Purely Rickart and dual purely Rickart objects in Grothendieck categories
The paper deals with a generalization of the notion of Rickart modules and their duals for Grothendieck categories. More precisely, the author defines the notions of purely Rickart objects and their duals, and studies their main properties. For instance, it is shown in Corollary 3.7 that every purely self-Rickart object has the direct pure intersection property. In addition, the author uses the new notion to characterize when a locally finitely generated Grothendieck categories is regular (Theorem 5.1). This happens if and only if every object is purely self-Rickart. The author also considers relative purely Baer objects in Grothendieck categories and use the results on Rickart objects to study them. Finally, several applications in concrete Grothendieck categories, like module categories and comodule categories are provided.
scientific article; zbMATH DE number 7397614
scientific article