Anticommutativity of Symmetric Elements under Generalized Oriented Involutions

Abstract

Let $R$ be a ring with $char(R)\ne 2$ whose unit group are denoted by $\mathcal{U}(R)$, $G$ a group with involution $\ast$, and $\sigma :G\to \mathcal{U}(R)$ a nontrivial group homomorphism, with $ker\sigma =N$, satisfying $x{x}^{\ast }\in N$ for all $x\in G$. Let $RG$ be the group ring of $G$ over $R$ and define the involution $\sigma \ast$ in $RG$ by ${\left({\sum }_{x\in G}{\alpha }_{x}x\right)}^{\sigma \ast }={\sum }_{x\in G}\sigma (x){\alpha }_x{x}^{\ast }$. In this paper, we will classify the group rings $RG$ such that $\mathcal{S}$ is anticommutative, where $\mathcal{S}$ is the largest subset of $(RG{)}^{+}=\left\{\alpha \in RG:{\alpha }^{\sigma \ast }=\alpha \right\}$ that can satisfy anticommutativity under $char(R)\ne 2$.

14 pages