Simple-Direct-Projective Modules

TL;DR: In this article, the dual notion of simple-direct-injective modules was introduced and studied, and it was shown that a ring R is artinian and serial with J2(R) = 0 if and only if every simple direct-projective right R-module is quasi-Projective.

Abstract: In this paper, we introduce and study the dual notion of simple-direct-injective modules. Namely, a right R-module M is called simple-direct-projective if, whenever A and B are submodules of M with B simple and M/A ≅ B ⊆⊕M, then A ⊆⊕M. Several characterizations of simple-direct-projective modules are provided and used to describe some well-known classes of rings. For example, it is shown that a ring R is artinian and serial with J2(R) = 0 if and only if every simple-direct-projective right R-module is quasi-projective if and only if every simple-direct-projective right R -module is a D3-module. It is also shown that a ring R is uniserial with J2(R) = 0 if and only if every simple-direct-projective right R-module is a C3-module if and only if every simple-direct-injective right R -module is a D3-module.