Split objects with respect to a fully invariant short exact sequence in abelian categories

TL;DR

This paper introduces and studies (dual) relative split objects in abelian categories, comparing them with (dual) relative Rickart objects, and explores their properties and applications in various categorical contexts.

Contribution

It presents new concepts of (dual) relative split objects and strongly relative split objects, analyzing their behavior and relationships in abelian categories and related structures.

Findings

Comparison between (dual) relative split and Rickart objects

Behavior of these objects under direct sums

Applications to Grothendieck, module, and comodule categories

Abstract

We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to direct sums and classes all of whose objects are (dual) relative split. We also introduce and study (dual) strongly relative split objects. Applications are given to Grothendieck categories, module and comodule categories.