Vol. 20 (2019), No. 1, pp. 345–352 DOI: 10.18514/MMN.2019.2586
A GENERALIZATION OF g-SUPPLEMENTED MODULES
BERNA KOS¸AR, CELIL NEBIYEV, AND AYTEN PEKIN Received 07 April, 2018
Abstract. In this work g-radical supplemented modules are defined which generalize g- supplemented modules. Some properties of g-radical supplemented modules are investigated.
It is proved that the finite sum of g-radical supplemented modules is g-radical supplemented. It is also proved that every factor module and every homomorphic image of a g-radical supplemen- ted module is g-radical supplemented. LetRbe a ring. ThenRRis g-radical supplemented if and only if every finitely generatedR-module is g-radical supplemented. In the end of this work, it is given two examples for g-radical supplemented modules separating with g-supplemented modules.
2010Mathematics Subject Classification: 16D10; 16D70
Keywords: small submodules, radical, supplemented modules, radical (generalized) supplemen- ted modules
1. INTRODUCTION
Throughout this paper all rings will be associative with identity and all modules will be unital left modules.
LetRbe a ring andM be anR -module. We will denote a submoduleN ofM byN M. LetM be anR-module andN M. IfLDM for every submoduleL ofM such thatM DNCL, thenN is called asmall submoduleofM and denoted byN M. LetM be anR -module andN M. If there exists a submoduleK ofM such thatM DNCK andN\KD0, thenN is called adirect summandof M and it is denoted by M DN˚K. For any module M;we haveM DM˚0.
RadM indicates the radical ofM. A submoduleN of an R -moduleM is called an essential submodule of M, denoted by N EM, in case K\N ¤0 for every submodule K¤0. Let M be an R -module and K be a submodule of M. K is called a generalized small (briefly, g-small) submodule ofM if for everyT EM withM DKCT implies thatT DM, this is written byKgM (in [6], it is called an e-small submoduleofM and denoted byKe M). It is clear that every small submodule is a generalized small submodule but the converse is not true generally.
LetM be anR module. M is called anhollow moduleif every proper submodule ofM is small inM.M is called alocal moduleifM has the largest submodule, i.e.
c 2019 Miskolc University Press
a proper submodule which contains all other proper submodules. Let U andV be submodules ofM. IfM DUCV andV is minimal with respect to this property, or equivalently,M DU CV andU\V V, then V is called asupplementofU in M. M is called asupplemented moduleif every submodule ofM has a supplement inM. LetM be anR-module andU; V M. IfM DUCV andM DUCT with T EV implies thatT DV, or equivalently,M DUCV andU\V gM, thenV is called ag-supplementofU inM.M is calledg-supplementedif every submodule ofM has a g-supplement inM. The intersection of maximal essential submodules of anR-moduleM is called ageneralized radicalofM and denoted byRadgM (in [6], it is denoted byRadeM). IfM have no maximal essential submodules, then we denoteRadgM DM:
Lemma 1 ([2,4,6]). Let M be an R -module and K; L; N; T M. Then the followings are hold.
.1/IfKNandN is generalized small submodule ofM, thenKis a generalized small submodule ofM.
.2/IfK is contained inN and a generalized small submodule ofN, thenK is a generalized small submodule in submodules ofM which contains submoduleN.
.3/LetS be an R-module andf WM !S be anR-module homomorphism. If KgM, thenf .K/gS.
.4/IfKgLandN gT, thenKCN gLCT.
Corollary 1. LetM1; M2; :::; MnM,K1gM1,K2gM2, ...,KngMn. ThenK1CK2C:::CKngM1CM2C:::CMn.
Corollary 2. Let M be an R -module and K N M . If N g M, then N=KgM=K.
Corollary 3. LetM be anR-module,KgM andLM. Then.KCL/ =Lg M=L.
Lemma 2. LetM be anR-module. ThenRadgM DP
LgML:
Proof. See [2].
Lemma 3. The following assertions are hold.
.1/IfM is anR module, thenRmgM for everym2RadgM. .2/IfNM, thenRadgN RadgM:
.3/IfK; LM, thenRadgKCRadgLRadg.KCL/ :
.4/Iff WM !Nis anR-module homomorphism, thenf RadgM
RadgN:
.5/IfK; LM, then RadgLKCLRadgKCL L :
Proof. Clear from Lemma1and Lemma2.
Lemma 4. LetM D ˚i2IMi:ThenRadgM D ˚i2IRadgMi:
Proof. Since Mi M, then by Lemma 3.2/, RadgMi RadgM and
˚i2IRadgMi RadgM: Let x 2 RadgM: Then by Lemma 3.1/, Rxg M:
Since x2M D ˚i2IMi, there exist i1; i2; :::; ik 2I andxi1 2Mi1, xi22Mi2, ..., xik2Mik such thatxDxi1Cxi2C:::Cxik. SinceRxgM , then by Lemma1.4/, under the canonical epimorphism it .tD1; 2; :::; k/ Rxit Dit.Rx/g Rxit: Thenxit2RadgMit .tD1; 2; :::; k/andxDxi1Cxi2C:::Cxik 2 ˚i2IRadgMi. HenceRadgM ˚i2IRadgMi and since˚i2IRadgMi RadgM,RadgM D
˚i2IRadgMi.
2. G-RADICAL SUPPLEMENTED MODULES
Definition 1. LetM be anR-module andU; V M. IfM DUCV andU\V RadgV, thenV is called a generalized radical supplement (briefly, g-radical supple- ment) ofU inM. If every submodule ofM has a generalized radical supplement in M, thenM is called a generalized radical supplemented (briefly, g-radical supple- mented) module.
Clearly we see that every g-supplemented module is g-radical supplemented. But the converse is not true in general. (See Example1and2.)
Lemma 5. LetM be anR-module andU; V M. ThenV is a g-radical supple- ment ofU inM if and only ifM DUCV andRmgV for everym2U\V.
Proof. .)/SinceV is a g-radical supplement ofU inM,M DUCV andU\ V RadgV. Let m2U \V. Since U\V RadgV, m2RadgV. Hence by Lemma3.1/,RmgV.
.(/SinceRmgV for everym2U\V, then by Lemma2,U\V RadgV
and henceV is a g-radical supplement ofU inM.
Lemma 6. LetM be an R-module, M1; U; X M andY M1. IfX is a g- radical supplement ofM1CU inM andY is a g-radical supplement of.U CX /\ M1inM1, thenXCY is a g-radical supplement ofU inM.
Proof. SinceX is a g-radical supplement ofM1CU inM, M DM1CU CX and.M1CU /\XRadgX:SinceY is a g-radical supplement of.U CX /\M1in M1,M1D.UCX /\M1CY and.UCX /\Y D.UCX /\M1\Y RadgY. Then M DM1CU CX D.U CX /\M1CY CU CX DU CXCY and, by Lemma3.3/,U\.XCY /.UCX /\YC.UCY /\XRadgYC.M1CU /\ X RadgY CRadgX Radg.XCY /. HenceXCY is a g-radical supplement
ofU inM.
Lemma 7. LetM DM1CM2. IfM1andM2are g-radical supplemented, then M is also g-radical supplemented.
Proof. LetU M. Then0 is a g-radical supplement of M1CM2CU inM. Since M1 is g-radical supplemented, there exists a g-radical supplement X of
.M2CU /\M1D.M2CUC0/\M1 inM1. Then by Lemma6, XC0DX is a g-radical supplement ofM2CU inM. SinceM2is g-radical supplemented, there exists a g-radical supplementY of.UCX /\M2inM2. Then by Lemma6,XCY
is a g-radical supplement ofU inM.
Corollary 4. LetM DM1CM2C:::CMk. IfMi is g-radical supplemented for everyi D1; 2; :::; k, thenM is also g-radical supplemented.
Proof. Clear from Lemma7.
Lemma 8. LetM be anR module,U; V M andKU. IfV is a g-radical supplement ofU inM, then.V CK/ =Kis a g-radical supplement ofU=KinM=K. Proof. Since V is a g-radical supplement of U in M, M DU CV and U \ V RadgV. Then M=K DU=KC.V CK/ =K and by Lemma 3.5/, .U=K/\ ..V CK/ =K/ D .U \VCK/ =K RadgVCK
=K RadgŒ.V CK/ =K.
Hence.VCK/ =K is a g-radical supplement ofU=KinM=K.
Lemma 9. Every factor module of a g-radical supplemented module is g-radical supplemented.
Proof. Clear from Lemma8.
Corollary 5. The homomorphic image of a g-radical supplemented module is g- radical supplemented.
Proof. Clear from Lemma9.
Lemma 10. Let M be a g-radical supplemented module. Then every finitely M generated module is g-radical supplemented.
Proof. Clear from Corollary4and Corollary5.
Corollary 6. LetRbe a ring. ThenRRis g-radical supplemented if and only if every finitely generatedR module is g-radical supplemented.
Proof. Clear from Lemma10.
Theorem 1. Let M be an R module. If M is g-radical supplemented, then M=RadgM is semisimple.
Proof. LetU=RadgM M=RadgM. SinceM is g-radical supplemented, there exists a g-radical supplementV ofU in M. ThenM DUCV andU\V RadgV. ThusM=RadgM DU=RadgMC VCRadgM
=RadgM and U=RadgM
\ V CRadgM
=RadgM
D U\V CRadgM
=RadgM RadgVCRadgM
=RadgM DRadgM=RadgM D0:
HenceM=RadgM DU=RadgM˚ VCRadgM
=RadgM andU=RadgM is a
direct summand ofM.
Lemma 11. LetM be a g-radical supplemented module andLM withL\ RadgM D0. ThenLis semisimple. In particular, a g-radical supplemented module M withRadgM D0is semisimple.
Proof. LetX L. Since M is g-radical supplemented, there exists a g-radical supplement T ofX inM. Hence M DXCT andX\T RadgT RadgM. SinceM DXCT andXL, by Modular Law,LDL\M DL\.XCT /DXC L\T. SinceX\T RadgM andL\RadgM D0,X\L\T DL\X\T L\RadgM D0. HenceLDX˚L\T andX is a direct summand ofL.
Proposition 1. LetM be a g-radical supplemented module. ThenM DK˚Lfor some semisimple moduleKand some moduleLwith essential generalized radical.
Proof. Let K be a complement of RadgM in M: Then by [5, 17.6], K˚RadgM EM. SinceK\RadgM D0, then by Lemma11,K is semisimple.
SinceM is g-radical supplemented, there exists a g-radical supplementLofKinM. HenceM DKCLandK\LRadgLRadgM. Then byK\RadgM D0, K\LD0. HenceM DK˚L:SinceM DK˚L, then by Lemma4,RadgM D RadgK˚RadgL. Hence K˚RadgM DK˚RadgL. Since K˚RadgLD K˚RadgM EM DK˚L, then by [1, Proposition 5.20],RadgLEL.
Proposition 2. LetM be anR module andU M. The following statements are equivalent.
.1/There is a decompositionM DX˚Y withX U andU\Y RadgY. .2/There exists an idempotente2End .M /withe .M /U and.1 e/ .U / Radg.1 e/ .M /.
.3/There exists a direct summandXofM withXU andU=XRadg.M=X /.
.4/ U has a g-radical supplementY such thatU\Y is a direct summand ofU. Proof. .1/).2/ For a decompositionM DX˚Y, there exists an idempotent e 2End .M / withX De .M /and Y D.1 e/ .M /. Sincee .M /DX U, we easily see that.1 e/ .U /DU\.1 e/ .M /. Then byY D.1 e/ .M /andU\Y RadgY,.1 e/ .U /DU \.1 e/ .M /DU \Y RadgY DRadg.1 e/ .M /.
.2/).3/ LetX De .M /and Y D.1 e/ .M /. Since e 2End .M / is idem- potent, we easily see thatM DX˚Y. ThenM DUCY. Sincee .M /DX U, we easily see that.1 e/ .U /DU\.1 e/ .M /. SinceM DUCY andU\Y D U\.1 e/ .M /D.1 e/ .U /Radg.1 e/ .M /DRadgY,Y is a g-radical sup- plement ofU inM. Then by Lemma8,M=X D.Y CX / =X is a g-radical supple- ment ofU=X inM=X. HenceU=XD.U=X /\.M=X /Radg.M=X /.
.3/).4/ Let M DX˚Y. Since X U, M DU CY. Let t 2U \Y and RtCT DY for an essential submoduleT ofY. Let..TCX / =X /\.L=X /D0for a submoduleL=XofM=X. Then.L\TCX / =XD..TCX / =X /\.L=X /D0and L\T CX DX. HenceL\T X and sinceX\Y D0,L\T \Y X\Y D0.
Since L\Y \T DL\T \Y D0 and T EY, L\Y D0. Since X L and
M DXCY, by Modular Law,LDL\M DL\.XCY /DXCL\Y DXC0D X. Hence L=XD0 and.TCX / =XEM=X. Since RtCT DY, R.tCX /C .TCX / =XD.RtCX / =XC.TCX / =XD.RtCT CX / =XD.Y CX / =XD M=X. Sincet2U,tCX2U=X Radg.M=X /and henceR .tCX /gM=X. Then byR.tCX /C.TCX / =XDM=Xand.TCX / =XEM=X,.TCX / =XD M=X and then XCT DM. Since XCT DM and T Y, by Modular Law, Y DY \M DY \.XCT /DX\Y CT D0CT DT. HenceRt gY and by Lemma5,Y is a g-radical supplement ofU inM. SinceM DX˚Y andXU, by Modular Law,U DU\M DU \.X˚Y /DX˚U \Y. HenceU \Y is a direct summand ofU.
.4/).1/ LetU DX˚U\Y for a submoduleX ofU. SinceY is a g-radical supplement of U in M, M DU CY andU \Y g Y. Hence M DU CY D
.X˚U \Y /CY DX˚Y.
Lemma 12. LetV be a g-radical supplement ofU in M:If U is a generalized maximal submodule ofM, thenU\V is a unique generalized maximal submodule ofV.
Proof. Since U is a generalized maximal submodule of M and V = .U\V /' .V CU / =U DM=U, U \V is a generalized maximal submodule of V. Hence RadgV U\V and sinceU\V RadgV,RadgV DU \V. ThusU\V is a
unique generalized maximal submodule ofV.
Definition 2. LetM be anR module. If every proper essential submodule ofM is generalized small inM orM has no proper essential submodules, thenM is called a generalized hollow module.
Clearly we see that every hollow module is generalized hollow.
Definition 3. LetM be anR module. If M has a large proper essential sub- module which contain all essential submodules ofM orM has no proper essential submodules, thenM is called a generalized local module.
Clearly we see that every local module is generalized local.
Proposition 3. LetM be anR module andRadgM ¤M. ThenM is general- ized hollow if and only ifM is generalized local.
Proof. .H)/Let M be generalized hollow and letL be a proper essential sub- module ofM. ThenLgM and by Lemma2,LRadgM. ThusRadgM is a proper essential submodule ofM which contain all proper essential submodules of M.
.(H/ Let M be a generalized local module, T be the largest proper essential submodule ofM andLbe a proper essential submodule ofM. LetLCSDM with SEM. IfS¤M, thenLCS T ¤M:ThusSDM andLgM:
Definition 4. LetM be anR-module andU; V M. If M DU CV andU \ V gM, then V is called a weak g-supplement ofU inM. If every submodule ofM has a weak g-supplement inM, then M is called a weakly g-supplemented module. (See [3]).
Clearly we can see that if M is a weakly g-supplemented module, thenM is g- semilocal (M=RadgM is semisimple, see [3]).
Proposition 4. Generalized hollow and generalized local modules are weakly g- supplemented, so are g-semilocal.
Proof. Clear from definitions.
Proposition 5. LetM be a g-radical supplemented module withRadgM gM. ThenM is weakly g-supplemented.
Proof. Clear from definitions.
Example 1. Consider theZ module Q: Since RadgQDRadQDQ, ZQ is g-radical supplemented. But, sinceZQis not supplemented and every nonzero sub- module ofZQis essential inZQ,ZQis not g-supplemented.
Example2. Consider theZ moduleQ˚Zp2 for a primep. It is easy to check thatRadgZp2 ¤Zp2. By Lemma4,Radg Q˚Zp2
DRadgQ˚RadgZp2¤ Q˚Zp2. SinceQandZp2 are g-radical supplemented, by Lemma7,Q˚Zp2 is g-radical supplemented. ButQ˚Zp2is not g-supplemented.
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Authors’ addresses
Berna Kos¸ar
Department of Mathematics, Ondokuz Mayıs University, 55270, Kurupelit-Atakum, Samsun, Tur- key
E-mail address:bernak@omu.edu.tr
Celil Nebiyev
Department of Mathematics, Ondokuz Mayıs University, 55270, Kurupelit-Atakum, Samsun, Tur- key
E-mail address:cnebiyev@omu.edu.tr
Ayten Pekin
Department of Mathematics, ˙Istanbul University, ˙Istanbul, Turkey E-mail address:aypekin@istanbul.edu.tr
