Modules close to SSP- and SIP-modules

TL;DR

This paper explores properties of SIP, SSP, and CS-Rickart modules, establishing conditions and connections with semisimple artinian rings, and characterizing rings via module properties.

Contribution

It provides new equivalent conditions for SIP and SSP modules and links ring-theoretic properties with module-theoretic conditions, especially for semisimple artinian rings.

Findings

R is semisimple artinian iff R_R is SIP and every right R-module has a SIP-cover

R is semiregular with J(R)=Z(R_R) iff finitely generated projective modules are SIP-CS and C2 modules

Connections established between ring properties and module classes like SIP, SSP, and CS-Rickart

Abstract

In this paper, we investigate some properties of SIP, SSP and CS-Rickart modules. We give equivalent conditions for SIP and SSP modules; establish connections between the class of semisimple artinian rings and the class of SIP rings. It shows that $R$ is a semisimple artinian ring if and only if $R_R$ is SIP and every right $R$-module has a SIP-cover. We also prove that $R$ is a semiregular ring and $J(R) = Z(R_R)$ if only if every finitely generated projective module is a SIP-CS module which is also a $C2$ module.