Semisimple-direct-injective modules

Authors : Adel ABYZOV, Muhammet Tamer KOŞAN, Truong Cong QUYNH, Daniel TAPKİN

Pages : 516-525

Doi:10.15672/hujms.730907

View : 16 | Download : 5

Publication Date : 2021-04-11

Article Type : Research Paper

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.
Keywords : C2 module, C3 module, SSP, Artinian serial ring, V ring, simple direct injective module