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.