Antisymmetric Elements in Group Rings II

TL;DR

This paper characterizes when $p$-antisymmetric elements in group rings commute, extending previous results to provide a more complete understanding of their algebraic structure.

Contribution

It offers a new characterization of commuting $p$-antisymmetric elements in group rings, completing earlier partial results.

Findings

Provides necessary and sufficient conditions for commutativity of $p$-antisymmetric elements.

Extends previous work to a more comprehensive classification.

Enhances understanding of involution-induced structures in group rings.

Abstract

Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $\vp : RG\to RG$ denote the $R$-linear extension of an involution $\vp$ defined on $G$. An element $x$ in $RG$ is said to be $\vp$-antisymmetric if $\vp (x) = -x$. A characterization is given of when the $\vp$-antisymmetric elements of $RG$ commute. This is a completion of earlier work.