Anticommutativity of Skew-symmetric Elements under Generalized Oriented Involutions

Let $R$ be a ring with $char(R)\mathrm{\ne }2$ whose unit group are denoted by $\mathcal{U}(R)$, $G$ a group, and $RG$ its group ring. Let $\ast$ be an involution in $G$, $\sigma :G\to \mathcal{U}(R)$ be a nontrivial group homomorphism, with $ker\sigma =N$, satisfying $x{x}^{\ast }\in N$ for all $x\in G$, and define the generalized oriented involution $\sigma \ast$ 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 ${\alpha }^{\sigma \ast }=-\alpha$, 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}.