SAS-Injective Modules

We introduce and investigate SAS-injective modules as a generalization of small injectivity. A right module 𝑀 over a ring 𝑅 is said to be SAS-𝑁 -injective (where 𝑁 is a right 𝑅 -module) if every right 𝑅 -homomorphism from a semiartinian small right submodule of 𝑁 into 𝑀 extends to 𝑁 . A module 𝑀 is said to be SAS-injective, if 𝑀 is SAS-𝑅 -injective. Some characterizations and properties of SAS-injective modules are given. Some results on small injectivity are extended to SAS-injectivity. 2020MSC


Introduction
Throughout  is an associative ring with identity and all modules are unitary -modules.If not otherwise specified, by a module (resp.homomorphism) we will mean a right -module (resp.right -homomorphism).For a submodule  of a module  , the notations  ≤ ,  ≪ ,  ≤  ,  ≤  , and  ≤ ⨁  mean, respectively, that  is a submodule, a small submodule, an essential submodule, a maximal submodule, and a direct summand of , respectively.If a is an element of right -module , then we use r(a) to denote the right annihilator of  in .Also, we use the symbols J(), soc() and Z() to denote the Jacobson radical, the socle and singular submodule of   , respectively.A module  is called semiartinian, if soc(/) ≠ 0, for any proper submodule  of M. For a right -module   , we denote by Sa() to the sum of all semiartinian submodules of .We refer the reader to [1,3,4,6,12], for general background materials.
A module  is called small-injective if every homomorphism from a small right ideal of  into  can be extended to a homomorphism from   into  [10].
In this article, a proper generalization of small-injectivity is introduced and investigated, namely SAS-injective modules.Let  be a right -module.A right -module  is said to be SAS--injective if every -homomorphism from a semiartinian small right submodule of  into  extends to .If  is SAS--injective, then we say that  is SAS-injective.Firstly, we give an example to show that SAS-injective modules need not be small-injective.Several properties of the class of SAS-injective modules are given.For example, we show that the class of SAS--injective modules is closed under isomorphic copies, direct products, finite direct sums and summands.Some characterizations of SAS-injective modules are given.We prove the equivalence of the following statements: (1) Every right -module is SAS-injective; (2) Every simple right -module is SAS-injective (3) Every semiartinian small submodule of any right -module SAS-injective; (4) Every semiartinian small right ideal of  is SAS-injective; (5) Every semiartinian small right ideal of  is a summand of ; (6) Sa(  ) ∩J() = 0. Conditions under which quotient of SAS-injective right -modules is SAS-injective are given.For instance, we prove that the equivalence of the following: (1) The class of SAS-injective right -modules is closed under quotient; (2) For any right -module , the sum of any two SAS-injective submodules of  is SAS-injective; (3) All semiartinian small submodules of   are projective.Finally, we give conditions such that the class of SAS-injective right -modules is closed under direct sums.For instance, we prove that the equivalence of the following conditions: (1) Sa(  ) ∩ J() is Noetherian ; (2) All direct sums of injective modules are SAS-injective; (3) The class of SAS-injective modules is closed under direct sums.

SAS-Injective Modules
As a generalization of small injective modules, we introduce the concept of SAS-injective modules.Definition 2.1.A right -module  is said to be SAS--injective (where  is a right -module), if any right -homomorphism :  →  extends to , where  is any semiartinian small submodule of .If  is SAS-injective, then  is said to be SAS-injective.

Examples 2.2.
(1) All small-injective modules are SAS-injective, but the converse is not true in general, for example: let  be the localization ring of ℤ at the prime , that is  = ℤ () = {   :  does not divide n}.Then  is not small injective with soc(  ) = 0 (see [13,Example 4]).Since soc(  ) = 0, we have that Sa(  ) = 0 and hence the zero ideal is the only semiartinian small right ideal in   .Thus   is SAS-injective and hence SAS-injectivity is a proper generalization of small injectivity.
(2) Clearly, if soc(  ) = 0, then 0 is the only semiartinian small submodule of  and hence every module is SAS-injective.Particularly, all ℤ-modules are SAS-injective.Some properties of SAS--injective modules are given in the following theorem.Theorem 2.3.Let ,  and  be right -modules.Then the following statements hold: (1) Let }   : ∈ } be a class of modules.Then the direct product ∏ ∈   is SAS--injective if and only if all   are SAS--injective.

Proof. Obvious. □
Corollary 2.4.The next statements hold: (1) A finite direct sum of SAS--injective modules is SAS--injective, for any module .Moreover, a finite direct sum of SAS-injective modules is SAS-injective.(2) A summand of an SAS-injective module is again SAS-injective.
Proof.(1) By applying Theorem 2.3 (1), when index  is taken to be a finite set.
(2) This directly by using Theorem 2.3 (5).□ If for any submodule  of a right -module , there exists an ideal  of  such that  = , then  is called a multiplication module [11, p. 3839 → Hom  (, ) ⟶ 0 is exact, for all submodule  of (  ) ∩ (), where  and  are the inclusion and canonical maps, respectively.
(4) Every semiartinian small submodule of  is a summand of .
(3) ⇒ (4) Let  be a semiartinian small submodule of .By hypothesis,  is SAS--injective and hence  ∘  =   for some a homomorphism :  ⟶ , where  is the inclusion and   is the identity homomorphism.Hence  is split and this implies that  is a summand of . is a small submodule of .By hypothesis  is a summand of  and hence  ⨁  =  for some submodule  of .Since  is a small submodule of , we have that  =  and hence  = 0. So,  = 0 and hence Sa() ∩ J() = 0. □ Corollary 2.9.For a ring , the next conditions are equivalent: (1) All right -modules are SAS-injective.
(2) All semiartinian small submodules of any right -module are SAS-injective.
(3) All semiartinian small right ideals of  are SAS-injective.
(4) All semiartinian small right ideals of  are summands of .