We give some characterizations ofµ-co-H-rings by weaklyµ-extending modules

Tomus 48 (2012), 183–196

ON µ-SINGULAR AND µ-EXTENDING MODULES

Yahya Talebi and Ali Reza Moniri Hamzekolaee

Abstract. LetM be a module andµbe a class of modules in Mod−R which is closed under isomorphisms and submodules. As a generalization of essential submodules Özcan in [8] defines aµ-essential submodule provided it has a non-zero intersection with any non-zero submodule inµ. We define and investigateµ-singular modules. We also introduceµ-extending and weakly µ-extending modules and mainly study weaklyµ-extending modules. We give some characterizations ofµ-co-H-rings by weaklyµ-extending modules.

Let Rbe a right non-µ-singular ring such that all injective modules are non-µ-singular, thenRis rightµ-co-H-ring if and only ifRis a QF-ring.

1. Introduction

Let R be a ring with identity. All modules we consider are unitary right R-modules and we denote the category of all such modules by Mod−R.

Letµbe a class of modules. For any moduleM thetrace ofµ inM is denoted by Tr(µ, M) =P{Imf:f ∈Hom(C, M), C ∈µ}. Dually thereject ofµ inM is denoted by Rej(M, µ) =T{Kerf :f ∈Hom(M, C), C∈µ}.

Let N be a submodule of M (N ≤ M). The notations N M, N ≤e M and N≤dM is used for a small submodule, an essential submodule and a direct summand of M, respectively. Soc(M) will denote the socle ofM. An R-module M is said to besmall, ifM ∼=LK for someR-modulesLandK. Dually,M is called singular if M ∼=N/K such thatK ≤e N. Every moduleM contains a largest singular submodule which is denoted byZ(M). ThenZ(M) = Tr(U, M) whereU denotes the class of all singular modules.

Simple modules split into four disjoint classes by combining the exclusive choices [injective or small] and [projective or singular]. Also note that if a moduleM is singular and projective, then it is zero.

Talebi and Vanaja in [10], define cosingular modules as a dual of singular modules. LetM be a module andMdenotes the class of all small modules. Then Z(M) = T{kerg | g ∈ Hom(M, L), L ∈ M} is a submodule of M. Then M is calledcosingular (non-cosingular) ifZ(M) = 0 (Z(M) =M). Every small module

2010Mathematics Subject Classification: primary 16S90; secondary 16D10, 16D70, 16D99.

Key words and phrases:µ-essential submodule,µ-singular module,µ-extending module, weakly µ-extending module.

Received April 6, 2011, revised June 25, 2012. Editor J. Trlifaj.

DOI: 10.5817/AM2012-3-183

is cosingular. The class of all cosingular modules is closed under submodules, direct sums and direct products ([10, Corollary 2.2]).

In Section 2, we give the definition of µ-singular modules and discuss some properties of such modules. It is proved that Ris a GCO-ring (i.e. every simple singular module is injective) if and only if for everyM-singular moduleN,Z(N) = N if and only if for every δ-singular module N, Z(N) = N (Corollary 2.14).

When we consider the class of all finitely cogenerated modules F C we prove that every finitely cogenerated R-module is projective if and only if for every F C-singular R-moduleN, Rej(N,F C) =N if and only ifR is semisimple Artinian (Corollary 2.16).

In Section 3, we defineµ-extending and weakly µ-extending modules and show that any direct summand of a weaklyµ-extending module and any homomorphic image of a weaklyµ-extending module are weaklyµ-extending modules (Proposi- tion 3.12 and Corollary 3.13).

In Section 4, we discuss when a direct sum of weaklyµ-extending modules is a weaklyµ-extending module. We show that a direct sum of a µ-singular module and a semisimple module is weakly µ-extending (Theorem 4.2).

In Section 5, we study rings in which every projective module isµ-extending. We call such ringsµ-co-H-ring. We show that a ringRisµ-co-H-ring if and only if every R-module is weaklyµ-extending (Theorem 5.3). LetR be a right non-µ-singular ring such that all injective modules are non-µ-singular, thenR is rightµ-co-H-ring if and only if Ris a QF-ring (Corollary 5.6).

In this paperµwill be a class in Mod−Rwhich is closed under isomorphisms and submodules, unless otherwise stated. We shall call any member ofµ, aµ-module.

In this article we denote the following classes:

S ={M ∈Mod−R, M is simple}, M={M ∈Mod−R, M is small}, δ={M ∈Mod−R, Z(M) = 0},

µ−Sing ={M ∈Mod−R, M isµ-singular}, F C={M ∈Mod−R, M is finitely cogenerated}.

2. µ-singular modules

Özcan in [8], investigate some properties ofµ-essential submodules. LetM be a module andN ≤M. ThenN is called aµ-essential submodule ofM, denoted byN ≤_µeM, ifN∩K6= 0 for any nonzero submoduleK ofM such thatK∈µ.

Now we list the properties ofµ-essential submodules. We omit the proofs because they are similar to those for essential submodules (see, [4]).

Lemma 2.1. Let M be a module.

a) Let N ≤L≤M. ThenN ≤µeM if and only if N≤µeL≤µeM. b) IfK₁≤µeL₁≤M,K₂≤µeL₂≤M, thenK₁∩K₂≤µeL₁∩L₂. c) If f: N→M is a homomorphism andK≤_µeM, thenf⁻¹(K)≤_µeN. d) IfN/L≤_µeK/L≤M/L, thenN≤_µeK.

e) LetK_i (i∈I) be an independent family of submodules ofM. IfK_i≤_µeL_i≤M for all i∈I, thenL

i∈IKi≤µeL

i∈ILi.

Definition 2.2. Let M be a module. M is called µ-singular ifM ∼=K/Lsuch that L≤µeK.

Every module M contains a largestµ-singular submodule which we denote by Zµ(M) = Tr(µ−Sing, M) whereµ−Sing is the class of all µ-singular modules.

Then Z(M) ≤ Z_µ(M). If M is a µ-singular module (i.e. Z_µ(M) = M) and a µ-module, then M is singular. For, let M ∈ µ and M ∼=K/L where L ≤µe K.

We claim that L ≤e K. Let 0 6= X ≤ K and assume that L∩X = 0. Then X ∼= (L⊕X)/L≤K/Land soX ∈µ. Since L≤_µe K we have a contradiction.

This proves thatM is singular. IfZ_µ(M) = 0, then M is callednon-µ-singular.

Proposition 2.3. Let M be a µ-singular module and f ∈HomR(R, M). Then Kerf ≤µeR.

Proof. By assumption, f(R)∼=L/K whereK≤µeL. SinceRis projective, there exists a homomorphism g:R → L such that πg = f where π is the natural epimorphismL→L/K. Then kerf =g⁻¹(K)≤µeR by Lemma 2.1.

Proposition 2.4. Let P be a projective module and X ≤ P. Then P/X is µ-singular if and only ifX ≤µeP.

Proof. If I≤RR andR/I isµ-singular, thenI≤µeRby Proposition 2.3. Now let P/X beµ-singular and assumeX µeP. LetF be a free module such that F =P⊕P⁰, P⁰ ≤F. ThenF/(X⊕P⁰)∼=P/X isµ-singular and X⊕P⁰µeF.

So we may assume that P is free i.e. P =L

Ri, each Ri is a copy of R. Then Ri/(Ri∩X)∼= (Ri+X)/X≤P/X isµ-singular. SoRi∩X ≤µeRi. This implies that (L

Ri)∩X ≤µeL

Ri=P, i. e. X≤µeP. Lemma 2.5. Let M be a module. Then Zµ(M) ={x∈M |xI= 0, I ≤µeR}.

Proof. LetxI = 0 for someI≤µeR. ThenR/Iisµ-singular. Definef:R/I→xR byr+I7→xr. Hence,x∈Tr(µ−Sing, M). Conversely assume thatx=x1+· · ·+ xn=f1(l1)+· · ·+fn(ln) andxi∈Imfiwherefi:Li→M such thatLiisµ-singular.

For each i we have liR ∼=R/ann(li) which implies that Ii = ann(li) ≤µe R by Proposition 2.4. TakeI=Tn

i=1Ii. ThenI≤µeR by Lemma 2.1 andxI = 0. This

completes the proof.

Proposition 2.6. A moduleM is non-µ-singular if and only ifHomR(N, M) = 0 for all µ-singular modulesN.

Proof. See [4, Proposition 1.20].

Proposition 2.7. Let M be a non-µ-singular module and N ≤M. ThenM/N is µ-singular if and only ifN ≤_µeM.

Proof. If M/N is µ-singular andxis a nonzero element ofM. ThenxI = 0 for someI≤µeR. So,xI ≤N. SinceM is non-µ-singular, we have xI6= 0 and thus xR∩N 6= 0. Therefore,N ≤µeM. Proposition 2.8. (1) The class of all non-µ-singular modules is closed under submodules, direct products,µ-essential extension and module extension.

(2) The class of all µ-singular modules are closed under submodules, factor modules and direct sums.

Proof. It follows from Lemma 2.5 and [4, Proposition 1.22].

Proposition 2.9. Assume that R is a right non-µ-singular ring, then:

(1) Z_µ(M/Z_µ(M)) = 0for any R-moduleM.

(2) An R-module M is µ-singular if and only if Hom_R(M, N) = 0 for all non-µ-singular modulesN.

(3) The class of all µ-singular modules is closed under module extension and µ-essential extension.

(4) The set of allµ-essential right ideals ofRdenoted byP(R)is closed under finite products.

Proof. The proof is easy by [4, Proposition 1.23] and Lemma 2.5.

Proposition 2.10. LetM be a simple module. Then M is either µ-singular or projective, but not both.

Proof. See [4, Proposition 1.24].

It is easy to see that a ringRis right non-µ-singular if and only if all projective rightR-modules are non-µ-singular.

From the properties ofµ-singular modules and Proposition 2.4 the following can be seen easily.

Proposition 2.11. For anR-moduleM the following are equivalent:

(1) M isµ-singular:

(2) M ∼=F/K with F a projective (free) module andK≤µeF;

(3) For everym∈M, the right annihilatorannr(m) isµ-essential inR.

Lemma 2.12. LetM be a module. IfZµ(M) = 0andK≤cM, thenZµ(M/K) = 0.

Theorem 2.13. Let µ be closed under factor modules. Then the following are equivalent:

(1) Everyµ-module is projective;

(2) For every singular moduleN,Rej(N, µ) =N;

(3) For everyµ-singular moduleN,Rej(N, µ) =N; (4) For every simple singular moduleN,Rej(N, µ) =N.

Proof. (1)⇒(2) LetN be a singular module andg:N →LwhereL∈µ. Then N/kerg ∈ µ. By (1), N/kerg is projective. Since N is singular, we have that N = kerg. Hence Rej(N, µ) =N.

(2)⇒(3) LetN be a µ-singular module andg:N →La homomorphism where L∈µ. ThenN/kerg∈µ. This implies that Rej(N/kerg, µ) = 0. SinceN/kerg is µ-singular and a µ-module, it is singular. Then by (2), N = kerg. Hence Rej(N, µ) =N.

(3)⇒(2) and (2)⇒(4) are clear.

(4)⇒(1) LetN be aµ-module. We claim thatN is semisimple. Letx∈N andK a maximal submodule of xR. ThenxR/K is a simpleµ-module. By (4), it cannot be singular. HencexR/K is projective. This implies thatK is a direct summand ofxR. HenceN is semisimple. By above process every simple submodule ofN is

projective. It follows thatN is projective.

If we consider the class M of all small modules, we have a characterization of GCO-rings. A ringR is called aGCO-ring if every simple singular module is injective.

Corollary 2.14. The following are equivalent for a ringR:

(1) Every small module is projective;

(2) Every singular module is non-cosingular;

(3) EveryM-singular module is non-cosingular;

(4) Ris a GCO-ring;

(5) Everyδ-singular module is non-cosingular.

Proof. (1)⇔(4) is by [7, Theorem 1.5]. (2)⇔(4) is by [9, Theorem 4.1].

Simple modules are either injective or small. Hence (1)–(4) are equivalent by Theorem 2.13.

(5)⇒(2) is clear.

(3)⇒(5) It is clear sinceM ⊆δ, everyδ-singular module isM-singular.

For the classδof all cosingular modules we have the following corollary.

Corollary 2.15. If the class δ is closed under the factor modules the following are equivalent:

(1) Every cosingular module is projective;

(2) For every singular moduleN,Rej(N, δ) =N; (3) For everyδ-singular module N,Rej(N, δ) =N;

(4) Ris a GCO-ring.

Proof. See [9, Theorems 4.1 and 4.2] and Theorem 2.13.

A module M is calledfinitely cogenerated if Soc(M) is finitely generated and essential submodule ofM. LetF Cbe the class of all finitely cogeneratedR-modules.

Note thatF C is closed under submodules. We next give a characterization of semi- simple Artinian rings which is taken from [8]. We give the proof for completeness.

Corollary 2.16. The following statements are equivalent for a ringR:

(1) Every finitely cogeneratedR-module is projective;

(2) For every singular moduleN,Rej(N,F C) =N;

(3) For everyF C-singular moduleN,Rej(N,F C) =N; (4) Ris semisimple Artinian.

Proof. (1)⇒(2)⇔(3) By Theorem 2.13.

(4)⇒(1) is clear.

(2)⇒(4) LetEbe an essential right ideal ofR. Suppose thatais an element of R buta /∈E. Let F be a right ideal ofRmaximal with respect to the properties that Eis contained inF anda /∈F. Then (aR+F)/F is simple singular. By (2), we have a contradiction. HenceR is semisimple Artinian.

A ringR is aquasi-Frobenius ring (briefly QF-ring) if and only if every right R-module is a direct sum of an injective module and a singular module. In this result we may takeµ-singular modules instead of singular as the following result shows.

Theorem 2.17. The following are equivalent for a ringR:

(1) Ris a QF-ring;

(2)Every rightR-module is a direct sum of an injective module and aµ-singular module.

Proof. (1)⇒(2) is clear.

(2) ⇒ (1) Let M be a projective R-module. Then M is a direct sum of an injective module and aµ-singular module. Since projectiveµ-singular modules are

zero,M is injective. ThenRis a QF-ring.

3. µ-extending modules

In this sectionµ-extending modules will be introduced. Then we define and study weakly µ-extending modules. It is proved that any factor module, any direct summand and any fully invariant submodule of a weakly µ-extending module are weaklyµ-extending.

Definition 3.1. LetM be a module. Then M is called an µ-extending module if for every submoduleN ofM there exists a direct summandD ofM such that N ≤µeD.

Clearly every essential submodule is µ-essential. Soµ-extending modules are a generalization of extending modules.

Note that by [5, Proposition 2.4], a moduleM is extending if and only if every closed submodule is direct summand. This may not be true for a µ-extending module.

Let M be a module and K a submodule of M. Then K is called aµ-closed submodule, denoted byK≤µcM, providedK≤µeL≤M impliesK=L, i.e.K doesn’t have any properµ-essential extension. A µ-closed submodule is closed but the converse is true whenM is a µ-module (see [8, Corollary 1.1]).

Proposition 3.2. The following statements hold for a module M.

(1)IfK≤µcM, then wheneverQ≤µeM such that K⊆Q, thenQ/K ≤µeM/K. (2)If L≤µeM, thenL/K≤µeM/K.

Proof. (1) SupposeK≤µcM. LetQ≤µeM such thatK⊆Q. LetP/K≤M/K be aµ-module such that (Q/K)∩(P/K) = 0. By Lemma 2.1(b),K=Q∩P ≤µeP and henceK=P. ThusQ/K≤µeM/K.

(2) It is clear by Lemma 2.1(d).

The following proposition is clear by definitions.

Proposition 3.3. LetM be aµ-extending module. Then everyµ-closed submodule is a direct summand.

We next give an equivalent condition for aµ-extending module.

Proposition 3.4. Let M be a module. ThenM is µ-extending if and only if for each submodule A of M there exists a decomposition M = M₁⊕M₂ such that A≤M₁ andA+M₂≤µeM₂.

Proof. Let M be µ-extending andA ≤M. Then there exists a decomposition M =M₁⊕M₂ such thatA≤µe M₁. Since {A, M2} is an independent family of submodules ofM the result follows from Lemma 2.1.

The converse is clear.

A moduleM is calledµ-uniformif every proper nonzero submodule isµ-essential in M.

Proposition 3.5. An indecomposable moduleM isµ-extending if and only if M isµ-uniform.

Definition 3.6. LetM be a module. ThenM is calledweaklyµ-extending if for every submoduleN ofM there exists a direct summandKofM such thatN ≤K andK/N isµ-singular.

The definition shows that everyµ-extending module is weaklyµ-extending. Also any µ-singular module is weaklyµ-extending.

Let M be aµ-singular module with unique composition seriesM ⊃U ⊃V ⊃0.

By [2], N=M ⊕(U/V) is not extending. ButN is weakly µ-extending.

We next give some equivalent conditions for weaklyµ-extending modules.

Proposition 3.7. The following are equivalent for a moduleM: (1) M is weaklyµ-extending;

(2) For every N ≤ M there exists a decomposition M = K⊕K⁰ such that N ≤K andM/(K⁰+N) isµ-singular;

(3) For everyN ≤M there exists a decomposition M/N=K/N⊕K⁰/N such that K≤d M andM/K⁰ isµ-singular;

(4)For everyN ≤M, there exists a direct summand K ofM such thatN ≤K and for any x∈K there is a right idealI withI≤µeR such thatxI ≤N.

Proposition 3.8. LetM be a non-µ-singular or projective module. Then,M is µ-extending if and only if M is weaklyµ-extending.

Proof. It is easy by Propositions 2.7 and 2.4.

Some special submodules of a weaklyµ-extending module are weaklyµ-extending.

Recall that a submodule N of M is called fully invariant iff(N)⊆N for each f ∈End(M). A moduleM is called aduo module, if every submodule ofM is fully invariant.

Proposition 3.9. Let N ≤ M be fully invariant and M a weakly µ-extending module. Then N is weaklyµ-extending.

Proof. LetL≤N ≤M. By assumption, there exists a decompositionM =K⊕K⁰ such that L ≤ K and K/L is µ-singular. Since N is fully invariant, we have N = (N ∩K)⊕(N ∩K⁰). Obviously, L ≤ N ∩K and (N ∩K)/L ≤ K/L is µ-singular. HenceN is weaklyµ-extending.

The Proposition 3.9 shows that every submodule of a duo module or of a multiplication weakly µ-extending module is weaklyµ-extending.

Proposition 3.10. Let M be a module andN a submodule of M.

(1)If M is weaklyµ-extending and the intersection ofN with any direct summand of M is a direct summand of N, thenN is weaklyµ-extending.

(2) IfN is weaklyµ-extending andD a direct summand of M such that (D+ N)/Dis non-µ-singular, thenD∩N is a direct summand ofN.

(3)If M is weaklyµ-extending and(D+N)/Dis non-µ-singular for any direct summandD of M, then N is weakly µ-extending if and only ifD∩N is a direct summand ofN for any direct summand D ofM.

Proof. (1) It is similar to the proof of Proposition 3.9.

(2) LetY =D∩N. SinceN is weaklyµ-extending, there is a direct summand K of N such that K/Y is µ-singular. By assumption, N/Y ∼= (D +N)/D is non-µ-singular. Hence, K/Y ≤ N/Y is both µ-singular and non-µ-singular. It follows that K=Y is a direct summand ofN.

(3) It is a consequence of (1) and (2).

The following proposition shows the equivalent condition of a cyclic submodule of a module to be weaklyµ-extending over a right weaklyµ-extending ring.

Proposition 3.11. Let R be a right weakly µ-extending ring and M a cyclic right R-module such that every nonzero direct summand of M contains a nonzero µ-module. Then the following are equivalent:

(1) M is non-µ-singular;

(2) Every cyclic submodule ofM is projective and weaklyµ-extending;

(3) Every cyclic submodule ofM is projective.

Proof. (1)⇒(2) Suppose thatM is non-µ-singular andN a cyclic submodule of M. Then there is a right idealI ofRsuch thatN ∼=R/I. SinceRisµ-extending and N is non-µ-singular, I is a µ-closed submodule of RR, hence I is a direct summand ofRR. ThusN is isomorphic to a direct summand ofRR. Therefore,N is projective and weakly µ-extending.

(2)⇒(3) It is clear.

(3)⇒(1) For anym∈Z_µ(M),mRis projective and is isomorphic toR/ann_r(m), where ann_r(m) is the right annihilator ofm. SinceR is right weaklyµ-extending and mR is µ-singular, then ann_r(m) ≤_µe R is a direct summand of R. Then, R= ann_r(m)⊕L. By assumption, ifL6= 0 then it contains a nonzeroµ-module.

Hence, annr(m) =R andm= 0. Hence,Zµ(M) = 0.

Any factor module of a µ-singular module isµ-singular and we show that any image of a weaklyµ-extending module is weaklyµ-extending. The direct summand of aµ-extending module may not beµ-extending. For weaklyµ-extending modules, we first show the following proposition and then show that any direct summand of a weaklyµ-extending module is weaklyµ-extending.

Proposition 3.12. LetM be a weaklyµ-extending module. Then any homomorphic image ofM is weaklyµ-extending.

Proof. Letf:M →N be an epimorphism andL a submodule ofN. Then there is a submoduleH ofM such thatL∼=H/Kerf. SinceM is weaklyµ-extending,

there are direct summandsK,K⁰ ofM such thatM =K⊕K⁰,H ≤K and that K/H is µ-singular. So N ∼=M/Kerf = (K/Kerf)⊕(K⁰+ Kerf)/Kerf and L∼=H/Kerf ≤K/Kerf. Since (K/Kerf)/(H/Kerf)∼=K/H is µ-singular,N

is weakly µ-extending.

Corollary 3.13. (1) Let M be a weakly µ-extending module. Then any direct summand ofM is weaklyµ-extending.

(2) Let M be a µ-extending module. Then any non-µ-singular homomorphic image ofM isµ-extending.

Corollary 3.14. The following are equivalent:

(1) Every(resp., finitely generated)module is weaklyµ-extending;

(2) Every(resp., finitely generated)projective module is weaklyµ-extending.

Proposition 3.15. Let R be a right non-µ-singular ring and f:M → M⁰ an epimorphism. Suppose that M⁰ is weakly µ-extending and Kerf is µ-singular injective, thenM is weaklyµ-extending.

Proof. LetN be a submodule ofM. First, we assume that Kerf ⊆N ≤M, then f(N)≤M⁰. SinceM⁰is weaklyµ-extending, there is a decomposition,M⁰=K⊕H such thatK/f(N) isµ-singular. SoM =f⁻¹(K) +f⁻¹(H). Since Kerf ≤f⁻¹(H) and Kerf is injective, thenf⁻¹(H) =T⊕Kerf for some submoduleT off⁻¹(H).

Thus M = f⁻¹(K) +T. Since f⁻¹(K)∩T ≤ f⁻¹(K)∩f⁻¹(H) = Kerf and f⁻¹(K)∩T ≤Kerf∩T = 0, we haveM =f⁻¹(K)⊕T andN ≤f⁻¹(K).

For anyx∈f⁻¹(K),f(x)∈K and there is an µ-essential right idealI ofR such that f(x)I ≤ f(N). It is easy to see that xI ≤ N and that f⁻¹(K)/N is µ-singular.

Now we assume that N does not contain Kerf. Set L = N + Kerf, then f(L) = f(N). As the case above, there is a decomposition M = f⁻¹(K)⊕T such that f⁻¹(K)/Lis µ-singular. Since Kerf isµ-singular, we have that (N+ Kerf)/N ∼= Kerf /(N∩Kerf) is µ-singular. SinceR is right non-µ-singular, by Proposition 2.9 we have thatf⁻¹(K)/N isµ-singular. In either case,M is weakly

µ-extending.

Proposition 3.16. LetRbe a right non-µ-singular ring andM a weaklyµ-extending module. ThenM =Z_µ(M)⊕T for some µ-extending submoduleT ofM and T is Z_µ(M)-injective.

Proof. IfZ_µ(M) = 0 orZ_µ(M) =M, it is clear.

Suppose that 0< Z_µ(M)< M. SinceM is weaklyµ-extending, there are direct summandsK,T ofM such thatM =K⊕T,Z_µ(M)≤K and thatK/Z_µ(M) is µ-singular. SoK isµ-singular. SinceZ_µ(M) =Z_µ(K)⊕Z_µ(T) =K⊕Z_µ(T), so Z_µ(M) =K andT is non-µ-singular. By Proposition 3.12,T isµ-extending.

Since for any submoduleN ofZµ(M), HomR(N, T) = 0, soTisZµ(M)-injective,

as required.

Corollary 3.17. LetR be a right non-µ-singular ring andM an injective module.

ThenZµ(M)is injective.

Corollary 3.18. Let R be a right non-µ-singular ring and M an indecompo- sable weakly µ-extending module. Then M is either a µ-singular module or a non-µ-singular µ-uniform module.

Proposition 3.19. LetM be a weaklyµ-extending module which contains maximal submodules. Then for any maximal submoduleN of M, eitherM/N isµ-singular or M =N⊕S for some simple submoduleS ofM.

Proof. Let N be a maximal submodule of M and suppose that M/N is not µ-singular. ThenN is a direct summand ofM, i.e.,M =N⊕Sfor some submodule

S ofM. Since S∼=M/N, so S is simple.

A moduleM is calledlocalif it has a largest submodule, i.e., a proper submodule which contains all other proper submodules. For a local moduleM, Rad(M), the Jacobson radical ofM is small inM.

Corollary 3.20. LetM be a local weakly µ-extending module. ThenM/Rad(M) isµ-singular.

Proposition 3.21. LetR be a right hereditary ring and M an injective module.

Then any factor module ofM is a direct sum of an injective module and aµ-singular injective module.

Proof. LetLbe any factor module ofM, then there is a submoduleN ofM such thatL∼=M/N. Since any injective module is weaklyµ-extending, there are direct summandsK, K⁰ofM such thatM =K⊕K⁰,N ≤Kand thatK/Nisµ-singular.

SoL∼=M/N =K/N⊕(K⁰+N)/N. SinceR is hereditary andM is injective, so M/N is injective. ThusK/N is aµ-singular injective module and (K⁰+N)/N is

injective.

4. Direct sum of weaklyµ-extending modules

A direct sum of µ-singular modules is also µ-singular. But a direct sum of µ-extending modules may not beµ-extending. Also a direct sum of weaklyµ-extending modules need not be weaklyµ-extending (see [1, Example 2.4]).

It may be interesting to see when a direct sum of weakly µ-extending modules is weakly µ-extending.

Proposition 4.1. Let M =L

i∈IMi be a distributive module. ThenM is weakly µ-extending if and only if each Mi is weaklyµ-extending for i∈I.

Proof. Let N be any submodule of M, thenN =L

i∈I(N∩Mi). SinceMi is weaklyµ-extending, there is direct summandHi≤dMi, such thatMi =Hi⊕H_i⁰ and (N∩Mi) ≤Hi and that Hi/(N∩Mi) is µ-singular for i∈ I. Hence M = (L

i∈IHi)⊕(L

i∈IH_i⁰) and (N = L

i∈I(N ∩Mi)) ≤ (H = L

i∈IHi). Since

H N =

L

i∈IHi

L

i∈I(N∩Mi)

∼=L

i∈I H_i

N∩Mi isµ-singular, soM is weakly µ-extending.

Theorem 4.2. LetM =M1⊕M2 withM1 beingµ-singular (µ-uniform)and M2

semisimple. ThenM is weaklyµ-extending.

Proof. LetN be any submodule ofM. ThenN+M₁=M₁⊕[(N+M₁)∩M₂].

Since M₂ is semisimple, then (N +M₁)∩M₂ is a direct summand of M₂ and thereforeN+M₁is a direct summand ofM. Note that (N+M₁)/N∼=M₁/(N∩M₁) isµ-singular, sinceM₁ isµ-singular (µ-uniform). SoM is weakly µ-extending.

Proposition 4.3. LetM =M₁⊕M₂ with M₁ being weaklyµ-extending and M₂ semisimple. Suppose that for any submoduleN ofM,N∩M₁ is a direct summand of N. ThenM is weaklyµ-extending.

Proof. Let N be any submodule of M. As in Theorem 4.2,N+M1 is a direct summand of M. By the hypothesis,N = (N∩M1)⊕K for some submoduleK of N. Since M1 is weaklyµ-extending, there is a direct summandT ofM1such thatT /(N∩M1) isµ-singular. ButN+M1= (N∩M1) +K+M1=M1⊕K, so (T⊕K)/N= (T⊕K)/[(N∩M1)⊕K]∼=T /(N∩M1)⊕K/K isµ-singular. Since T⊕K is a direct summand ofN+M1and hence a direct summand of M, then

M is weakly µ-extending.

Proposition 4.4. LetM =M₁⊕M₂ with M₁ being weaklyµ-extending and M₂ injective. Suppose that for any submodule N of M, we have N∩M₂ is a direct summand ofN, thenM is weaklyµ-extending.

Proof. LetN ≤M. By the hypothesis, there is a submoduleN⁰ of N such that N = (N ∩M₂)⊕N⁰. Note thatN⁰∩M₂ = 0 and hence (M₂+N⁰)/N⁰ ∼=M₂ is an injective module, so there is a submodule M⁰ of M containing N⁰ such that M/N⁰ = [(M2+N⁰)/N⁰]⊕(M⁰/N⁰). Thus it is easy to see thatM =M2⊕M⁰ and thatM⁰ ∼=M/M2∼=M1. HenceM⁰ is weaklyµ-extending. There are direct summands K, K⁰ of M⁰ such that M⁰ = K⊕K⁰ and that K/N⁰ is µ-singular.

Since N ∩M2 is a submodule of an injective module M2, so there is a direct summandH ofM2such thatH/(N∩M2) isµ-singular. Following from the fact that (H⊕K)/[(N∩M2)⊕N⁰]∼= [H/(N∩M2)]⊕(K/N⁰) and thatH⊕K≤dM,

thenM is weakly µ-extending.

Proposition 4.5. Let M = M1⊕M2 such that M1 is weakly µ-extending and M2 is an injective module. Then M is weaklyµ-extending if and only if for every submoduleN ofM such thatN∩M₂6= 0, there is a direct summandK ofM such that K/N isµ-singular.

Proof. Suppose that for every submoduleN ofM such thatN∩M₂6= 0, there is a direct summandKofM such thatK/N isµ-singular. LetN be a submodule ofM such thatN∩M₂= 0. Then, since (M₂+N)/N∼=M₂is an injective module, there is a submoduleM⁰ofM containingN such thatM/N= (M⁰/N)⊕((M₂+N)/N).

It is easy to see thatM =M⁰⊕M2. SinceM⁰∼=M/M2∼=M1is weaklyµ-extending, there is a direct summandK ofM⁰, hence ofM, such thatK/N isµ-singular. So

M is weakly µ-extending. The converse is obvious.

5. Rings whose projective modules are µ-extending

In [6], a ringR is called aright co-H-ringif every projective rightR-module is extending. It is known that a ring Ris a right co-H-ring if and only if R is right

P-extending (i.e., any direct sum ofR_Ris extending). In this section we introduce rings in which all projective right modules are µ-extending. We call such rings µ-co-H-rings. It is easy to check that a ringR is µ-co-H-ring if and only if any direct sum ofR_Risµ-extending.

Lemma 5.1. Let Rbe a ring. A projectiveR-moduleM is weaklyµ-extending if and only if every factor module of M is a direct sum of a µ-singular module and a projective module.

Proof. Suppose that M is weaklyµ-extending. LetM⁰ be any factor module of M, then there is a submodule N of M such thatM/N∼=M⁰. SinceM is weakly µ-extending, then there are direct summandsK, K⁰ ofM such thatM =K⊕K⁰ andK/N isµ-singular. ThusM/N= (K/N)⊕((K⁰+N)/N). As M is projective, K⁰∼= (K⁰+N)/N is projective. Conversely, letN be any submodule ofM, then M/N is a direct sum of a µ-singular module and a projective module. We may assume thatM/N =S/N⊕T /N, whereS/N isµ-singular andT /N is projective.

ThenM =S+T and asM/S∼=T /N is projective,S is a direct summand ofM.

ThusM is weaklyµ-extending.

Lemma 5.2. Let R be any right non-µ-singular ring. Then the following are equivalent:

(1) All modules are weaklyµ-extending;

(2) All projective modules are weaklyµ-extending;

(3) All non-µ-singular modules are µ-extending.

Proof. (1)⇔(2) By Corollary 3.14.

(1)⇔(3) This is a consequence of Propositions 3.8 and 3.12 and the fact that over a right non-µ-singular ring all projective modules are non-µ-singular.

As an immediate consequence of Lemmas 5.1, 5.2 and Proposition 3.8, we have:

Theorem 5.3. Let R be any ring, then the following are equivalent:

(1) Ris a right µ-co-H-ring;

(2) All rightR-modules are weaklyµ-extending;

(3) All projective rightR-modules areµ-extending;

(4) All projective rightR-modules are weaklyµ-extending;

(5) Every factor module of any projective module is a direct sum of a µ-singular module and a projective module.

Theorem 5.4. Let R be a right non-µ-singular ring, consider the following:

(1) Ris a right µ-co-H-ring;

(2) Every non-µ-singular module is projective;

(3) Every module is weaklyµ-extending;

(4) Every non-µ-singular module is µ-extending.

Then(1)⇔(3)⇔(4)and(1)⇒(2).

Proof. (1)⇒(2) Suppose thatRis a rightµ-co-H-ring andM a non-µ-singular module. Then there is a projective moduleP and an epimorphismf:P →M. Set K=Kerf, thenK is aµ-closed submodule ofP. SinceP isµ-extending, thenK

is a direct summand ofP and henceM is isomorphic to a direct summand ofP.

ThusM is projective.

(1)⇔(3) By Theorem 5.3.

(1)⇒(4) It is clear by Propositions 3.8 and 3.12.

(4) ⇒(1) Since a ring R is right non-µ-singular if and only if all projective modules are non-µ-singular, by (4), all projective modules areµ-extending andR

is a rightµ-co-H-ring.

Corollary 5.5. LetR be a ring such that all µ-singular modules are projective, then Ris a right µ-co-H-ring if and only ifR is semisimple.

Proof. Suppose that R is a right µ-co-H-ring. LetM be anR-module module andN a submodule of M, then by Theorem 5.3, M is weakly µ-extending, i.e., there is a direct summandK ofM such thatN ≤K and K/N isµ-singular. By hypothesis,K/N is projective, soN is a direct summand ofK and hence a direct summand ofM. ThusM is semisimple andRis semisimple.

The converse is obvious.

It is known from [3, Theorem 24.20] that a ringR is a QF-ring if and only if all projective modules are injective if and only if all injective modules are projective.

Obviously every QF-ring R is a left and right µ-co-H-ring. As an immediate consequence of Theorem 5.4, we have:

Corollary 5.6. LetRbe a right non-µ-singular ring such that all injective modules are non-µ-singular. ThenR is a right µ-co-H-ring if and only ifR is a QF-ring.

References

[1] Chatters, A. W., Khuri, S. M.,Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc.21(2) (1980), 434–444.

[2] Dung, N. V., Huynh, D. V., Smith, P. F., Wisbauer, R.,Extending Modules, Pitman, London, 1994.

[3] Faith, C.,Algebra II: Ring Theory, Springer–Verlag Berlin–Heidelberg–New York, 1976.

[4] Goodearl, K. R.,Ring Theory, Marcel Dekker, New York – Basel, 1976.

[5] Mohamed, S. H., Müller, B. J.,Continuous and Discrete Modules, London Math. Soc.147 (1990).

[6] Oshiro, K.,Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J.13(1984), 310–338.

[7] Özcan, A. Ç.,On GCO–modules and M–small modules, Comm. Fac. Sci. Univ. Ankara Ser.

A151(2) (2002), 25–36.

[8] Özcan, A. Ç., On µ–essential and µ–M–singular modules, Proceedings of the Fifth China–Japan–Korea Conference, Tokyo, Japan, 2007, pp. 272–283.

[9] Özcan, A. Ç.,The torsion theory cogenerated byδ–M–small modules and GCO–modules, Comm. Algebra35(2007), 623–633.

[10] Talebi, Y., Vanaja, N.,The torsion theory cogenerated by M–small modules, Comm. Algebra 30(3) (2002), 1449–1460.

Department of Mathematics, Faculty of Mathematical Science, University of Mazandaran, Babolsar, Iran

E-mail:[email protected] [email protected]