrete MathematicalSciences amp Cryptography vol 11 no 4 pp 447ndash456 2008

[7] S Spartalis ldquoOn the number of 119867Vminus119903119894119899119892119904 with 119875-hyperopera-tionsrdquoDiscrete Mathematics vol 155 no 1ndash3 pp 225ndash231 1996

[8] P Corsini Prolegomena of Hypergroup Theory Rivista di Mate-matica Pura ed Applicata Aviani Tricesimo Italy 1993

[9] M K Sen and U Dasgupta ldquoSome aspects of119866119867-ringsrdquoAnnals

of the Alexandru Ioan Cuza UniversitymdashMathematics vol 56no 2 pp 253ndash272 2010

[10] C Namnak N Triphop and Y Kemprasit ldquoHomomorphismsof somemultiplicative hyperringsrdquo Set-ValuedMathematics andApplications vol 1 no 2 pp 145ndash152 2008

[11] D M Olson and V K Ward ldquoA note on multiplicativehyperringrdquo Italian Journal of Pure andAppliedMathematics vol1 pp 77ndash84 1997

[12] R Procesi and R Rota ldquoComplementary multiplicative hyper-ringsrdquoDiscreteMathematics vol 308 no 2-3 pp 188ndash191 2008

[13] R Procesi and R Rota ldquoOn some classes of hyperstructuresrdquoDiscrete Mathematics vol 208-209 pp 485ndash497 1999

[14] R Rota ldquoStrongly distributive multiplicative hyperringsrdquo Jour-nal of Geometry vol 39 no 1-2 pp 130ndash138 1990

[15] R Rota ldquoSugli Iperanelli Moltiplicativirdquo Rendiconti di Matem-atica Series VII vol 2 no 4 pp 711ndash724 1982

[16] R Procesi Ciampi andR Rota ldquoPolynomials overmultiplicativehyperringsrdquo Journal of Discrete Mathematical Sciences amp Cryp-tography vol 6 no 2-3 pp 217ndash225 2003

[17] T Vougiouklis Hyperstructures and Their RepresentationsMonographs in Mathematics Hardonic Press 1994

[18] U Dasgupta ldquoOn prime and primary hyperideals of a multi-plicative hyperringrdquoAnalele Stiintifice ale Universitatii Al I Cuzadin IasimdashMatematica vol 58 no 1 pp 19ndash37 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Algebra 7

consider the polynomial 119892(119909) = sum 119887119896119909119896isin 119877[119909] Then from

the definition of lowast 119891(119909) lowast 119892(119909) = sum 119888119896119909119896 1198880isin 11988601198870and

119888119896isin 1198860119887119896+ 1198861119887119896minus1+ 1198862119887119896minus2+ sdot sdot sdot + 119886

1198961198870for 119896 isin N Thus

1198901199090isin 119891(119909) lowast 119892(119909) (due to relation and since 119890 isin 119886

01198870) So

119891(119909) is hyperinvertible in 119877[119909] that is 119891(119909) isin 119880ℎ(119877[119909])

The (left right) 119867V-ideal of a multiplicative 119867V-ring 119878generated by 119860 isin 119875(119878) is the smallest (resp left right) 119867V-ideal of 119878 containing119860which is denoted by (resp ⟨119860⟩

119897 ⟨119860⟩119903)

⟨119860⟩ The principal (left right) 119867V-ideal of the multiplicative119867V-ring 119878 generated by an element 119886 of 119878 denoted by (resp⟨119886⟩119897 ⟨119886⟩119903) ⟨119886⟩ is the (resp left right)119867V-ideal (resp ⟨119886⟩119897

⟨119886⟩119903) ⟨119886⟩ of the multiplicative119867V-ring 119878

If the multiplicative 119867V-ring 119878 has an 119894-set then for any119886 isin 119878 ⟨119886⟩

119897= H119897(119886) = ⋃sum

119899

119894=1119860119894 119860119894isin H119897(119886) 119899 isin N and

⟨119886⟩119903=H119903(119886) = ⋃sum

119899

119894=1119860119894 119860119894isinH119903(119886) 119899 isin N

Definition 19 A commutative multiplicative 119867V-ring 119878 withan 119894-setE is called amultiplicative119867V-field (resp an inversivemultiplicative 119867V-field) if 119878 ⟨0

119878⟩ sube 119880(119878) (resp 119878

⟨0119878⟩ sube 119880

ℎ(119878)) If a multiplicative119867V-field (resp an inversive

multiplicative 119867V-field) is a multiplicative hyperring thenwe call it a multiplicative hyperfield (resp an inversivemultiplicative hyperfield)

Proposition 20 Let 119865 be a polynomially structured inversivemultiplicative hyperfield Then a polynomial 119891(119909) = sum119886

119896119909119896isin

119865[119909]0119865[119909] is hyperinvertible in119865[119909] if and only if ord 119891 = 0

Proof If 119865 is an inversive multiplicative hyperfield then byProposition 18 any polynomial119891(119909) = sum119886

119896119909119896isin 119865[119909]0

119865[119909]

is hyperinvertible in 119865[119909] if and only if 1198860= 0119865 Hence the

result follows

Definition 21 A commutative multiplicative 119867V-ring 119878 iscalled a principal C-ideal 119867V-ring if every C-ideal of119878 is a principal 119867V-ideal A principal C-ideal 119867V-ringwhich is a (strong) 119867V-domain is called a principal C-ideal(strong) 119867V-domain

Proposition 22 Let 119865 be a polynomially structured inversivemultiplicative hyperfieldThen themultiplicative119867V-ring119865[119909]of polynomials over 119865 is a principalC-ideal119867V-ring

Proof Let 119890 be a hyperidentity of the inversive multiplicativehyperfield 119865 Then the polynomial 1198901199090 is a hyperidentity in119865[119909] and thus 119865[119909] = ⟨1198901199090⟩ (since 119865 is commutative) Solet 119868 be any proper C-ideal of 119865[119909] If 119868 = 0

119865[119909] then 119868 is

the principal hyperideal ⟨0119865[119909]⟩ (since for119865 having absorbing

zero 0119865[119909]

is absorbing in 119865[119909]) Suppose that 119868 = 0119865[119909]

Then take a nonzero polynomial 119891(119909) = sum119886119896119909119896isin 119868 such

that ord119891(119909) le ord 119905(119909) for any 119905(119909) isin 119868lowast Let us writeord119891(119909) = 119898 Then 119886

119898= 0119865and 119886119896= 0119865for any 0 le 119896 lt 119898

Consider then the polynomial 119892(119909) = sumlowast119896isinN0119887119896119909119896isin 119865[119909]

where 119887119896= 119886119898+119896

Then clearly 119891(119909) isin (119890119909119898) lowast 119892(119909) Alsoby Proposition 18 119892(119909) is hyperinvertible in 119865[119909] Thus thereexists ℎ(119909) isin 119865[119909] such that 1198901199090 isin 119892(119909) lowast ℎ(119909) Now119891(119909) isin (119890119909

119898) lowast 119892(119909) rArr 119891(119909) lowast ℎ(119909) sube [119890119909

119898lowast 119892(119909) lowast ℎ(119909)]

119903

Again 119891(119909) lowast ℎ(119909) sube 119868 (since 119891(119909) isin 119868) So [119890119909119898 lowast 119892(119909) lowastℎ(119909)]119903cap 119868 = 120601 Hence [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868 (since

119868 is a C-ideal and every C-ideal is a right C-ideal) Now[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap [119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119903= 120601 whence

[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897cap 119868 = 120601 (since [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]

119903sube 119868)

Consequently [119890119909119898 lowast 119892(119909) lowast ℎ(119909)]119897sube 119868 (since 119868 is a left C-

ideal) Then 119890119909119898 isin (119890119909119898) lowast (1198901199090) sube 119890119909119898 lowast (119892(119909) lowast ℎ(119909)) =[119890119909119898lowast 119892(119909) lowast ℎ(119909)]

119897sube 119868 Thus ⟨119890119909119898⟩ sube 119868

Now let ℎ(119909) = sum 119887119896119909119896isin 119868lowast be arbitrary Suppose that

ord ℎ = 119899 Then 119887119899= 0119865and 119887119896= 0119865for any 0 le 119896 lt 119899

By choice of 119891(119909) from 119868 here 119898 le 119899 So one can definea polynomial 119892(119909) = sum 119888

119896119909119896isin 119865[119909] where 119888

119896= 0119865for all

0 le 119896 le 119899 minus 119898 minus 1 and 119888119896= 119887119898+119896

for all 119896 ge 119899 minus 119898 Clearlythen ℎ(119909) isin 119890119909119898 lowast 119892(119909) sube ⟨119890119909119898⟩ whence 119868 sube ⟨119890119909119898⟩ Thus119868 = ⟨119890119909

119898⟩

Remark 23 In a ring an invertible element can never be adivisor of zero This not true in general for a multiplicativehyperring In fact on the commutative group of integers(Z +) if we define a hyperoperation ∘ by stating that 119909 ∘ 119910 =0 119909119910 for all 119909 119910 isin Z then (Z + ∘) is a commutative mul-tiplicative hyperring with a hyperidentity 1 Every nonzeroelement of (Z + ∘) is a zero divisor and 1 minus1 isin Z are inparticular hyperunits of (Z + ∘) To get a parity with thering theory in this regard we perceive the notion of stronghyperinvertibility of an element of amultiplicative hyperring

Definition 24 A hyperinvertible element of a multiplicativehyperring with a hyperidentity is said to be strongly hyperin-vertible (or a strong hyperunit) if it is not a zero divisor inthat multiplicative hyperring A commutative multiplicativehyperring with absorbing zero and a hyperidentity 119890 is said tobe a strongly inversive multiplicative hyperfield if each of itsnonzero elements is a strong hyperunit

Example 25 Let 119877 = 119886radic2 + 119887radic3 119886 119887 isin Q and 119860 =radic2radic3 Then with respect to usual addition + of reals(119877 +) is a commutative group with identity 0 On 119877 ∘ is ahyperoperation defined by

(119886radic2 + 119887radic3) ∘ (119888radic2 + 119889radic3)

= (119886radic2 + 119887radic3) sdot 119905 sdot (119888radic2 + 119889radic3) 119905 isin 119860

= 119901radic2 + 2119902radic3 3119902radic2 + 119901radic3

(9)

where 119901 = 2119886119888 + 3119887119889 and 119902 = 119887119888 + 119886119889Then (119877 + ∘) is a strongly inversivemultiplicative hyper-

field which is polynomially structured

Definition 26 A local (C-local) multiplicative 119867V-ring is acommutative multiplicative119867V-ring with an 119894-set which hasa unique maximal119867V-ideal (respC-ideal)

Proposition 27 Let 119865 be a polynomially structured stronglyinversive multiplicative hyperfieldThen (i) 119865[119909] is a principalC-ideal strong 119867V-domain (ii) for any 119891 isin 119865[119909] 0

119865[119909]

there exist a hyperinvertible element 119892 isin 119865[119909] and 119898 isin

N cup 0 such that 119891 isin (119890119909119898) lowast 119892 where 119890 is a hyperidentity

8 Algebra

in 119865 and (iii) 119865[119909] is a local as well as aC-local multiplicative119867V-ring

Proof (i) Here 119865 is a strongly inversive multiplicative hyper-field So 119865 is a strong hyperdomain Thus by Proposition 13the multiplicative 119867V-ring 119865[119909] is a strong 119867V-domainAgain by Proposition 22 119865[119909] is a principal C-ideal 119867V-ring (since 119865 is an inversive multiplicative hyperfield) So byDefinition 21 119865[119909] is a principalC -ideal strong119867V-domain

(ii) Let 119891 isin 119865[119909] 0119865[119909] and 119869 = ⟨119891⟩ Then 119869 =H

119903(119891)

So for any ℎ isin 119869lowast there exist119860119894= [119891lowast119891

1198941lowast1198911198942lowastsdot sdot sdotlowast119891

119894119898119894]119903isin

H119903(119891) (119894 = 1 2 119896 119896119898

119894isin N) such that ℎ isin sum119896

119894=1119860119894

Thus ℎ = sum119896119894=1119905119894for some 119905

119894isin 119860119894 Since 119865[119909] is a strong119867V-

domain (by Proposition 13) with absorbing zero so 0119865[119909]isin

119860119894= [119891 lowast 119891

1198941lowast 1198911198942lowast sdot sdot sdot lowast 119891

119894119898119894]119903rArr [119891 lowast 119891

1198941lowast 1198911198942lowast

sdot sdot sdot lowast 119891119894119898119894]119903= 0119865[119909] Hence without any loss of generality

(since 0119865[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119865[119909])) we may

assume that 0119865[119909]notin 119860119894for any 119894 (since ℎ = 0

119865[119909]) Then

for any 119894 = 1 2 119896 and 119895 = 1 2 119898119894 119891119894119895= 0119865[119909]

So by Proposition 11(ii) ord 119905

119894= ord(119860

119894) ge ord119891 and so

ord ℎ = minord 119905119894 119894 ge ord119891 Thus ord119869lowast ge ord119891 So

by Proposition 17 119891 is a nonzero polynomial in the C-idealC(119869) such that ord119891 le ord ℎ for all ℎ isin C(119869)lowast So there existsan invertible element 119892 isin 119865[119909] and an integer119898 ge 0 such that119891 isin (119890119909

119898) lowast 119892 where 119890 is a hyperidentity in 119865 (see proof of

Proposition 22)(iii) For any integer119898 ge 1 since 119890119909119898 isin (119890119909) lowast (119890119909119898minus1) so

⟨119890119909119898⟩ sube ⟨119890119909⟩ Again for any119867V-ideal 119868 of the multiplicative

119867V-ring119865[119909] if119891 isin 119868lowast is such that ord119891 le ord119892 for all119892 isin 119868lowast

then 119868 sube ⟨119890119909119898⟩ where 119898 = ord119891 So ⟨119890119909⟩ is the uniquemaximal119867V-ideal in 119865[119909] Thus 119865[119909] is a local multiplicative119867V-ring Now 119890119909 isin C

1(as defined in Proposition 15) So

ord(119890119909) le ord119891 for any 119891 isin Clowast1 Thus C

1being a C-ideal

of 119865[119909] we have C1= ⟨119890119909⟩ (see proof of Proposition 22)

Hence 119865[119909] is aC-local multiplicative119867V-ring

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vougiouklis ldquoThe fundamental relation in hyperrings Thegeneral hyperfieldrdquo in Algebraic Hyperstructures and Appli-cations (Xanthi1990) pp 203ndash211 World Science PublisherTeaneck NJ USA 1991

[2] M R Darafsheh and B Davvaz ldquo119867V-ring of fractionsrdquo ItalianJournal of Pure and Applied Mathematics no 5 pp 25ndash34 1999

[3] B Davvaz and V Leoreanu-FoteaHyperring Theory and Appli-cations International Academic Press Palm Harbor Fla USA2007

[4] S Spartalis A Dramalides and T Vougiouklis ldquoOn 119867V-groupringsrdquoAlgebras Groups and Geometries vol 15 no 1 pp 47ndash541998

[5] T Vougiouklis ldquo119867V-groups defined on the same setrdquo DiscreteMathematics vol 155 no 1ndash3 pp 259ndash265 1996

[6] R Procesi and R Rota ldquoMultiplicative H119907-rings and com-

plementary hyperstructuresrdquo Journal of Discrete MathematicalSciences amp Cryptography vol 11 no 4 pp 447ndash456 2008

[7] S Spartalis ldquoOn the number of 119867Vminus119903119894119899119892119904 with 119875-hyperopera-tionsrdquoDiscrete Mathematics vol 155 no 1ndash3 pp 225ndash231 1996

[8] P Corsini Prolegomena of Hypergroup Theory Rivista di Mate-matica Pura ed Applicata Aviani Tricesimo Italy 1993

[9] M K Sen and U Dasgupta ldquoSome aspects of119866119867-ringsrdquoAnnals

of the Alexandru Ioan Cuza UniversitymdashMathematics vol 56no 2 pp 253ndash272 2010

[10] C Namnak N Triphop and Y Kemprasit ldquoHomomorphismsof somemultiplicative hyperringsrdquo Set-ValuedMathematics andApplications vol 1 no 2 pp 145ndash152 2008

[11] D M Olson and V K Ward ldquoA note on multiplicativehyperringrdquo Italian Journal of Pure andAppliedMathematics vol1 pp 77ndash84 1997

[12] R Procesi and R Rota ldquoComplementary multiplicative hyper-ringsrdquoDiscreteMathematics vol 308 no 2-3 pp 188ndash191 2008

[13] R Procesi and R Rota ldquoOn some classes of hyperstructuresrdquoDiscrete Mathematics vol 208-209 pp 485ndash497 1999

[14] R Rota ldquoStrongly distributive multiplicative hyperringsrdquo Jour-nal of Geometry vol 39 no 1-2 pp 130ndash138 1990

[15] R Rota ldquoSugli Iperanelli Moltiplicativirdquo Rendiconti di Matem-atica Series VII vol 2 no 4 pp 711ndash724 1982

[16] R Procesi Ciampi andR Rota ldquoPolynomials overmultiplicativehyperringsrdquo Journal of Discrete Mathematical Sciences amp Cryp-tography vol 6 no 2-3 pp 217ndash225 2003

[17] T Vougiouklis Hyperstructures and Their RepresentationsMonographs in Mathematics Hardonic Press 1994

[18] U Dasgupta ldquoOn prime and primary hyperideals of a multi-plicative hyperringrdquoAnalele Stiintifice ale Universitatii Al I Cuzadin IasimdashMatematica vol 58 no 1 pp 19ndash37 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Algebra

in 119865 and (iii) 119865[119909] is a local as well as aC-local multiplicative119867V-ring

Proof (i) Here 119865 is a strongly inversive multiplicative hyper-field So 119865 is a strong hyperdomain Thus by Proposition 13the multiplicative 119867V-ring 119865[119909] is a strong 119867V-domainAgain by Proposition 22 119865[119909] is a principal C-ideal 119867V-ring (since 119865 is an inversive multiplicative hyperfield) So byDefinition 21 119865[119909] is a principalC -ideal strong119867V-domain

(ii) Let 119891 isin 119865[119909] 0119865[119909] and 119869 = ⟨119891⟩ Then 119869 =H

119903(119891)

So for any ℎ isin 119869lowast there exist119860119894= [119891lowast119891

1198941lowast1198911198942lowastsdot sdot sdotlowast119891

119894119898119894]119903isin

H119903(119891) (119894 = 1 2 119896 119896119898

119894isin N) such that ℎ isin sum119896

119894=1119860119894

Thus ℎ = sum119896119894=1119905119894for some 119905

119894isin 119860119894 Since 119865[119909] is a strong119867V-

domain (by Proposition 13) with absorbing zero so 0119865[119909]isin

119860119894= [119891 lowast 119891

1198941lowast 1198911198942lowast sdot sdot sdot lowast 119891

119894119898119894]119903rArr [119891 lowast 119891

1198941lowast 1198911198942lowast

sdot sdot sdot lowast 119891119894119898119894]119903= 0119865[119909] Hence without any loss of generality

(since 0119865[119909] + 119860 = 119860 for any 119860 isin 119875lowast(119865[119909])) we may

assume that 0119865[119909]notin 119860119894for any 119894 (since ℎ = 0

119865[119909]) Then

for any 119894 = 1 2 119896 and 119895 = 1 2 119898119894 119891119894119895= 0119865[119909]

So by Proposition 11(ii) ord 119905

119894= ord(119860

119894) ge ord119891 and so

ord ℎ = minord 119905119894 119894 ge ord119891 Thus ord119869lowast ge ord119891 So

by Proposition 17 119891 is a nonzero polynomial in the C-idealC(119869) such that ord119891 le ord ℎ for all ℎ isin C(119869)lowast So there existsan invertible element 119892 isin 119865[119909] and an integer119898 ge 0 such that119891 isin (119890119909

119898) lowast 119892 where 119890 is a hyperidentity in 119865 (see proof of

Proposition 22)(iii) For any integer119898 ge 1 since 119890119909119898 isin (119890119909) lowast (119890119909119898minus1) so

⟨119890119909119898⟩ sube ⟨119890119909⟩ Again for any119867V-ideal 119868 of the multiplicative

119867V-ring119865[119909] if119891 isin 119868lowast is such that ord119891 le ord119892 for all119892 isin 119868lowast

then 119868 sube ⟨119890119909119898⟩ where 119898 = ord119891 So ⟨119890119909⟩ is the uniquemaximal119867V-ideal in 119865[119909] Thus 119865[119909] is a local multiplicative119867V-ring Now 119890119909 isin C

1(as defined in Proposition 15) So

ord(119890119909) le ord119891 for any 119891 isin Clowast1 Thus C

1being a C-ideal

of 119865[119909] we have C1= ⟨119890119909⟩ (see proof of Proposition 22)

Hence 119865[119909] is aC-local multiplicative119867V-ring

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vougiouklis ldquoThe fundamental relation in hyperrings Thegeneral hyperfieldrdquo in Algebraic Hyperstructures and Appli-cations (Xanthi1990) pp 203ndash211 World Science PublisherTeaneck NJ USA 1991

[2] M R Darafsheh and B Davvaz ldquo119867V-ring of fractionsrdquo ItalianJournal of Pure and Applied Mathematics no 5 pp 25ndash34 1999

[3] B Davvaz and V Leoreanu-FoteaHyperring Theory and Appli-cations International Academic Press Palm Harbor Fla USA2007

[4] S Spartalis A Dramalides and T Vougiouklis ldquoOn 119867V-groupringsrdquoAlgebras Groups and Geometries vol 15 no 1 pp 47ndash541998

[5] T Vougiouklis ldquo119867V-groups defined on the same setrdquo DiscreteMathematics vol 155 no 1ndash3 pp 259ndash265 1996

[6] R Procesi and R Rota ldquoMultiplicative H119907-rings and com-

plementary hyperstructuresrdquo Journal of Discrete MathematicalSciences amp Cryptography vol 11 no 4 pp 447ndash456 2008

[7] S Spartalis ldquoOn the number of 119867Vminus119903119894119899119892119904 with 119875-hyperopera-tionsrdquoDiscrete Mathematics vol 155 no 1ndash3 pp 225ndash231 1996

[8] P Corsini Prolegomena of Hypergroup Theory Rivista di Mate-matica Pura ed Applicata Aviani Tricesimo Italy 1993

[9] M K Sen and U Dasgupta ldquoSome aspects of119866119867-ringsrdquoAnnals

of the Alexandru Ioan Cuza UniversitymdashMathematics vol 56no 2 pp 253ndash272 2010

[10] C Namnak N Triphop and Y Kemprasit ldquoHomomorphismsof somemultiplicative hyperringsrdquo Set-ValuedMathematics andApplications vol 1 no 2 pp 145ndash152 2008

[11] D M Olson and V K Ward ldquoA note on multiplicativehyperringrdquo Italian Journal of Pure andAppliedMathematics vol1 pp 77ndash84 1997

[12] R Procesi and R Rota ldquoComplementary multiplicative hyper-ringsrdquoDiscreteMathematics vol 308 no 2-3 pp 188ndash191 2008

[13] R Procesi and R Rota ldquoOn some classes of hyperstructuresrdquoDiscrete Mathematics vol 208-209 pp 485ndash497 1999

[14] R Rota ldquoStrongly distributive multiplicative hyperringsrdquo Jour-nal of Geometry vol 39 no 1-2 pp 130ndash138 1990

[15] R Rota ldquoSugli Iperanelli Moltiplicativirdquo Rendiconti di Matem-atica Series VII vol 2 no 4 pp 711ndash724 1982

[16] R Procesi Ciampi andR Rota ldquoPolynomials overmultiplicativehyperringsrdquo Journal of Discrete Mathematical Sciences amp Cryp-tography vol 6 no 2-3 pp 217ndash225 2003

[17] T Vougiouklis Hyperstructures and Their RepresentationsMonographs in Mathematics Hardonic Press 1994

[18] U Dasgupta ldquoOn prime and primary hyperideals of a multi-plicative hyperringrdquoAnalele Stiintifice ale Universitatii Al I Cuzadin IasimdashMatematica vol 58 no 1 pp 19ndash37 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of