Pseudo semi-projective modules and endomorphism rings

A module $M$ is called pseudo semi-projective if, for all $\alpha,\beta\in \mathrm{End}_R(M)$ with $\mathrm{Im}(\alpha)=\mathrm{Im}(\beta)$, there holds $\alpha\, \mathrm{End}_R(M)=\beta\, \mathrm{End}_R(M)$. In this paper, we study some properties of pseudo semi-projective modules and their endomorphism rings. It is shown that a ring $ R$ is a semilocal ring if and only if each semiprimitive finitely generated right $R$-module is pseudo semi-projective. Moreover, we show that if $M$ is a coretractable pseudo semi-projective module with finite hollow dimension, then $\mathrm{End}_R(M)$ is a semilocal ring and every maximal right ideal of $\mathrm{End}_R(M)$ has the form $\{s \in \mathrm{End}_R(M) | \mathrm{Im}(s) + \mathrm{Ker}(h)\ne M\}$ for some endomorphism $h$ of $M$ with $h(M)$ hollow.