Coherent and ideal actions in ideally exact categories

TL;DR

This paper introduces internal coherent and ideal actions within ideally exact categories, generalizing ring and algebra actions, and explores their properties and connections to semidirect products.

Contribution

It defines and analyzes internal coherent and ideal actions in ideally exact categories, extending classical algebraic concepts.

Findings

Every ideal action is coherent.

The converse holds in certain ideally exact contexts.

Connections to semidirect products are established.

Abstract

In the context of ideally exact categories, we introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts. Furthermore, a connection with G. Janelidze's notion of semidirect product in ideally exact categories is analysed.