On two generalizations of semi-projective modules: SGQ-projective and $pi$-semi-projective

Let $R$ be a ring and $M$ a right $R$-module with $S=End_R(M)$. A module $M$ is called semi-projective if for any epimorphism $f:Mrightarrow N$, where $N$ is a submodule of $M$, and for any homomorphism $g: Mrightarrow N$, there exists $h:Mrightarrow M$ such that $fh=g$. In this paper, we study SGQ-projective and $pi$-semi-projective modules as two generalizations of semi-projective modules. A module $M$ is called an SGQ-projective module if for any $phiin S$, there exists a right ideal $X_phi$ of $S$ such that $D_S(Im phi)=phi Soplus X_phi$ as right $S$-modules. We call $M$ a $pi$-semi-projective module if for any $0neq sin S$, there exists a positive integer $n$ such that $s^nneq 0$ and any $R$-homomorphism from $M$ to $s^nM$ can be extended to an endomorphism of $M$. Some properties of this class of modules are investigated.