Anticommutativity of Skew-symmetric Elements under Generalized Oriented Involutions

TL;DR

This paper classifies group rings with a generalized oriented involution where skew-symmetric elements form an anticommutative set, extending previous results and providing a comprehensive understanding of their algebraic structure.

Contribution

It introduces a classification of group rings with generalized oriented involutions where skew-symmetric elements are anticommutative, generalizing earlier specific cases.

Findings

Characterization of group rings with anticommutative skew-symmetric elements

Extension of previous results to generalized oriented involutions

Identification of conditions for anticommutativity in these rings

Abstract

Let $R$ be a ring with $char(R)\neq2$ whose unit group are denoted by $\mathcal{U}(R)$, $G$ a group, and $RG$ its group ring. Let $*$ be an involution in $G$, $\sigma:G\rightarrow\mathcal{U}(R)$ be a nontrivial group homomorphism, with $ker\ \sigma=N$, satisfying $xx^*\in N$ for all $x\in G$, and define the generalized oriented involution $\sigma*$ in $RG$ by $\left( \sum_{x\in G}\alpha_xx\right)^{\sigma*}=\sum_{x\in G}\sigma(x)\alpha_xx^*$. An element $\alpha\in RG$ is called skew-symmetric if $\alpha^{\sigma *}=-\alpha$, and the set of all skew-symmetric elements are denoted by $(RG)^-$. In this paper, we will classify the group rings $RG$ such that $(RG)^-$ is anticommutative, generalizing, and obtaining as consequence, the main result of \cite{GP13a}.