Coherent and ideal actions in ideally exact categories

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.