Separable functors and firm modules

TL;DR

This paper extends the theory of separable functors and ring extensions to nonunital rings using firm modules, providing new results and a local version of Maschke's theorem.

Contribution

It introduces nonunital analogues of classical separability and semisimplicity results within the framework of firm modules.

Findings

Established nonunital versions of classical separability results

Proved a locally unital Maschke's theorem for group rings

Developed a theory applicable to nonunital rings in the context of firm modules

Abstract

We develop a theory of separable ring extensions and separable functors for nonunital rings in the setting of firm modules. We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings.