Pure semisimple and K\"{o}the group rings

TL;DR

This paper characterizes when abelian and certain non-abelian group rings over division rings are K"othe rings, providing new criteria and examples for pure semisimplicity in various characteristics.

Contribution

It offers a complete characterization of K"othe group rings, including abelian and non-abelian cases, and extends Maschke-type results for pure semisimplicity.

Findings

Characterization of abelian group rings as K"othe rings

Criteria for non-abelian group rings over division rings

Examples illustrating the theoretical results

Abstract

In this article we provide a complete characterization of abelian group rings which are K\"{o}the rings. We also provide characterizations of (possibly non-abelian) group rings over division rings which are K\"{o}the rings, both in characteristic zero and in prime characteristic, and prove a Maschke type result for pure semisimplicity of group rings. Furthermore, we illustrate our results by several examples.