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

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.