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.