Pure Direct Injectıve Objects in Grothendıeck Categorıes

Abstract

We study generalizations of the concept of direct-injectivity (respectively pure-direct-injectivity) from module categories to abelian categories (respectively Grothendieck categories). We examine for which categories or under what conditions direct-injective objects are injective or quasi-injective. Also we examine for which categories or under what conditions pure-direct-injective objects are injective, quasi-injective, pure-injective or direct-injective. We investigate classes all of whose objects are direct-injective (respectively pure-direct-injective). We also give applications of some results to module categories and comodule categories.