research article generalized -radical supplemented...

Research ArticleGeneralized oplus-Radical Supplemented Modules

Burcu NiGancJ Tuumlrkmen1 and Ali Pancar2

1 Faculty of Art and Science Amasya University Ipekkoy 05100 Amasya Turkey2Department of Mathematics Faculty of Art and Science Ondokuz Mayis University Kurupelit 55139 Samsun Turkey

Correspondence should be addressed to Burcu Nisancı Turkmen burcunisanciehotmailcom

Received 28 October 2013 Accepted 16 December 2013 Published 14 January 2014

Academic Editors A V Kelarev and S Yang

Copyright copy 2014 B Nisancı Turkmen and A Pancar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Calısıcı and Turkmen called a module 119872 generalized oplus-supplemented if every submodule has a generalized supplement thatis a direct summand of 119872 Motivated by this it is natural to introduce another notion that we called generalized oplus-radicalsupplemented modules as a proper generalization of generalized oplus-supplemented modules In this paper we obtain variousproperties of generalized oplus-radical supplemented modules We show that the class of generalized oplus-radical supplemented modulesis closed under finite direct sumsWe attain that over a Dedekind domain amodule119872 is generalized oplus-radical supplemented if andonly if119872119875(119872) is generalized oplus-radical supplemented We completely determine the structure of these modules over left 119881-ringsMoreover we characterize semiperfect rings via generalized oplus-radical supplemented modules

1 Introduction

Throughout the whole text all rings are to be associativeunit and all modules are left unitary We specially mention[1ndash4] among books concerning the structures of modulesand rings We shall write 119873 le 119872 (119873 ≪ 119872) if 119873 is asubmodule of 119872 (small in 119872) By Rad(119872) we denote theradical of 119872 Let 119880119881 le 119872 119881 is called a supplement of119880 in 119872 if it is minimal with respect to 119872 = 119880 + 119881 119881 isa supplement of 119880 in 119872 if and only if 119872 = 119880 + 119881 and119880 cap 119881 ≪ 119881 [4] A module 119872 is called supplemented ifevery submodule of119872 has a supplement and it is called oplus-supplemented if every submodule of119872 has a supplement thatis a direct summand of119872 [5] Clearly every oplus-supplementedmodule is supplemented In [6] Zoschinger introduced anotion of modules whose radical has supplements calledradical supplemented Xue defined generalized supplementedmodules as another generalization of supplemented modules[7] Let119880 be any submodule of119872 If there exists a submodule119881 of119872 such that119872 = 119880+119881 and119880cap119881 sube Rad(119881)119881 is calleda generalized supplement of 119880 in119872119872 is called generalizedsupplemented if every submodule of 119872 has a generalizedsupplement in 119872 Also Calısıcı and Turkmen called amodule 119872 generalized oplus-supplemented if every submodule

has a generalized supplement that is a direct summand of119872 as a generalization of oplus-supplemented modules [8] Soit is natural to introduce another notion that we calledgeneralizedoplus-radical supplementedmodules Amodule119872 iscalled generalized oplus-radical supplemented if every submodulecontaining radical has a generalized supplement that is adirect summand of119872

In this paper we obtain various properties of generalizedoplus-radical supplemented modules as a proper generalizationof generalized oplus-supplemented modules We prove the fol-lowing indications

(i) Every generalized oplus-radical supplemented modulehas a radical direct summand

(ii) 119875(119872) is a generalized oplus-radical supplemented mod-ule for every 119877-module119872

(iii) The class of generalized oplus-radical supplementedmodules is closed under finite direct sums

(iv) If119872 is a generalized oplus-radical supplementedmodulethen 119872119880 is a generalized oplus-radical supplementedmodule for every fully invariant submodule 119880 of119872

(v) Let 119872119894119894isin119868

be any infinite collection of generalized oplus-radical supplemented modules and let119872 = oplus

119894isin119868119872119894

Hindawi Publishing CorporationISRN AlgebraVolume 2014 Article ID 603851 4 pageshttpdxdoiorg1011552014603851

2 ISRN Algebra

If119872 is a duomodule then119872 is generalized oplus-radicalsupplemented

(vi) Over a Dedekind domain a module119872 is generalizedoplus-radical supplemented if and only if 119872119875(119872) isgeneralized oplus-radical supplemented

(vii) Every generalized oplus-radical supplemented 119877-moduleis injective if and only if 119877 is left Noetherian 119881-ring

(viii) A ring 119877 is semiperfect if and only if every finitelygenerated free 119877-module is generalized oplus-radicalsupplemented

2 Generalized oplus-RadicalSupplemented Modules

Definition 1 A module 119872 is called generalized oplus-radicalsupplemented if every submodule containing radical has ageneralized supplement that is a direct summand of119872

Recall that a module 119872 is called radical if 119872 has nomaximal submodules that is Rad(119872) = 119872 For a module119872 119875(119872) will indicate the sum of all radical submodules of119872 If 119875(119872) = 0119872 is called reduced Note that 119875(119872) is thelargest radical submodule of119872 [4]

Now we have the following simple fact which lays a keyrole in our study

Lemma 2 Let119872 be an 119877-module If Rad (119872) = 119872 then119872is a generalized oplus-radical supplemented module

Proof Since Rad(119872) = 119872 119872 has the trivial generalizedsupplement 0 in 119872 Consequently 119872 is a generalized oplus-radical supplemented module

Corollary 3 119875(119872) is a generalized oplus-radical supplementedmodule for every 119877-module119872

Proposition 4 Every generalized oplus-radical supplementedmodule has a radical direct summand

Proof Let119872 be a generalized oplus-radical supplemented mod-ule Then there exist submodules 119881 and 1198811015840 of 119872 such that119872 = Rad(119872) + 119881 Rad(119872) cap 119881 sube Rad(119881) and119872 = 119881 oplus 1198811015840Since Rad(119872) = Rad(119881)oplusRad(1198811015840) by [4Theorem 216] then119872 = Rad(1198811015840) + 119881 Applying the modular law we obtain thatRad(1198811015840) = 1198811015840

Recall from [4] that a submodule119880 of an 119877-module119872 iscalled fully invariant if 119891(119880) is contained in 119880 for every 119877-endomorphism 119891 of119872 A module119872 is called duo if everysubmodule of119872 is fully invariant [9]

Theorem 5 The following statements hold over a ring 119877

(1) Let 119872119894119894isin12119899

be any finite collection ofmodules and119872 = 119872

1oplus 1198722oplus sdot sdot sdot oplus 119872

119899 Then 119872 is generalized

oplus-radical supplemented if for each 1 le 119894 le 119899 119872119894is

generalized oplus-radical supplemented

(2) Let 119872119894119894isin119868

be any infinite collection of modules and =oplus119894isin119868119872119894 If119872 is a duo module then119872 is generalized oplus-

radical supplemented if for each 119894 isin 119868119872119894is generalized

oplus-radical supplementedProof (1) Let 119872

119894be a generalized oplus-radical supplemented

module for each 1 le 119894 le 119899 To prove that119872 is a generalizedoplus-radical supplemented module it is sufficient by inductionon 119899 to prove this is the case when 119899 = 2 Hence suppose that119899 = 2 Let 119873 be any submodule of 119872 with Rad(119872) sube 119873Then 119872 = 119872

1+ 1198722+ 119873 so that 119872

1+ 1198722+ 119873 has a

generalized supplement 0 in119872 Since119872 = 1198721oplus 1198722 then

Rad(1198722) sube 119873+119872

1 It follows that Rad(119872

2) sube 119872

2cap(119873+119872

1)

has a generalized supplement119867 in1198722such that119867 is a direct

summand of1198722 By [10 Proposition 24]119867 is a generalized

supplement of119873 +1198721in119872 Moreover Rad(119872

1) sube 119873 + 119867

Since 1198721is a generalized oplus-radical supplemented module

1198721sube (119873+119867)has a generalized supplement119866 in119872

1such that

119866 is a direct summand of1198721 Again applying [10 Proposition

24]119867 + 119866 is a generalized supplement of119873 in119872 It is clearthat 119867 + 119866 is a direct summand of 119872 Therefore 119872 is ageneralized oplus-radical supplemented module(2) Let119880 be any submodule such that Rad(119872) sube 119880 Then

Rad(119872119894) sube 119880 cap 119872

119894for every 119894 isin 119868 By this hypothesis there

exist submodules119881119894and1198811015840119894of119872119894such that119872

119894= (119880cap119872

119894)+119881119894

(119880cap119872119894)cap119881119894sube Rad(119881

119894) and119872

119894= 119881119894oplus1198811015840

119894for every 119894 isin 119868 Since

119880 is a fully invariant submodule of119872 then119880 = oplus119894isin119868(119880cap119872

119894)

Let119881 = oplus119894isin119868119881119894and1198811015840 = oplus

119894isin1198681198811015840

119894 Then there exist submodules

119880 and 119881 of 119872 such that 119872 = oplus119894isin119868119872119894= [oplus119894isin119868(119880 cap 119872

119894)] +

(oplus119894isin119868119881119894) = 119880+119881119880cap119881 = oplus

119894isin119868(119880cap119872

119894)cap (oplus119894isin119868119881119894) = oplus119894isin119868[(119880cap

119872119894) cap 119881119894] sube oplus119894isin119868Rad(119881

119894) = Rad(119881) by [4 Theorem 216] and

119881 oplus 1198811015840= (oplus119894isin119868119881119894) oplus (oplus

119894isin1198681198811015840

119894) = oplus

119894isin119868119872119894= 119872 Thus119872 is a

generalized oplus-radical supplemented module

Proposition 6 If119872 is a generalized oplus-radical supplementedmodule then 119872119880 is a generalized oplus-radical supplementedmodule for every fully invariant submodule 119880 of119872

Proof Let 119880 be any fully invariant submodule of119872 and let119870119880 be any submodule of 119872119880 with Rad(119872119880) sube 119870119880Since (Rad(119872) + 119880)119880 sube Rad(119872119880) we have Rad(119872) sube 119870By the hypothesis we have119872 = 119870 + 119881 119870 cap 119881 sube Rad(119881)and119872 = 119881oplus1198811015840 for some submodules119881 and1198811015840 of119872 By [11Lemma 23] (119881 + 119880)119880 is generalized supplement of 119870119880 in119872119880 Since 119880 is a fully invariant submodule of119872 we have119880 = (119880cap119881) + (119880 cap119881

1015840) and [(119881 +119880)119880] cap [(1198811015840 +119880)119880] = 0

Then119872119880 = [(119881 + 119880)119880] oplus [(1198811015840 + 119880)119880] Hence119872119880 is ageneralized oplus-radical supplemented module

Corollary 7 Let 119877 be a Dedekind domain and let119872 be an 119877-module119872 is a generalized oplus-radical supplemented module ifand only if119872119875(119872) is a generalized oplus-radical supplementedmodule

Proof We know that 119875(119872) is a fully invariant submodule of119872 So by Proposition 6119872119875(119872) is a generalized oplus-radicalsupplemented module Conversely suppose that119872119875(119872) isa generalized oplus-radical supplemented module Since 119877 is aDedekind domain we have119872 = 119875(119872)oplus119873 for some submod-ule 119873 of119872 By the hypothesis 119873 is a generalized oplus-radical

ISRN Algebra 3

supplemented module Hence 119872 is a generalized oplus-radicalsupplemented module by Theorem 5(1) and Corollary 3

Example 8 Consider the left Z-module Q According toLemma 2Q is a generalizedoplus-radical supplementedmoduleOn the other handQ is not oplus-supplemented because it is nottorsion

Theorem 9 Let 119872 be a module Suppose that Rad (119872) issmall in119872 Then119872 is a generalized oplus-radical supplementedmodule if and only if it is oplus-supplemented

Proof (rArr) Let 119873 be any submodule of119872 Then Rad(119872) subeRad(119872) + 119873 sube 119872 Since 119872 is a generalized oplus-radicalsupplemented module we have 119872 = Rad(119872) + 119873 + 119871(Rad(119872)+119873)cap119871 sube Rad(119871) and119872 = 119871oplus1198711015840 for submodules119871 1198711015840le 119872 Since Rad(119872) ≪ 119872 we have 119872 = 119873 + 119871

119873 cap 119871 ≪ 119871 Therefore119872 is oplus-supplemented(lArr) is trivial

Corollary 10 Every generalizedoplus-radical supplementedmod-ule over a left Bass ring is oplus-supplemented

Proposition 11 Every direct summand of 119875(119872) is a general-ized oplus-radical supplemented module for every 119877-module119872

Proof Let 119873 be any direct summand of 119875(119872) Then thereexists a submodule 119871 of 119875(119872) such that 119875(119872) = 119873oplus119871 Since119875(119872) = Rad(119875(119872)) = Rad(119873)oplusRad(119871) by [4Theorem216]then 119873 = Rad(119873) oplus [119873 cap Rad(119871)] = Rad(119873) By Lemma 2119873 is generalized oplus-radical supplemented

Proposition 12 Let 119872 be a generalized oplus-radical supple-mentedmodule Suppose that a cofinite fully invariant submod-ule 119870 of119872 is a direct summand of119872 Then 119870 is generalizedoplus-radical supplemented

Proof Let119880 be any submodule of119870with Rad(119870) sube 119880 By thehypothesis we have119872 = 119870 oplus 119871 for some finitely generatedsubmodule 119871 of 119872 Then Rad(119871) ≪ 119871 Clearly Rad(119872) sube119880 + Rad(119871) So there exist submodules 119881 and 1198811015840 of119872 suchthat 119872 = 119880 + Rad(119871) + 119881 (119880 + Rad(119871)) cap 119881 sube Rad(119881)and119872 = 119881 oplus 1198811015840 Since Rad(119871) ≪ 119871 we have119872 = 119880 + 119881119880cap119881 sube Rad(119881) and119872 = 119881oplus1198811015840 It follows that119870 = 119880+(119870cap119881) and 119880 cap (119870 cap 119881) sube Rad(119872) Since 119870 is a fully invariantsubmodule of119872 then 119870 = (119870 cap 119881) oplus (119870 cap 1198811015840) Note that119880 cap (119870 cap 119881) sube Rad(119870 cap 119881) Therefore 119870 is generalized oplus-radical supplemented

Lemma 13 Let 119872 be a module and 119873 a submodule suchthat Rad (119872) sube 119873 If 119873 is a direct summand of 119872 thenRad (119872) = Rad (119873) In particular if Rad (119872) is a directsummand of119872 then Rad (119872) = 119875(119872)

Proof Let 119873 be any direct summand of 119872 such thatRad(119872) sube 119873 Then there exists a submodule 1198731015840 of 119872such that 119872 = 119873 oplus 1198731015840 By [4 Theorem 216] Rad(119872) =Rad(119873) oplus Rad(1198731015840) Since Rad(119872) sube 119873 Rad(119872) =[Rad(119873) oplus Rad(1198731015840)] cap 119873 = Rad(119873) oplus [Rad(1198731015840) cap 119873] =Rad(119873) Now we take 119873 = Rad(119872) under the similar

condition Rad(119872) = Rad(Rad(119872)) that is Rad(119872) isradical Consequently Rad(119872) = 119875(119872)

Let 119877 be a ring and let119872 be an 119877-module We considerthe following condition(D3) If119872

1and119872

2are direct summands of119872 with119872 =

1198721+1198722 then119872

1cap1198722is also a direct summand of119872

Proposition 14 Let 119872 be a generalized oplus-radical supple-mented module with (D

3) and let 119873 be a submodule with

Rad (119872) sube 119873 If119873 is a direct summand of119872119873 is generalizedoplus-radical supplemented

Proof Let 119880 be a submodule of 119873 such that Rad(119873) sube 119880By Lemma 13 Rad(119872) = Rad(119873) Since119872 is a generalizedoplus-radical supplemented module there exist submodules119881 and 1198811015840 of 119872 such that 119872 = 119880 + 119881 119880 cap 119881 sube Rad(119881)and119872 = 119881 oplus 1198811015840 Then 119873 = 119880 + (119873 cap 119881) Since119872 satisfies(D3) 119873 cap 119881 is a direct summand of119872 Then there exists a

submodule119883 of119872 such that119872 = (119873cap119881)oplus119883 It follows that119880cap(119873cap119881) sube Rad(119873cap119881) and119873 = (119873cap119881)oplus(119873cap119883)Therefore119873 is a generalized oplus-radical supplemented module

A module 119872 is called local if 119872Rad(119872) is simple andRad(119872) is small in119872 [10]Themodule119872 is called119908-local if119872Rad(119872) is simple [12]

The following fact is clear

Corollary 15 Every 119908-local module is generalized oplus-radicalsupplemented

Lemma 16 Let 119872 be an indecomposable module If 119872 isa generalized oplus-radical supplemented module then 119872 =Rad (119872) or119872 is 119908-local

Proof Suppose that Rad(119872) =119872 Then 119872 contains at leastone maximal submodule say 119870 Since Rad(119872) sube 119870 thenthere exist submodules119881 and 1198811015840 of119872 such that119872 = 119870+119881119870 cap 119881 sube Rad(119881) and119872 = 119881 oplus 1198811015840 We have 119881 = 119872 because119872 is indecomposable Since119881 is a generalized supplement of119870 119881 is 119908-local by [12 Lemma 33] So119872 is 119908-local

Theorem 17 Let 119877 be a local commutative ring and 119872 be auniform 119877-module Every submodule of 119872 is generalized oplus-radical supplemented if and only if it is uniserial

Proof (rArr) By [13 Lemma 62] it suffices to show that everyfinitely generated submodule of 119872 is local Let 119873 be anyfinitely generated submodule of 119872 By assumption 119873 isindecomposable So by Lemma 16 119873 is 119908-local Since 119873 isfinitely generated119873 is local(lArr) Since119872 is uniserial every submodule of119872 is hollow

by [3 217] So it is easy to see that every submodule of119872 isa generalized oplus-radical supplemented module

Corollary 18 Let 119877 be a local commutative ring Supposethat every submodule of119864(119877 Rad (119877)) is generalizedoplus-radicalsupplemented where 119864(119877 Rad (119877)) is the injective hull of thesimple module 119877 Rad (119877) Then 119877 is a uniserial ring

4 ISRN Algebra

Proof Since 119864(119877Rad(119877)) is uniform the hypothesis impliesthat 119864(119877Rad(119877)) is uniserial by Theorem 17 It follows from[13 Lemma 62] that 119877 is a uniserial ring

Theorem19 The following statements are equivalent for a ring119877

(1) 119877 is semiperfect(2) Every finitely generated free119877-module is generalized oplus-

radical supplemented

Proof By [14Theorem 21] andTheorem 9 the proof is clear

Lemma 20 Let119872 be a module Suppose that Rad (119872) = 0Then119872 is a generalized oplus-radical supplemented module if andonly if it is semisimple

Proof This is clear by [15 Proposition 33]

A ring119877 is called left119881-ring if every simple left119877-moduleis injective The ring 119877 is called an SSI ring if semisimpleleft 119877-module is injective Let 119877 be a commutative ring 119877 isregular if and only if every simple left 119877-module is injective[13 Theorem 214]

Proposition 21 Let 119877 be a left 119881-ring and119872 an 119877-moduleThen119872 is generalized oplus-radical supplemented if and only if119872is semisimple

Corollary 22 Let 119877 be a commutative regular ring and 119872an 119877-module Then119872 is generalized oplus-radical supplementedif and only if119872 is semisimple

Proposition 23 The following statements are equivalent for aring 119877

(1) Every generalized oplus-radical supplemented 119877-module isinjective

(2) 119877 is left Noetherian 119881-ring

Proof By [16 Proposition 53] the proof is clear

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Kasch Modules and Rings vol 17 Academic Press LondonUK 1982

[2] A V Kelarev Ring Constructions and Applications vol 9WorldScientific River Edge NJ USA 2002

[3] J Clark C Lomp N Vanaja and R Wisbauer Lifting ModulesSupplements and Projectivity in Module Theory Frontiers inMathematics Birkhauser Basel Switzerland 2006

[4] R Wisbauer Foundations of Module and Ring Theory vol 3Gordon and Breach Philadelphia Pa USA 1991

[5] S H Mohamed and B J Muller Continuous and DiscreteModules vol 147 of London Mathematical Society Lecture NoteCambridge University Press Cambridge UK 1990

[6] H Zoschinger ldquoModuln die in jeder Erweiterung ein Komple-ment habenrdquo Mathematica Scandinavica vol 35 pp 267ndash2871974

[7] W Xue ldquoCharacterizations of semiperfect and perfect ringsrdquoPublicacions Matematiques vol 40 no 1 pp 115ndash125 1996

[8] H Calısıcı and E Turkmen ldquoGeneralized oplus-supplementedmodulesrdquo Algebra and Discrete Mathematics vol 10 no 2 pp10ndash18 2010

[9] A Idelhadj and R Tribak ldquoOn some properties of oplus-supplemented modulesrdquo International Journal of Mathematicsand Mathematical Sciences no 69 pp 4373ndash4387 2003

[10] Y Wang and N Ding ldquoGeneralized supplemented modulesrdquoTaiwanese Journal of Mathematics vol 10 no 6 pp 1589ndash16012006

[11] E Turkmen and A Pancar ldquoOn cofinitely Rad-supplementedmodulesrdquo International Journal of Pure and Applied Mathemat-ics vol 53 no 2 pp 153ndash162 2009

[12] E Buyukasik and C Lomp ldquoOn a recent generalization ofsemiperfect ringsrdquo Bulletin of the Australian MathematicalSociety vol 78 no 2 pp 317ndash325 2008

[13] D W Sharpe and P Vamos Injective Modules CambridgeUniversity Press Cambridge UK 1972

[14] D Keskin P F Smith andW Xue ldquoRings whosemodules areoplus-supplemented modulesrdquo Acta Mathematica Hungarica vol 83pp 161ndash169 1999

[15] K Varadarajan ldquoDual Goldie dimensionrdquo Communications inAlgebra vol 7 no 6 pp 565ndash610 1979

[16] C Lomp ldquoOn semilocal modules and ringsrdquo Communicationsin Algebra vol 27 no 4 pp 1921ndash1935 1999

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2 ISRN Algebra

If119872 is a duomodule then119872 is generalized oplus-radicalsupplemented

(vi) Over a Dedekind domain a module119872 is generalizedoplus-radical supplemented if and only if 119872119875(119872) isgeneralized oplus-radical supplemented

(vii) Every generalized oplus-radical supplemented 119877-moduleis injective if and only if 119877 is left Noetherian 119881-ring

(viii) A ring 119877 is semiperfect if and only if every finitelygenerated free 119877-module is generalized oplus-radicalsupplemented

2 Generalized oplus-RadicalSupplemented Modules

Definition 1 A module 119872 is called generalized oplus-radicalsupplemented if every submodule containing radical has ageneralized supplement that is a direct summand of119872

Recall that a module 119872 is called radical if 119872 has nomaximal submodules that is Rad(119872) = 119872 For a module119872 119875(119872) will indicate the sum of all radical submodules of119872 If 119875(119872) = 0119872 is called reduced Note that 119875(119872) is thelargest radical submodule of119872 [4]

Now we have the following simple fact which lays a keyrole in our study

Lemma 2 Let119872 be an 119877-module If Rad (119872) = 119872 then119872is a generalized oplus-radical supplemented module

Proof Since Rad(119872) = 119872 119872 has the trivial generalizedsupplement 0 in 119872 Consequently 119872 is a generalized oplus-radical supplemented module

Corollary 3 119875(119872) is a generalized oplus-radical supplementedmodule for every 119877-module119872

Proposition 4 Every generalized oplus-radical supplementedmodule has a radical direct summand

Proof Let119872 be a generalized oplus-radical supplemented mod-ule Then there exist submodules 119881 and 1198811015840 of 119872 such that119872 = Rad(119872) + 119881 Rad(119872) cap 119881 sube Rad(119881) and119872 = 119881 oplus 1198811015840Since Rad(119872) = Rad(119881)oplusRad(1198811015840) by [4Theorem 216] then119872 = Rad(1198811015840) + 119881 Applying the modular law we obtain thatRad(1198811015840) = 1198811015840

Recall from [4] that a submodule119880 of an 119877-module119872 iscalled fully invariant if 119891(119880) is contained in 119880 for every 119877-endomorphism 119891 of119872 A module119872 is called duo if everysubmodule of119872 is fully invariant [9]

Theorem 5 The following statements hold over a ring 119877

(1) Let 119872119894119894isin12119899

be any finite collection ofmodules and119872 = 119872

1oplus 1198722oplus sdot sdot sdot oplus 119872

119899 Then 119872 is generalized

oplus-radical supplemented if for each 1 le 119894 le 119899 119872119894is

generalized oplus-radical supplemented

(2) Let 119872119894119894isin119868

be any infinite collection of modules and =oplus119894isin119868119872119894 If119872 is a duo module then119872 is generalized oplus-

radical supplemented if for each 119894 isin 119868119872119894is generalized

oplus-radical supplementedProof (1) Let 119872

119894be a generalized oplus-radical supplemented

module for each 1 le 119894 le 119899 To prove that119872 is a generalizedoplus-radical supplemented module it is sufficient by inductionon 119899 to prove this is the case when 119899 = 2 Hence suppose that119899 = 2 Let 119873 be any submodule of 119872 with Rad(119872) sube 119873Then 119872 = 119872

1+ 1198722+ 119873 so that 119872

1+ 1198722+ 119873 has a

generalized supplement 0 in119872 Since119872 = 1198721oplus 1198722 then

Rad(1198722) sube 119873+119872

1 It follows that Rad(119872

2) sube 119872

2cap(119873+119872

1)

has a generalized supplement119867 in1198722such that119867 is a direct

summand of1198722 By [10 Proposition 24]119867 is a generalized

supplement of119873 +1198721in119872 Moreover Rad(119872

1) sube 119873 + 119867

Since 1198721is a generalized oplus-radical supplemented module

1198721sube (119873+119867)has a generalized supplement119866 in119872

1such that

119866 is a direct summand of1198721 Again applying [10 Proposition

24]119867 + 119866 is a generalized supplement of119873 in119872 It is clearthat 119867 + 119866 is a direct summand of 119872 Therefore 119872 is ageneralized oplus-radical supplemented module(2) Let119880 be any submodule such that Rad(119872) sube 119880 Then

Rad(119872119894) sube 119880 cap 119872

119894for every 119894 isin 119868 By this hypothesis there

exist submodules119881119894and1198811015840119894of119872119894such that119872

119894= (119880cap119872

119894)+119881119894

(119880cap119872119894)cap119881119894sube Rad(119881

119894) and119872

119894= 119881119894oplus1198811015840

119894for every 119894 isin 119868 Since

119880 is a fully invariant submodule of119872 then119880 = oplus119894isin119868(119880cap119872

119894)

Let119881 = oplus119894isin119868119881119894and1198811015840 = oplus

119894isin1198681198811015840

119894 Then there exist submodules

119880 and 119881 of 119872 such that 119872 = oplus119894isin119868119872119894= [oplus119894isin119868(119880 cap 119872

119894)] +

(oplus119894isin119868119881119894) = 119880+119881119880cap119881 = oplus

119894isin119868(119880cap119872

119894)cap (oplus119894isin119868119881119894) = oplus119894isin119868[(119880cap

119872119894) cap 119881119894] sube oplus119894isin119868Rad(119881

119894) = Rad(119881) by [4 Theorem 216] and

119881 oplus 1198811015840= (oplus119894isin119868119881119894) oplus (oplus

119894isin1198681198811015840

119894) = oplus

119894isin119868119872119894= 119872 Thus119872 is a

generalized oplus-radical supplemented module

Proposition 6 If119872 is a generalized oplus-radical supplementedmodule then 119872119880 is a generalized oplus-radical supplementedmodule for every fully invariant submodule 119880 of119872

Proof Let 119880 be any fully invariant submodule of119872 and let119870119880 be any submodule of 119872119880 with Rad(119872119880) sube 119870119880Since (Rad(119872) + 119880)119880 sube Rad(119872119880) we have Rad(119872) sube 119870By the hypothesis we have119872 = 119870 + 119881 119870 cap 119881 sube Rad(119881)and119872 = 119881oplus1198811015840 for some submodules119881 and1198811015840 of119872 By [11Lemma 23] (119881 + 119880)119880 is generalized supplement of 119870119880 in119872119880 Since 119880 is a fully invariant submodule of119872 we have119880 = (119880cap119881) + (119880 cap119881

1015840) and [(119881 +119880)119880] cap [(1198811015840 +119880)119880] = 0

Then119872119880 = [(119881 + 119880)119880] oplus [(1198811015840 + 119880)119880] Hence119872119880 is ageneralized oplus-radical supplemented module

Corollary 7 Let 119877 be a Dedekind domain and let119872 be an 119877-module119872 is a generalized oplus-radical supplemented module ifand only if119872119875(119872) is a generalized oplus-radical supplementedmodule

Proof We know that 119875(119872) is a fully invariant submodule of119872 So by Proposition 6119872119875(119872) is a generalized oplus-radicalsupplemented module Conversely suppose that119872119875(119872) isa generalized oplus-radical supplemented module Since 119877 is aDedekind domain we have119872 = 119875(119872)oplus119873 for some submod-ule 119873 of119872 By the hypothesis 119873 is a generalized oplus-radical

ISRN Algebra 3















1and119872


1198721+1198722 then119872
















4 ISRN Algebra

















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Volume 2014




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Algebra









ISRN Algebra 3















1and119872


1198721+1198722 then119872
















4 ISRN Algebra

















References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 ISRN Algebra

















References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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