Submodules of pseudo semi-projective modules

A module $M$ is called pseudo semi-projective if, for all endomorphisms $\alpha ,\beta$ of $M$ with $\mathrm{\text{Im}}(\alpha )=\mathrm{\text{Im}}(\beta )$, then $\alpha \mathrm{\text{End}}(M)=\beta \mathrm{\text{End}}(M)$. In this paper, we study submodules of pseudo semi-projective modules. It is shown that if every (finitely generated) submodule of a semiprojective right $R$-module is pseudo semi-projective, then every factor ring of $R$ is right (semi-)hereditary. Moreover, we show that if $R$ is left perfect and finitely generated submodules of pseudo semi-projective right $R$-modules are pseudo semi-projective, then $R$ has a decomposition of abelian groups $R_{\mathbb{Z}}=S\oplus J(R)$, where $S$ is a semisimple subring of $R$ containing 1.