Pure-Direct-Injective Modules

Abstract

In this paper, we study the class of modules having the property that if any pure submodule is isomorphic to a direct summand of such a module then the pure submodule is itself a direct summand. These modules are termed as pure-direct-injective modules (or pure-C2 modules). We have characterized the rings whose pure-C2 modules satisfy certain conditions, such as being C2, (pure-) injective, projective, or dual-Utumi. For instance, it is proved that if R is a right Noetherian ring over which every pure-C2 right R-module is pure-injective, then R is Kroll-Schmidt semiperfect. The rings over which every pure-C2 module is injective, projective and dual-Utumi are exactly the semisimple rings. Also, it is shown that a ring R is right perfect if and only if every projective right R- module is pure-C2.