Projectivity relative to closed (neat) submodules

Abstract

An [Formula: see text]-module [Formula: see text] is called closed (neat) projective if, for every closed (neat) submodule [Formula: see text] of every [Formula: see text]-module [Formula: see text], every homomorphism from [Formula: see text] to [Formula: see text] lifts to [Formula: see text]. In this paper, we study closed (neat) projective modules. In particular, the structure of a ring over which every finitely generated (cyclic, injective) right [Formula: see text]-module is closed (neat) projective is studied. Furthermore, the relationship among the proper classes which are induced by closed submodules, neat submodules, pure submodules and [Formula: see text]-pure submodules are investigated.