ON TWO GENERALIZATIONS OF SEMI-PROJECTIVE MODULES: SGQ-PROJECTIVE AND π-SEMI-PROJECTIVE

Let R be a ring and M a right R-module with S = EndR(M). A module M is called semi-projective if for any epimorphism f : M → N, where N is a submodule of M, and for any homomorphism g : M → N, there exists h : M → M such that fh = g. In this paper, we study SGQ-projective andπ-semiprojective modules as two generalizations of semi-projective modules. A module M is called an SGQ-projective module if for any φ ∈ S, there exists a right ideal Xφ of S such that DS(Imφ) = φS ⊕ Xφ as right S-modules. We call M a π-semi-projective module if for any 0 6= s ∈ S, there exists a positive integer n such that s n 6= 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.