Semisimple-direct-injective modules

Year 2021, Volume: 50 Issue: 2, 516 - 525, 11.04.2021

Adel Abyzov Muhammet Tamer Koşan , Truong Cong Quynh , Daniel Tapkin

https://doi.org/10.15672/hujms.730907

Cited By: 1

https://izlik.org/JA79XU29GY

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

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