Rings whose injective hulls are dual square free

Abstract

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with $M=A+B$ and $M/A\cong M/B.$ The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is dual-square-free. In this article we consider certain classes of rings R over which $E(R_R)$ is a DSF-module. For example, if R is a right hereditary ring such that $E(R_R)$ is DSF, then R is right noetherian, right distributive, every right ideal of R is two-sided, and every subfactor of R_R is quasi-continuous. Also, if R is semilocal and $E(R_R)$ is a DSF-module with small radical, then R is basic, semiperfect and right self-injective. As an immediate application, if R is a right perfect ring such that $E(R/J^2)$ is DSF as a right R-module then R is right Artinian. If in addition we assume that either $E(R_R)$ or $E{(}_{R}R)$ is a DSF-module then the ring R becomes quasi-Frobenius.

Keywords:

• Distributive modules
• quasi-duo rings
• square-free and dual-square-free modules

2010 Mathematics Subject Classification:

• Primary 16D40
• 16D50
• 16D60
• Secondary 16L30
• 16L60
• 16P20
• 16P40
• 16P60

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Correction to: Rings whose injective hulls are dual square free