Submodules of pseudo semi-projective modules

A module [Formula: see text] is called pseudo semi-projective if, for all endomorphisms [Formula: see text] of [Formula: see text] with [Formula: see text], then [Formula: see text]. In this paper, we study submodules of pseudo semi-projective modules. It is shown that if every (finitely generated) submodule of a semiprojective right [Formula: see text]-module is pseudo semi-projective, then every factor ring of [Formula: see text] is right (semi-)hereditary. Moreover, we show that if [Formula: see text] is left perfect and finitely generated submodules of pseudo semi-projective right [Formula: see text]-modules are pseudo semi-projective, then [Formula: see text] has a decomposition of abelian groups [Formula: see text], where [Formula: see text] is a semisimple subring of [Formula: see text] containing 1.