Strongly Rickart objects in abelian categories: Applications to strongly regular and strongly Baer objects

TL;DR

This paper explores the theory of strongly Rickart objects in abelian categories, establishing properties, behavior under (co)products, and functor transfer, with applications to various algebraic categories.

Contribution

It introduces the use of strongly relative Rickart objects to analyze strongly relative regular and Baer objects, providing new insights and general properties.

Findings

Established properties of strongly relative Rickart objects

Analyzed behavior under (co)products and functors

Applied theory to Grothendieck, module, and comodule categories

Abstract

We show how the 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. For each of them, 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.