LetMbe an eg-radical supplemented module

Vol. 22 (2021), No. 2, pp. 799–805 DOI: 10.18514/MMN.2021.3661

ESSENTIAL G-RADICAL SUPPLEMENTED MODULES

CELIL NEBIYEV AND HASAN H ¨USEYIN ¨OKTEN Received 18 January, 2021

Abstract. LetMbe anR-module. If every essential submodule ofMhas a g-radical supplement in M, thenM is called an essential g-radical supplemented (or briefly eg-radical supplemen- ted) module. In this work, some properties of these modules are investigated. It is proved that every factor module and every homomorphic image of an eg-radical supplemented module are eg-radical supplemented. LetMbe an eg-radical supplemented module. Then every finitely M-generatedR-module is eg-radical supplemented.

2010Mathematics Subject Classification: 16D10; 16D70

Keywords: essential submodules, g-small submodules, generalized radical, g-supplemented mod- ules

1. INTRODUCTION

Throughout this paper all rings are associative with identity and all modules are unital left modules.

Let R be a ring and M be an R-module. We denote a submodule N of M by N ≤M. A module M is said to besimple if M have no submodules with distinct from 0 andM. The sum of all simple submodules of a moduleMis called thesocle of M and denoted by Soc(M). M is called a semisimple module, ifM is a direct sum of simple modules(it is equivalent to Soc(M) =M). LetMbe anR-module and N≤M. IfL=Mfor every submoduleLofMsuch thatM=N+L, thenNis called a small (orsuperfluous) submodule of M and denoted by N≪M. A submodule N of an R-moduleM is called anessential submodule, denoted byN⊴M, in case K∩N̸=0 for every submoduleK̸=0, or equvalently,N∩L=0 forL≤Mimplies thatL=0. LetM be anR-module andK be a submodule ofM. K is called agen- eralized small(brieflyg-small) submodule ofMif for every essential submodule T ofMwith the propertyM=K+T implies thatT =M, we denote this byK≪_gM (in [13], it is called an e-small submoduleofM and denoted byK≪_eM). Let M be an R-module and U,V ≤M. If M=U+V andV is minimal with respect to this property, or equivalently, M=U+V andU∩V ≪V, thenV is called a sup- plement ofU inM. M is said to be supplemented if every submodule ofM has a supplement in M. Mis said to be essential supplemented(brieflye-supplemented)

© 2021 Miskolc University Press

if every essential submodule ofM has a supplement in M. LetM be anR-module andU,V ≤M. If M=U+V andM=U+T withT ⊴V implies that T =V, or equivalently,M=U+V andU∩V ≪_gV, thenV is called ag-supplementofU in M. Mis said to beg-supplementedif every submodule ofM has a g-supplement in M. Mis said to beessential g-supplementedif every essential submodule ofMhas a g-supplement in M. The intersection of all maximal submodules of anR-module M is called theradical ofM and denoted by Rad(M). IfM have no maximal sub- modules, then we denote Rad(M) =M.Let M be anR-module andU,V ≤M. If M=U+VandU∩V≤Rad(V), thenVis called ageneralized(Radical)supplement (brieflyRad-supplement)ofU inM. Mis said to begeneralized(Radical) supple- mented(brieflyRad-supplemented)if every submodule ofMhas a Rad-supplement in M. The intersection of all essential maximal submodules of an R-moduleM is called the generalized radical (brieflyg-radical) of M and denoted by RadgM (in [13] it is denoted by RadeM). If M have no essential maximal submodules, then we denote RadgM=M. LetMbe anR-module andU,V ≤M. IfM=U+V and U∩V ≤RadgV, thenV is called ag-radical supplementofU inM. Mis said to be g-radical supplemented if every submodule ofMhas a g-radical supplement inM.

LetMbe anR-module andK≤V≤M. We sayV lies above KinMifV/K≪M/K.

More details about supplemented modules are in [2,11]. More details about essen- tial supplemented modules are in [7,8]. More informations about g-small submodules and g-supplemented modules are in [3,4]. The definition of essential g-supplemented modules and some properties of them are in [5]. More details about generalized (Rad- ical) supplemented modules are in [10,12]. The definition of g-radical supplemented modules and some properties of them are in [4].

Lemma 1([4, Lemma 3]). The following assertions are hold.

(1) If N≤M, thenRadgN≤RadgM.

(2) If K,L≤M, thenRadgK+RadgL≤Radg(K+L).

(3) If f:M−→N is an R-module homomorphism, then f(RadgM)≤RadgN.

2. ESSENTIAL G-RADICAL SUPPLEMENTED MODULES

Definition 1([6, Definition 1]). Let Mbe an R-module. If every essential sub- module ofMhas a g-radical supplement inM, thenMis called an essential g-radical supplemented (or briefly eg-radical supplemented) module.

Clearly we can see that every essential g-supplemented module is eg-radical sup- plemented. But the converse is not true in general (see Example1and Example2).

Every g-radical supplemented module is eg-radical supplemented.

Proposition 1. Let M be an eg-radical supplemented R-module. If every nonzero submodule of M is essential in M, then M is g-radical supplemented.

Proof. Clear from definitions. □

Lemma 2. Let M be an eg-radical supplemented module. Then M/RadgM have no proper essential submodules.

Proof. LetU/RadgM⊴M/RadgM. ThenU⊴Mand sinceMis eg-radical supple- mented,Uhas a g-radical supplementVinM. HereM=U+VandU∩V≤RadgV≤ RadgM. ThenM/RadgM= (U+V)/RadgM=U/RadgM+ (V+RadgM)/RadgM and U/RadgM ∩ (V+RadgM)/RadgM = (U∩V+RadgM)/RadgM = RadgM/RadgM=0. Hence M/RadgM=U/RadgM⊕(V+RadgM)/RadgM and sinceU/RadgM⊴M/RadgM, U/RadgM=M/RadgM. Thus M/RadgM have no

proper essential submodules. □

Corollary 1. Let M be an eg-radical supplemented module. Then M/RadgM is semisimple.

Proof. Since M is eg-radical supplemented, by Lemma 2, M/RadgM have no proper essential submodules. Then by [11, Section 21.1], Soc(M/RadgM) =M/RadgM

andM/RadgMis semisimple. □

Corollary 2. Let M be an essential g-supplemented module. Then M/RadgM is semisimple.

Proof. Clear from Corollary1. □

Lemma 3. Let M be an R-module, U ⊴M and N≤M. If U+N has a g-radical supplement in M and N is eg-radical supplemented, then U has a g-radical supple- ment in M.

Proof. LetX be a g-radical supplement ofU+N inM. SinceU⊴M,U+X ⊴ M and (U+X)∩N ⊴N. Since N is eg-radical supplemented, (U+X)∩N has a g-radical supplementY in N. Since X is a g-radical supplement of U+N inM, M=U+N+X and(U+N)∩X ≤RadgX. SinceY is a g-radical supplement of (U+X)∩NinN,N= (U+X)∩N+Y and(U+X)∩Y= (U+X)∩N∩Y≤RadgY. ThenM=U+N+X=U+ (U+X)∩N+Y+X =U+X+Y and, by Lemma1, U∩(X+Y)≤(U+X)∩Y+ (U+Y)∩X≤RadgY+ (U+N)∩X≤RadgY+RadgX

≤Radg(X+Y). HenceX+Y is a g-radical supplement ofUinM. □ Lemma 4. Let M=M₁+M₂. If M₁and M₂are eg-radical supplemented, then M is also eg-radical supplemented.

Proof. Let U ⊴M. Then U+M₁ ⊴M and since U+M₁+M₂ has a trivial g-radical supplement 0 inMandM₂is eg-radical supplemented, by Lemma3,U+M1

has a g-radical supplement inM. SinceM₁is eg-radical supplemented andU ⊴M, by Lemma3again,U has a g-radical supplement inM. HenceMis eg-radical sup-

plemented. □

Corollary 3. Let M=M₁+M₂+· · ·+Mn. If Miis eg-radical supplemented for every i=1,2, . . . ,n, then M is also eg-radical supplemented.

Proof. Clear from Lemma4. □ Lemma 5. Let f:M→N be an R-module epimorphism, U,V≤M andKer(f)≤ U . If V is a g-radical supplement of U in M, then f(V)is a g-radical supplement of

f(U)in N.

Proof. SinceV is a g-radical supplement ofU in M, M=U+V andU∩V ≤ RadgV. ThenN= f(M) = f(U+V) = f(U) +f(V). Letx∈ f(U)∩f(V). Then there existu∈U andv∈V withx= f(u) =f(v). Here f(v−u) =f(v)−f(u) =0 andv−u∈Ker(f)≤U. Thenv=v−u+u∈U and sincev∈V,v∈U∩V. Hence x= f(v)∈ f(U∩V) and f(U)∩f(V)≤ f(U∩V). Here clearly we can see that f(U∩V)≤ f(U)∩f(V)and f(U)∩f(V) = f(U∩V). SinceU∩V ≤RadgV, by Lemma 1, f(U)∩f(V) = f(U∩V)≤ f(RadgV)≤Radgf(V). Hence f(V) is a

g-radical supplement of f(U)inN, as desired. □

Lemma 6. Every homomorphic image of an eg-radical supplemented module is eg-radical supplemented.

Proof. LetMbe an eg-radical supplemented R-module and f:M→N be anR- module epimorphism. LetU⊴N. By [11, Section 17.3 (3)], f⁻¹(U)⊴Mand since M is eg-radical supplemented, f⁻¹(U) has a g-radical supplementV inM. Since ker(f)≤ f⁻¹(U), by Lemma5, f(V)is a g-radical supplement of f f⁻¹(U)

=U inN. HenceNis eg-radical supplemented, as desired. □ Corollary 4. Every factor module of an eg-radical supplemented module is eg- radical supplemented.

Proof. Clear from Lemma6. □

Lemma 7. Let M be an eg-radical supplemented R-module. Then every finitely M-generated R-module is eg-radical supplemented.

Proof. LetN be a finitelyM-generatedR-module. Then there exist a finite index set Λand an R-module epimorphism f: M^(Λ)→N. Since M is eg-radical supple- mented, by Corollary3, M^(Λ)is eg-radical supplemented. Then by Lemma6, N is

eg-radical supplemented, as desired. □

Proposition 2. Let R be a ring. Then the R-moduleRR is eg-radical supplemented if and only if every finitely generated R-module is eg-radical supplemented.

Proof. (=⇒)Clear from Lemma7.

(⇐=)Clear, sinceRRis finitely generated. □

Definition 2. LetMbe anR-module andX≤M. IfXis a g-radical supplement of an essential submodule ofM, then X is called an eg-radical supplement submodule inM.

Let M be an R-module. It is defined the relation β^∗ on the set of submodules of anR-module M by Xβ^∗Y if and only ifY+K=M for every K ≤M such that X+K=M andX+T =M for everyT ≤Msuch thatY+T =M(see [1]). It is defined the relationβ^∗_gon the set of submodules of anR-moduleMbyXβ^∗_gY if and only ifY+K=Mfor everyK⊴Msuch thatX+K=MandX+T =Mfor every T ⊴Msuch thatY+T =M(see [9]).

Lemma 8. Let M be an R-module. If every essential submodule of M isβ^∗_g equi- valent to an eg-radical supplement submodule in M, then M is eg-radical supplemen- ted.

Proof. LetX⊴M. By hypothesis, there exists an eg-radical supplement submod- ule V in M with Xβ^∗_g V. Let V be a g-radical supplement of an essential submoduleUinM. ThenM=U+V andU∩V ≤RadgV. SinceU⊴M, by hypo- thesis, there exists an eg-radical supplement submoduleYinMwithUβ^∗_gY. LetYbe a g-radical supplement of S in M and S ⊴ M. Then M = S +Y and S∩Y ≤RadgY. Since Xβ^∗_g V and M=U+V, M =X+U and since Uβ^∗_gY and X⊴M,M=X+Y. AssumeX∩Y≰RadgY. Then there exists an essential maximal submoduleTofY such thatX∩Y+T=Y. By using [2, Lemma 1.24], we can see that M = S + Y = X ∩ Y + S + T = Y + X ∩ (S+T) = U + X ∩ (S+T)

=X+U∩(S+T) =V+U∩(S+T) =U∩V+S+T. Since T is an essential maximal submodule ofY, by_S+T^M =^Y+S+T_S+T ∼=_Y∩(S+T)^Y =_S∩Y+T^Y =^Y_T andS+T⊴M, S+T is an essential maximal submodule ofMand henceU∩V ≤RadgV ≤S+T. ThenM=U∩V+S+T =S+T. This is a contradiction. HenceX∩Y≤RadgY and Y is a g-radical supplement ofX inM. ThusMis eg-radical supplemented. □ Corollary 5. Let M be an R-module. If every essential submodule of M is β^∗ equivalent to an eg-radical supplement submodule in M, then M is eg-radical sup- plemented.

Proof. Clear from Lemma8. □

Corollary 6. Let M be an R-module. If every essential submodule of M lies above an eg-radical supplement submodule in M, then M is eg-radical supplemented.

Proof. Clear from Corollary5. □

Corollary 7. Let M be an R-module. If every essential submodule of M is an eg-radical supplement submodule in M, then M is eg-radical supplemented.

Proof. Clear from Corollary6. □

Lemma 9. Let M be an R-module. If every submodule of M isβ^∗equivalent to an eg-radical supplement submodule in M, then M is g-radical supplemented.

Proof. LetX≤M. By hypothesis, there exists an eg-radical supplement submod- uleV inMwithXβ^∗V. LetV be a g-radical supplement of an essential submoduleU

inM. ThenM=U+V andU∩V≤RadgV. By hypothesis, there exists an eg-radical supplement submoduleY inMwithUβ^∗Y. LetY be a g-radical supplement of an es- sential submoduleSinM. ThenM=S+Y andS∩Y ≤RadgY. SinceXβ^∗V and M=U+V,M=X+Uand sinceUβ^∗Y,M=X+Y. AssumeX∩Y ≰RadgY. Then there exists an essential maximal submoduleT ofY such thatX∩Y+T =Y. By us- ing [2, Lemma 1.24], we can see thatM=S+Y=X∩Y+S+T =Y+X∩(S+T) = U+X∩(S+T) =X+U∩(S+T) =V+U∩(S+T) =U∩V+S+T. SinceT is an essential maximal submodule ofY, by _S+T^M = ^Y+S+T_S+T ∼= _Y∩(S+T)^Y = _S∩Y+T^Y = ^Y_T andS+T ⊴M,S+T is an essential maximal submodule ofMand henceU∩V ≤ RadgV ≤S+T. ThenM=U∩V+S+T =S+T. This is a contradiction. Hence X∩Y ≤RadgY andY is a g-radical supplement of X in M. Thus M is g-radical

supplemented. □

Corollary 8. Let M be an R-module. If every submodule of M lies above an eg- radical supplement submodule in M, then M is g-radical supplemented.

Proof. Clear from Lemma9. □

Corollary 9. Let M be an R-module. If every submodule of M is an eg-radical supplement submodule in M, then M is g-radical supplemented.

Proof. Clear from Lemma9. □

Example1. Consider theZ-moduleQ.Since RadgQ=Rad(Q) =Q,_ZQis eg- radical supplemented. But, since_ZQ is not supplemented and every nonzero sub- module of_ZQis essential in_ZQ,_ZQis not essential g-supplemented.

Example2. Consider theZ-moduleQ⊕Zp² for a primep. It is easy to check that RadgZp² ̸=Zp². By [4, Lemma 4], Radg Q⊕Zp²

=RadgQ⊕RadgZp² ̸=Q⊕Zp². Since QandZp² are eg-radical supplemented, by Lemma4, Q⊕Zp² is eg-radical supplemented. ButQ⊕Zp² is not essential g-supplemented.

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Authors’ addresses

Celil Nebiyev

(Corresponding author) Ondokuz Mayıs University, Department of Mathematics, Atakum, Sam- sun, Turkey

E-mail address:cnebiyev@omu.edu.tr

Hasan H ¨useyin ¨Okten

Amasya University, Technical Sciences Vocational School, Amasya, Turkey E-mail address:hokten@gmail.com