具有定向态射的群环中的反对称元
Antisymmetric elements in group rings with an orientation morphism
摘要：
设$R$是交换环，$G$是群，$RG$是它的群环.令$ϕ_σ : RG\to RG$表示由$ϕ_σ (\sum r_{g}g) = \sum r_{g} σ(g) g^{-1}$定义的对合，其中$σ:G\to \{\pm 1\}$是群同态（称为定向态射）。如果$ϕ_σ (x) =-x$，则称$RG$中的元素$x$是反对称的。给出了群$G$的一个充分的刻画，并给出了$RG$的反对称元素可换的定向。
英文摘要：
Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $ϕ_σ : RG\to RG$ denote the involution defined by $ϕ_σ (\sum r_{g}g) = \sum r_{g} σ(g) g^{-1}$, where $σ:G\to \{\pm 1\}$ is a group homomorphism (called an orientation morphism). An element $x$ in $RG$ is said to be antisymmetric if $ϕ_σ (x) =-x$. We give a full characterization of the groups $G$ and its orientations for which the antisymmetric elements of $RG$ commute.
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