,n, thenMis also essential radical supplemented

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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 22 (2021), No. 1, pp. 427–434 DOI: 10.18514/MMN.2021.3215

ESSENTIAL RADICAL SUPPLEMENTED MODULES

CELIL NEBIYEV Received 3 February, 2020

Abstract. In this work, (amply) essential radical supplemented modules are defined and some properties of these modules are investigated. LetMbe anR-module andM=M1+M2+· · ·+ Mn. IfMi is essential radical supplemented for everyi=1,2, . . . ,n, thenMis also essential radical supplemented. It is proved that every factor module and every homomorphic image of an essential radical supplemented module are essential radical supplemented. LetMbe an essential radical supplementedR-module. Then every finitelyM-generatedR-module is essential radical supplemented.

2010Mathematics Subject Classification: 16D10; 16D70

Keywords: small submodules, essential submodules, radical, generalized (radical) supplemented modules

1. INTRODUCTION

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

LetMbe anR-module andN≤M. IfL=Mfor every submoduleLofMsuch that M=N+L, thenN is called asmall (orsuperfluous) submoduleofMand denoted byNM. A submoduleN of anR-moduleMis called anessential submoduleof Mand denoted byNEMin caseK∩N6=0 for every submoduleK6=0, or equivalently,N∩L=0 for L≤Mimplies thatL=0. Let Mbe anR-module and K be a submodule ofM. K is called ageneralized small(briefly,g-small) submoduleofM if for every essential submoduleT ofMwith the propertyM=K+T implies that T=M, then we writeK_gM. It is clear that every small submodule is a generalized small submodule but the converse is not true generally. LetMbe anR-module and U,V ≤M. IfM=U+V andV is minimal with respect to this property, or equivalently,M=U+V andU∩V V, thenV is called asupplementofU inM. M is called asupplemented moduleif every submodule ofMhas a supplement inM. Let M be anR-module andU≤M. If for everyV ≤M such that M=U+V,U has a supplementV⁰withV⁰ ≤V, we sayU hasample supplementsinM. If every submodule ofMhasample supplementsinM, thenMis called anamply supplemented module. If every essential submodule ofMhas a supplement inM, thenMis called

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an essential supplemented (or briefly, e-supplemented) module. If every essential submodules ofMhas ample supplements inM, thenMis called anamply essential supplemented (or briefly, amply e-supplemented) module. Let Mbe an R−module andU,V ≤M. If M=U+V andM=U+T withT EV implies that T =V, or equivalently,M=U+V andU∩V _gV, thenV is called ag-supplementofUinM.

Mis said to be g-supplementedif every submodule ofMhas a g-supplement inM.

The intersection of all maximal submodules of anR-moduleMis called theradical of M and denoted by RadM. If M have no maximal submodules, then we denote RadM=M.The intersection of essential maximal submodules of anR-module M is called thegeneralized radicalofMand denoted by RadgM. IfMhave no essential maximal submodules, then we denote RadgM=M.LetMbe anR-module and U,V ≤M. IfM=U+V andU∩V ≤RadV, thenV is called ageneralized(radical)supplement(or briefly,Rad-supplement) ofU inM. Mis called ageneralized (radical) supplemented (or briefly, Rad-supplemented) module if every submodule ofMhas a Rad-supplement inM. LetMbe anR-module andU≤M. If for every V ≤Msuch thatM=U+V,Uhas a Rad-supplementV⁰withV⁰≤V, we sayUhas ample Rad-supplementsinM. If every submodule ofMhas ample Rad-supplements inM, thenMis called anamply generalized(radical)supplemented(or briefly,amply Rad-supplemented) module. LetM be anR-module. We say submodules X andY ofMareβ^∗ equivalent,Xβ^∗Y, if and only ifY+K=Mfor everyK≤Msuch that X+K=M and X+T =M for every T ≤M such thatY+T =M. Let M be an R-moduleX≤Y ≤M. IfY/XM/X, then we sayY lies above X inM.

More information about (amply) supplemented modules are in [3,9,10] and [11].

More information about (amply) essential supplemented modules are in [5,6]. More results about g-small submodules and g-supplemented modules are in [4,7]. The definitions of (amply) generalized supplemented modules and some properties of them are in [8,10]. Some properties of (amply) generalized supplemented modules are also in [2]. The definition ofβ^∗ equivalence relation and some properties of this relation are in [1].

In this paper, we define (amply) essential radical supplemented modules and in- vestigate some properties about these modules. We constitute relationships between essential radical supplemented modules and amply essential radical supplemented modules by Proposition3and Proposition4. We also constitute relationships between essential radical supplemented modules andπ-projective modules by Lemma12. We give two examples for essential radical supplemented modules separating with essential supplemented modules at the end of this paper.

Lemma 1. Let M be an R-module and K≤N≤M. If K is a generalized small submodule of N, then K is a generalized small submodule in submodules of M which contain N.

Proof. See [4, Lemma 1 (2)].

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Lemma 2. Let M be an R-module. ThenRadgM=∑L_gML.

Proof. See [4, Lemma 5 and Corollary 5].

Lemma 3. Let V be a Rad-supplement of U in M. ThenRadV =V∩RadM.

Proof. LetT be any maximal submodule ofV. Since

M/(U+T) = (U+T+V)/(U+T)∼=V/(U∩V+T) =V/T,

thenU+T is a maximal submodule ofM. Hence RadM≤U+T andV∩RadM≤ U∩V+T =T. Thus V∩RadM≤RadV and since RadV ≤V∩RadM, RadV =

V∩RadM.

2. ESSENTIAL RADICAL SUPPLEMENTED MODULES

Definition 1. Let M be anR-module. If every essential submodule of M has a Rad-supplement inM, thenMis called anessential radical supplemented(or briefly, e-Rad-supplemented)module.

Clearly we see that every essential supplemented module is essential radical supplemented. But the converse is not true in general. (See Examples1and2).

Definition 2. LetMbe anR-module andX≤M. IfX is a Rad-supplement of an essential submodule inM, thenXis called anessential radical supplement(or briefly, e-Rad-supplement)submoduleinM.

Lemma 4. Let M be an R-module, V be an e-Rad-supplement in M and x∈V . Then Rx_gM if and only if Rx_gV .

Proof. (=⇒)Let Rx_gM. SinceV is an e-Rad-supplement in M, there exists U EM such thatV is a Rad-supplement ofU inM. LetRx+T =V withT EV. ThenM=U+V =U+T+Rx, and sinceRx_gMand(U+T)EM,U+T =M.

Letx=u+t with u∈U andt∈T. Sincex,t∈V, thenu=x−t∈V. ThenV = Rx+T ≤Ru+Rt+T =Ru+T≤V andRu+T =V. Sinceu∈U∩V≤RadV, then RuV andT =V. HenceRx_gV.

(⇐=)Clear from Lemma1.

Corollary 1. Let M be an R-module and V be an e-Rad-supplement in M. Then RadgV =V∩RadgM.

Proof. Let x∈RadgV. Here Rx _gV and by Lemma 1, Rx_gM. Then by Lemma2,Rx≤RadgMandx∈V∩RadgM.

Let y∈V∩RadgM. Then y∈V andRy_gM. By Lemma 4, Ry_gV. By Lemma2,Ry≤RadgV andy∈RadgV.

Hence RadgV =V∩RadgM.

Proposition 1. Let M be an essential radical supplemented module. Then M/RadM have no proper essential submodules.

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Proof. Let_RadM^K be any essential submodule of_RadM^M . Since _RadM^K E_RadM^M ,KEM and sinceMis essential radical supplemented,Khas a Rad-supplementV inM. Then M=K+V andK∩V ≤RadV. SinceM=K+V, _RadM^M = _RadM^K +^V^+RadM_RadM . Since K∩V ≤RadM, then _RadM^K ∩^V^+RadM_RadM =^K∩V_RadM^+RadM =0 and _RadM^M = _RadM^K ⊕^V^+RadM_RadM . Since _RadM^M =_RadM^K ⊕^V^+RadM_RadM and_RadM^K E _RadM^M , _RadM^K = _RadM^M . Hence _RadM^M have no

proper essential submodules.

Lemma 5. Let M be an R-module, U be an essential submodule of M and M₁≤M.

If M₁is e-Rad-supplemented and U+M₁has a Rad-supplement in M, then U has a Rad-supplement in M.

Proof. Let X be a Rad-supplement ofU+M₁ in M. Then M=U+M₁+X andX∩(U+M₁)≤RadX. SinceU EM, (U+X)EM and(U+X)∩M₁EM₁. SinceM₁is e-Rad-supplemented,(U+X)∩M₁has a Rad-supplementY inM₁. This case M₁= (U+X)∩M₁+Y and(U+X)∩Y = (U+X)∩M₁∩Y ≤RadY. Then M=U+M₁+X =U+X+ (U+X)∩M₁+Y =U+X+Y andU∩(X+Y)≤ (U+X)∩Y + (U+Y)∩X ≤ (U+M₁)∩X + (U+X)∩Y ≤ RadX +RadY ≤ Rad(X+Y). HenceX+Y is a Rad-supplement ofUinM.

Corollary 2. Let M be an R-module, U be an essential submodule of M and Mi≤ M for every i=1,2, . . . ,n. If Mi is e-Rad-supplemented for every i=1,2, . . . ,n and U+M₁+M₂+· · ·+Mn has a Rad-supplement in M, then U has a Rad-supplement in M.

Proof. Clear from Lemma5.

Lemma 6. Let M=M₁+M₂. If M₁and M₂ are e-Rad-supplemented, then M is also e-Rad-supplemented.

Proof. LetUEM. Then 0 is a Rad-supplement ofU+M₁+M₂inM. SinceM₂ is e-Rad-supplemented and(U+M₁)EM, by Lemma 5,U+M₁ has a Rad-supplement inM. Since M₁ is e-Rad-supplemented andU EM, by Lemma 5,U has a Rad-supplement inM. HenceMis e-Rad-supplemented.

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

Lemma 7. Every factor module of an e-Rad-supplemented module is e-Rad-supplemented.

Proof. LetM be an e-Rad-supplementedR−module and ^M_K be any factor module ofM. Let ^U_K E^M_K. ThenUEM and sinceMis e-Rad-supplemented,U has a Rad-supplementV inM. SinceK≤U, by the proof of [8, Proposition 2.6(1)],^V^+K_K is a Rad-supplement of ^U_K in ^M_K. Hence ^M_K is e-Rad-supplemented.

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Corollary 4. Every homomorphic image of an e-Rad-supplemented module is e-Rad-supplemented.

Lemma 8. Let M be an e-Rad-supplemented R-module. Then every finitely M-generated R-module is e-Rad-supplemented.

Proof. LetN be a finitelyM-generatedR-module. Then there exist a finite index setΛand anR-module epimorphism f:M^(Λ)−→N. SinceMis e-Rad-supplemented, by Corollary3,M^(Λ)is e-Rad-supplemented. Then by Corollary4,Nis e-Rad-supp-

lemented.

Proposition 2. Let R be a ring. ThenRR is essential radical supplemented if and only if every finitely generated R-module is essential radical supplemented.

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

Proof. Let U be an essential submodule of M. By hypothesis there exists an e-Rad-supplement submodule X inM such thatUβ^∗X. SinceX is an e-Rad-supplement submodule inM, there exists an essential submoduleY ofMsuch thatX is a Rad-supplement ofY inM. This caseM=X+Y andX∩Y ≤RadX. SinceY EM, by hypothesis, there exists an e-Rad-supplement submoduleV inMsuch thatYβ^∗V. SinceUβ^∗X andM=X+Y, thenM=U+Y and sinceYβ^∗V, M=U+V. Let x∈U∩V andRx+T =MwithT ≤M. ThenU∩V+T=Mand sinceM=U+V, M=U+V∩T =X+V∩T. SinceM=V+T =X+V∩T,M=V+X∩T. Then by Yβ^∗V, M=Y+X∩T. Since M=X+T =Y +X∩T, M=X∩Y+T. Let x=y+t, withy∈X∩Y andt∈T. SinceRx+T =M,Ry+T =Malso holds. By y∈X∩Y≤RadX≤RadM,RyMand sinceRy+T =M,T =M. HenceRxM andx∈RadM. SinceV is a Rad-supplement inM, then by Lemma3,V∩RadM= RadV. Sincex∈V andx∈RadM,x∈V∩RadM=RadV andU∩V≤RadV. Hence V is a Rad-supplement ofUinMandMis essential radical supplemented.

Corollary 5. Let M be an R-module. If every essential submodule of M lies above an e-Rad-supplement submodule in M, then M is essential radical supplemented.

3. AMPLY ESSENTIAL RADICAL SUPPLEMENTED MODULES

Definition 3. Let M be an R-module. If every essential submodule has ample Rad-supplements inM, thenMis called anamply essential radical supplemented(or briefly,amply e-Rad-supplemented)module.

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Lemma 10. Let M be an amply e-Rad-supplemented module. Then every factor module of M is amply e-Rad-supplemented.

Proof. Let M/K be any factor module of M, U/K EM/K andU/K+V/K= M/KwithV/K≤M/K. SinceU/KEM/K,UEM. SinceU/K+V/K=M/K, U+V =M. BecauseMis amply e-Rad-supplemented,Uhas a Rad-supplementV⁰ inMwithV⁰ ≤V. By the proof of [8, Proposition 2.6(1)],^V

0+K

K is a Rad-supplement of^U_K in^M_K. In addition to this,^V

0+K

K ≤^V_K. HenceM/Kis amply e-Rad-supplemented.

Corollary 6. Let M be an amply e-Rad-supplemented module. Then every homomorphic image of M is amply e-Rad-supplemented.

Lemma 11. Let M be an R-module. If every submodule of M is e-Rad-supplemented, then M is amply e-Rad-supplemented.

Proof. LetM=U+V withUEMandV ≤M. By hypothesis,V is e-Rad-supplemented. SinceU EM, U∩V EV. SinceV is e-Rad-supplemented, U∩V has a Rad-supplementK inV. HereU∩V+K=V andU∩K=U∩V∩K ≤RadK.

ThenM=U+V =U+U∩V+K=U+KandU∩K≤RadK. HenceMis amply

e-Rad-supplemented.

Proposition 3. Let R be any ring. Then every R-module is e-Rad-supplemented if and only if every R-module is amply e-Rad-supplemented.

Proof. (=⇒)LetMbe anyR-module. Since everyR-module is e-Rad-supplemented, every submodule ofMis e-Rad-supplemented. Then by Lemma11,Mis amply e-Rad-supplemented.

(⇐=)Clear.

Lemma 12. Let M be aπ-projective and e-Rad-supplemented R-module. Then M is amply e-Rad-supplemented.

Proof. LetUEM,M=U+V andX be a Rad-supplement ofU inM. Since M isπ-projective andM=U+V, there exists anR-module homomorphism f:M→M such that Imf ⊂V and Im(1−f)⊂U. So, we haveM= f(M) + (1−f) (M) = f(U) +f(X) +U=U+f(X). Suppose thata∈U∩f(X). Sincea∈ f(X), then there existsx∈Xsuch thata= f(x). Sincea=f(x) = f(x)−x+x=x−(1−f) (x) and (1−f) (x)∈U, we have x=a+ (1−f) (x) ∈U. Thus x∈U∩X and so, a= f(x) ∈ f(U∩X). Therefore we have U∩ f(X) ≤ f(U∩X)≤ f(RadX)≤ Radf(X). This means that f(X)is a Rad-supplement ofUinM. Moreover, f(X)⊂

V. ThereforeMis amply e-Rad-supplemented.

Corollary 7. If M is a projective and e-Rad-supplemented module, then M is an amply e-Rad-supplemented module.

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Proposition 4. Let R be a ring. The following assertions are equivalent.

(i) RR is e-Rad-supplemented (ii) _RR is amply e-Rad-supplemented.

(iii) Every finitely generated R-module is e-Rad-supplemented.

(iv) Every finitely generated R-module is amply e-Rad-supplemented.

Proof. (i)⇐⇒(ii)Clear from Corollary7, sinceRRis projective.

(i) =⇒(iii)Clear from Lemma8.

(iii) =⇒(iv) Let M be a finitely generatedR-module. Then there exist a finite index setΛand anR-module epimorphism f:R^(Λ)−→M. Since every finitely generated R-module is e-Rad-supplemented,R^(Λ) is e-Rad-supplemented. SinceRR is projective,R^(Λ)is also projective. Then by Corollary7, R^(Λ)is amply e-Rad-supplemented. Since f:R^(Λ)−→Mis anR-module epimorphism, by Corollary6,Mis also amply e-Rad-supplemented.

(iv) =⇒(i)Clear.

Example1. Consider theZ-moduleQ.Since RadQ=Q,_ZQis essential radical supplemented. But, since_ZQis not supplemented and every nonzero submodule of

ZQis essential in_ZQ,_ZQis not essential supplemented.

Example2. Consider theZ-moduleQ⊕Zpfor a primep. It is easy to check that Rad(Q⊕Zp) =Q6=Q⊕Zp. Since QandZp are essential radical supplemented, by Lemma6,Q⊕Zpis essential radical supplemented. ButQ⊕Zpis not essential supplemented.

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Author’s address

Celil Nebiyev

Ondokuz Mayıs University, 55270, Kurupelit, Atakum, Samsun, Turkey E-mail address:cnebiyev@omu.edu.tr