On A Class of Soc-Injective Modules

Let R be a ring. The class of SA-injective right R-modules (SAIR) is introduced as a class of soc-injective right R-modules. Let N be a right R-module. A right R-module M is said to be SA-N-injective if every R-homomorphism from a semi-artinian submodule of N into M extends to N. A module M is called SA-injective, if M is SA-R-injective. We characterize rings over which every right module is SA-injective. Con ditions under which the class SAIR is closed under quotient (resp. direct sums, pure homomorphic images) are given. The definability of the class SAIR is studied. Finally, relations between SA-injectivity and certain generalizations of injectivity are given.