Anticommutativity of Symmetric Elements under Generalized Oriented Involutions

TL;DR

This paper classifies group rings over rings with involution where the symmetric elements form an anticommutative set, extending understanding of algebraic structures with generalized involutions.

Contribution

It provides a classification of group rings with generalized oriented involutions where symmetric elements are anticommutative, a novel extension in ring theory.

Findings

Identifies conditions for anticommutativity of symmetric elements

Classifies group rings satisfying these conditions

Extends previous involution-related algebraic results

Abstract

Let $R$ be a ring with $char(R)\neq2$ whose unit group are denoted by $\mathcal{U}(R)$, $G$ a group with involution $*$, and $\sigma:G\rightarrow\mathcal{U}(R)$ a nontrivial group homomorphism, with $ker\ \sigma=N$, satisfying $xx^*\in N$ for all $x\in G$. Let $RG$ be the group ring of $G$ over $R$ and define the involution $\sigma*$ in $RG$ by $\left( \sum_{x\in G}\alpha_xx\right)^{\sigma*}=\sum_{x\in G}\sigma(x)\alpha_xx^*$. In this paper, we will classify the group rings $RG$ such that $\mathcal{S}$ is anticommutative, where $\mathcal{S}$ is the largest subset of $(RG)^+=\left\{ \alpha\in RG: \alpha^{\sigma *}=\alpha\right\}$ that can satisfy anticommutativity under $char(R)\neq2$.