Pure-direct-projective modules

In this paper, we introduce and study the pure-direct-projective modules, that is the modules M every pure submodule A of which with M/A isomorphic to a direct summand of M is a direct summand of M. We characterize rings over which every right R-module is pure-direct-projective. We examine for which rings or under what conditions pure-direct-projective right R-modules are direct-projective, projective, quasi-projective, pure-projective, flat or injective. We prove that over a Noetherian ring every injective module is pure-direct-projective and a right hereditary ring R is right Noetherian if and only if every injective right R-module is pure-direct-projective. We obtain some properties of pure-direct-projective right R-modules which have DPSP and DPIP.