# Reversibility of a module with respect to the bifunctors Hom and‎ ‎Rej

Document Type : Research Paper

Authors

1 Isfahan University of Technology

2 Department of of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111,

Abstract

Let $M_R$ be a non-zero‎
‎module and ${\mathcal F}‎: ‎\sigma[M_R]\times \sigma[M_R]‎ ‎\rightarrow$ Mod-$\Bbb{Z}$ a bifunctor‎. ‎The‎
‎$\mathcal{F}$-reversibility of $M$ is defined by ${\mathcal‎ ‎F}(X,Y)=0 \Rightarrow {\mathcal F}(Y,X)=0$ for all non-zero $X,Y$‎
‎in $\sigma[M_R]$‎. ‎Hom (resp‎. ‎Rej)-reversibility of $M$ is‎
‎characterized in different ways‎. ‎Among other things‎, ‎it is shown‎
‎that $R_R$ {\rm($_RR$)} is Hom-reversible if and only if $R =‎ ‎\bigoplus_{i=1}^n R_i$ such that each $R_i$ is a perfect ring with‎
‎a unique simple module (up to isomorphism)‎. ‎In particular‎, ‎for a‎
‎duo ring‎, ‎the concepts of perfectness and Hom-reversibility‎
‎coincide‎.

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Main Subjects

### History

• Receive Date: 09 August 2012
• Revise Date: 08 July 2013
• Accept Date: 09 July 2013