S-Injective Modules and Rings
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Advances in Pure Mathematics, 2014, 4, 25-33 Published Online January 2014 (http://www.scirp.org/journal/apm) http://dx.doi.org/10.4236/apm.2014.41004 S-Injective Modules and Rings Nasr A. Zeyada1,2 1Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt 2Department of Mathematics, Faculty of Science, King Abdulaziz University, KSA Email: [email protected] Received October 7, 2013; revised November 7, 2013; accepted November 15, 2013 Copyright © 2014 Nasr A. Zeyada. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intellectual property Nasr A. Zeyada. All Copyright © 2014 are guarded by law and by SCIRP as a guardian. ABSTRACT We introduce and investigate the concept of s-injective modules and strongly s-injective modules. New characte- rizations of SI-rings, GV-rings and pseudo-Frobenius rings are given in terms of s-injectivity of their modules. KEYWORDS S-Injective Module; Kasch Ring; Pseudo Frobenius Ring 1. Introduction In this paper we introduce and investigate the notion of s-injective Modules and Rings. A right R-module M is called right s-N-injective, where N is a right R-module, if every R-homomorphism fK: → M extends to N , where K is a submodule of the singular submodule ZN( ) . M is called strongly s-injective if M is s-N-injective for every right R-module N. The connection between this new injectivity condition and other injectivity conditions has been established, and examples are provided to distinguish s-injectivity from other injectivity concepts such as mininjectivity, soc-injectivity. Several properties of this new class of injectivity are highlighted. Throughout this paper all rings are associative with identity, and all modules are unitary R-modules. For a right R-module M R , we denoted JM( ) , soc( M ) , and ZM( ) by the Jacobson radical, the socle and the singular submodule of M , respectively. Sr , Sl , Zr , Zl and J are used to indicate the right socle, the left socle, the right singular ideal, the left singular ideal, and the Jacobson radical of R , respectively. For a submodule N of M , the notations NM⊆ess , NM⊆max and NM⊆⊕ mean, respectively, that N is essential, maximal, and direct summand. If X is a subset of a right R-module M , right annihilators will be denoted by r( X) = rR ( X) =∈={ r R: xr 0 for all x ∈ X} , with a similar definition of left annihilators • • lX( ) = lR ( X) . Multiplication maps x→ ax and x→ xa will be denoted by a and a , respectively. If M and N are right R-modules, then M is called N-injective if every R-homomorphism from a submodule of N into M can be extended to an R-homomorphism from N into M. Mod-R indicates the category of right R-modules. We refer to [1-3] for all the undefined notions in this paper. 2. Strongly S-Injective Modules Definition 1 A right R-module M is called s-N-injective if every R-homomorphism fK: → M extends to N , where K is a submodule of the singular submodule ZN( ) . M is called s-injective if M is s-R-injective. M is called strongly s-injective, if M is s-N-injective for all right R-modules N . For example every nonsingular R-module is strongly s-injective. In particular, the ring of integers Z is strongly s-injective, but not injective. Proposition 1 1) Let N be a right R-module and {MiIi : ∈ } a family of right R-modules. Then the direct product ∏M i iI∈ OPEN ACCESS APM 26 N. A. ZEYADA is s-N-injective if and only if each M i is s-N-injective, iI∈ . 2) Let M, N , and K be right R-modules with KN⊆ . If M is s-N-injective, then M is s-K-injective. 3) Let M, N , and K be right R-modules with KN≅ . If K is s-M-injective, then N is s-M-injective. 4) Let N be a right R-module and {Aii : ∈ I} a family of right R-modules. Then N is sA--⊕ i injective iI∈ if and only if N is sA--i injective, ∀∈iI. 5) A right R-module M is s-injective if and only if M is s-P-injective for every projective right R-module P . 6) Let M, N , and K be right R-modules with NM⊆⊕ . If M is s-K-injective, then N is s-K- injective. 7) If A , B , and M are right R-modules, ABRR≅ , and M is s-A-injective, then M is s-B-injective. Proof. The proofs of 1) through 4) are routine. 5). This follows from 4). 6). Let fL: → N be an R-homomorphism where L is singular submodule of N . Then the map ι fL: → can be extended to an R-homomorphism gK: → M, where ι : NM→ the inclusion map. Now, the map π gK: → N is an extension of f , where π : MN→ the natural projection map into N . 7). Let fA: RR→ B be an R-isomorphism, and gK: → MR an R-homomorphism where K is a singular submodule of B . The restriction of f to ZA( ) induces an isomorphism fZA( RR) :. ZA( ) → ZB( R) By hypothesis, the map gfK : → MR can be extended to an R-homomorphism η :.AMRR→ Now, the map η fBM−1 : → is an extension of g . □ The next two corollaries are immediate consequences of the above proposition. Corollary 1 Let N be a right R-module. Then the following statements are true: 1) A finite direct sum of s-N-injective modules is again s-N-injective. In particular a finite direct sum of s-injective (strongly s-injective) modules is again s-injective (strongly s-injective). 2) A summand of s-N-injective (s-injective, strongly s-injective) module is again s-N-injective (s-injective, strongly s-injective) module. Corollary 2 1) Let M be a right R-module and 1 =+++ee12 en in R , where the ei are orthogonal idempotents. Then M is s-injective if and only if M is seR--i injective for each i , 1.≤≤in 2) Assume that e and f are idempotents of R , eR≅ fR, and M is s−− eR injective. Then M is s-fN-injective. Proposition 2 If N is a finitely generated right R-module, then the following conditions are equivalent: 1) Any direct sum of s-N-injective modules is s-N-injective. 2) Any direct sum of injective modules is s-N-injective. 3) ZN( ) is noetherian. Proof. 1) ⇒ 2). Clear. 2) ⇒ 3). Consider a chain UU12⊆⊆ of singular submodules of N and UU=1≤ii. Let E( NUi ) be the injective hull of NUi , i ≥ 1, and f: U→ ⊕ E( NUi ) be a map defined by fn( ) =( nU + i ) . Since, 1≤i ˆ ⊕E( NUi ) is s-N-injective, f can be extended to an R-homomorphism f:. N→ ⊕ E( NUi ) Since N 1≤i 1≤i ˆ ⊆ ⊆ is finitely generated, f( N) ⊕ E( NUi ) for some n, then f( U) ⊕ E( NUi ) and UU= jn+ for 1≤≤in 1≤≤in every j ≥ 1. Hence ZN( ) is noetherian. 3) ⇒ 1). Let EE= ⊕ i be a direct sum of s-N-injective modules, and fU: → ER be a homomorphism of iI∈ right R-modules where U⊆ ZN( ) . Since ZN( ) is noetherian, fU( ) ⊆ ⊕ Ei for some finite subset FI⊆ . iF∈ Since finite direct sums of s-N-injective modules is s-N-injective, f can be extended to an R-homomorphism fNˆ :.→ E □ The second singular submodule of a right R-module M , denoted by ZM2 ( ) , is defined by the equality Z2 ( M) ZM( ) = ZMZM( ( )) . We see that a ZM2 ( ) is closed submodule of M and MZ2 ( M) is non-singular. A right R-module is Goldie torsion if ZG2 ( ) = G. Lemma 1 Let M and N be right R-modules such that ZM2 ( ) is injective. Then every homomorphism OPEN ACCESS APM N. A. ZEYADA 27 fK: → M where K⊆ ZN2 ( ) extends to N . Proof. Let M= ZM2 ( ) ⊕ T where ZM2 ( ) is injective and ZT( ) = 0 . If fK: → M is a homomor- phism where K⊆ ZN2 ( ) such that fZK( ( )) = 0 , then f( K) ≅ K Ker( f ) is singular. So f extendable ≠∈ ≅ to N . Now suppose that 0 fk( ) T, so f( kR) kR Ker( f kR ) is singular which is a contradiction. Thus fK( ) T= 0 . Suppose that fk( ) = a + b where aZM∈ 2 ( ) and bT∈ , bR f( kR) = 0 . So ra( ) ⊆ rb( ) and the kernal of the map aR→ bR by ar br is essential in aR which is a contradiction. Then every □ homomorphism fK: → M where K⊆ ZN2 ( ) extends to N . Proposition 3 The following statements are equivalent: 1) M is strongly s-injective. 2) M is sI--( M) injective, where IM( ) is the injective hull of M . 3) MEK= ⊕ , where K is nonsingular and E is injective with ZM( ) ⊆ess E. 4) ZM2 ( ) is injective. 5) M is G-injective for every Goldie torsion module G . 6) M is I-injective, where I= IZ( 2 ( M)) is the injective hull of ZM2 ( ) . Proof. 1) ⇒ 2). Clear. 2) ⇒ 3). If ZM( ) = 0 , we are done. Assume that ZM( ) ≠ 0 and consider the following diagram: 0 ↓ 0 →ZM( ) →i1 D i2 ↓ M where ι1 and i2 are inclusion maps and D is injective closure of ZM( ) in IM( ) .