Projectivity relative to closed (neat) submodol

An R-module F is called closed (neat) projective if, for every closed (neat) submodule L of every R-module M, every homomorphism from F to M/L lifts to M. In this paper, we study closed (neat) projective modules. In particular, the structure of a ring over which every finitely generated (cyclic, injective) right R-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 C-pure submodules are investigated.