Normality in group rings

TL;DR

This paper characterizes when the group ring KG is normal with respect to a specific involution, based on the properties of the group G, the ring K, and the involution structure.

Contribution

It provides a complete description of the group rings that are normal under a certain involution, linking algebraic properties of G and K with involution behavior.

Findings

Characterization of normality in group rings under involution

Conditions on G and K for normality to hold

Explicit description of involution-compatible group rings

Abstract

Let $KG$ be the group ring of a group $G$ over a commutative ring $K$ with unity. The rings $KG$ are described for which $xx^\sigma=x^\sigma x$ for all $x=\sum_{g\in G}\alpha_gg\in KG$, where \quad $x\mapsto x^\sigma=~\sum_{g\in G}\alpha_gf(g)\sigma(g)$\quad is an involution of $KG$; here $f: G\to U(K)$ is a homomorphism and $\sigma$ is an anti-automorphism of order two of $G$.