S-Injective Modules and Rings

We introduce and investigate the concept of s-injective modules and strongly s-injective modules. New characterizations of SI-rings, GV-rings and pseudo-Frobenius rings are given in terms of s-injectivity of their modules.


Introduction
In this paper we introduce and investigate the notion of s-injective Modules  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][2][3] for all the undefined notions in this paper.

4) Let N be a right R-module and { }
: i A i I ∈ a family of right R-modules. Then N is -- Proof. The proofs of 1) through 4) are routine. 5). This follows from 4). 6).
can be extended to an R-homomorphism : the inclusion map. Now, the map π : be an R-isomorphism, and : .
By hypothesis, the map : can be extended to an R-homomorphism : .
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).
. So ( ) ( ) r a r b ⊆ and the kernal of the map aR bR → by ar br  is essential in aR which is a contradiction. Then every homomorphism : The following statements are equivalent:  i f hi f hgi = = . Hence M is strongly s-injective.

Corollary 3 Let M a Goldie torsion right R-module. Then M is injective if and only if M is strongly s-injective.
Proposition 4 For a ring R , the following conditions are equivalent: , , and in this case E and T are relatively injective.

OPEN ACCESS APM
N. A. ZEYADA 28 1) ⇒ 7) Since a finite direct sum of s-N-injective is s-N-injective for every right R-module N (Corollary 1), so every finitely generated free right R-module is strongly s-injective. But a direct summand of strongly s-injective is strongly s-injective (Corollary 1). Therefore every finitely generated projective module is strongly s-injective. The converse is clear. □ The following examples show that the two classes of strongly s-injective rings and of soc-injective rings are different.

Example 1 Let
is the subring of Q generated by 1 and , S then R is a von Neumann regular ring with ( ) soc R S = , and hence 2 0 r Z = and R is strongly s-injective. However, the map : which is impossible), and so R is not a soc-injective ring. L L L = ⊕ ⊕ is essential in K . But this gives that K L is noetherian which is impossible because is essential in H . Thus U and H U are noetherian. Hence H is noetherian. □ It is well-known that, a ring R is right noetherian if and only if all direct sums of injective right R-modules are injective. In the next Proposition we obtain a characterization of ring which has ACC on essential right ideals in terms of strongly s-injective right R-modules.

Proposition 6
The following conditions on a ring R are equivalent: 1) Every direct sum of strongly s-injective right R-modules is strongly s-injective.
2) Every direct sum of injective right R-modules is strongly s-injective.
3) Every finitely generated right R-module has ACC on essential submodules. : .
Proof. 1) Let M be s-injective, and : be an extension of the restriction map .
Clearly, h is an extension of f , and so M is injective by the Baer's Criterion.
2) Let T be a right ideal of R . Since R is right r Z -semiperfect, then ,

Corollary 4 A ring R is right nonsingular if and only if every right R-module is s-injective.
A ring R is called a right (left) SI ring if every singular right (left) R-module is injective. SI rings were initially introduced and investigated by Goodearl [5] Theorem 1 The following statements are equivalent: 1) R is right SI ring.

2) Every right R-module is strongly s-injective.
3) Every singular right R-module is strongly s-injective. Proof. Clear from Proposition 3.
2). Let M be a s-N-injective and K be a singular submodule of ( ) is a monomorphism and the maps N η and M η are isomorphisms ( we may assume that injective. The converse is similarly. □ As for right self-injectivity, right strongly s-injectivity turns out to be a Morita invariant.

1) Every homomorphic image of a s-M-injective right R-module is s-M-injective. 2) Every homomorphic image of an injective right R-module is s-M-injective.
3) Every singular submodule of M is projective. Proof. 1) ⇒ 2) Obvious. 2) ⇒ 3) Consider the following diagram: Where K is a singular submodule of M , E and N are right R-modules with E injective, η an R-epimorphism, and f an R-homomorphism. Since N is s-injective, f can be extended to an R-homomorphism : .
Since M is projective, g can be lifted to an R-homomorphism : and hence K is projective.
3) ⇒ 1) Let N and L be right R-modules with : N L η → an R-epimorphism, K is a singular submodule of M and N is s-M-injective. Consider the following diagram: Since N is s-injective, g can be extended to an R-homomorphism 3) ⇒ 4). Let K be a right ideal of R . Then and E is a summand of R and generated by an idempotent. Hence R is 2 - . Let M be a strongly s-injective and : Using Proposition 3 let : R g R M → be an extension of the restriction map .
Clearly, h is an extension of f , and so M is injective by the Baer's Criterion. □ Theorem 3 The following are equivalent for a ring R : 1) R is a right PF-ring.
2) R is Z r -semiperfect, right strongly s-injective ring with essential right socle. 3) R is semiperfect, right min-C2, right strongly s-injective ring with essential right socle. 4) R is semiperfect with , right strongly s-injective ring with essential right socle. 5) R is right finitely cogenerated, right min-C2, right strongly s-injective ring. 6) R is a right Kasch, right strongly s-injective ring. 7) R is a right strongly s-injective ring and the dual of every simple left R-module is simple.

OPEN ACCESS APM
The right ideal T may be considered as an R/J-module. Let : Since T is injective as an R/J-module so h extends to : g R J T → . The map π : g R T → , where π is the natural epimorphism π : R R J → , extends f and T is injective. Therefore R is right selfinjective and R is right PF . 1) ⇒ 5). Clear. 5) ⇒ 1). Since R is right strongly s-injective ring, it follows from Proposition 3 that 0 aR = and every simple right ideal in T is a summand of T . Since R is right finitely cogenerated, T has a finitely generated essential socle. Thus T is semisimple. Hence R is injective and R is right PF − ring.
If aR is a nonsingular simple right ideal, then ( ) 0 r a eR =  for some simple right ideal eR with 2 e e = . But R is 2 C , so eR aR ≅ and aR is a summand. Thus R is right min-CS, so by [3,Theorem 4.8] R is semiperfect with essential right socle. Hence R is right PF by 3).

□
The following is an example of a right perfect, left Kasch ring and right strongly s-injective which is not right self-injective ring.
Example 3 Let K be a field and let R be the ring of all upper triangular, countably infinite square matrices over R with only finitely many off-diagonal entries. Let S be the K-subalgebra of R generated by 1 and ( ). = whereas S is neither left perfect nor right self-injective because it is not right finite dimensional.
Remark 1 Note that the ring of integers  is an example of a commutative noetherian strongly s-injective ring which is not quasi-Frobenius. Definition 3 A ring R is called right CF-ring (FGF-ring) if every cyclic (finitely generated) right R-module embeds in a free module. It is not known whether right CF-rings (FGF-rings) are right artinian (quasi-Frobenius). In the next result we provide a positive answer if we assume in addition that the ring R is strongly right s-injective.

S-CS Modules and Rings
A module M is said to satisfy C1-condition or called CS module if every submodule of M is essential in a direct summand of M .

Definition 4 A right R-module M is called s-CS module if every singular submodule of M is essential in a summand of M .
For example, every nonsingular module is s-CS. In particular, the ring of integers Z is s-CS but not CS Proposition 14 For a right R-module M , the following statements are equivalent: 1) The second singular submodule