Strongly Rickart objects in abelian categories

TL;DR

This paper introduces and explores (dual) strongly relative Rickart objects in abelian categories, analyzing their properties, behavior under (co)products, and applications to various module and comodule categories.

Contribution

It develops a new theoretical framework for strongly relative Rickart objects and investigates their transfer properties and applications in different categorical contexts.

Findings

Established general properties of strongly relative Rickart objects.

Analyzed behavior with respect to (co)products and functors.

Provided applications to Grothendieck, module, and comodule categories.

Abstract

We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories.