Pure-direct-projective modules

Abstract

In this paper, we introduce and study the pure-direct-projective modules, that is the modules [Formula: see text] every pure submodule [Formula: see text] of which with [Formula: see text] isomorphic to a direct summand of [Formula: see text] is a direct summand of [Formula: see text]. We characterize rings over which every right [Formula: see text]-module is pure-direct-projective. We examine for which rings or under what conditions pure-direct-projective right [Formula: see text]-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 [Formula: see text] is right Noetherian if and only if every injective right [Formula: see text]-module is pure-direct-projective. We obtain some properties of pure-direct-projective right [Formula: see text]-modules which have [Formula: see text] and [Formula: see text].