Semisimple-direct-injective modules

Abstract

The notion of simple-direct-injective modules which are a generalization of injective modules unifies $C2$ and $C3$-modules. In the present paper, we introduce the notion of the semisimple-direct-injective module which gives a unified viewpoint of $C2$, $C3$, SSP properties and simple-direct-injective modules. It is proved that a ring $R$ is Artinian serial with the Jacobson radical square zero if and only if every semisimple-direct-injective right $R$-module has the SSP and, for any family of simple injective right $R$-modules $\{S_i\}_{\mathcal{I}}$, $\oplus_{\mathcal{I}}S_i$ is injective. We also show that $R$ is a right Noetherian right V-ring if and only if every right $R$-module has a semisimple-direct-injective envelope if and only if every right $R$-module has a semisimple-direct-injective cover.