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.