On splitting rings for Azumaya skew group rings

Abstract

Let $B$ be a ring with 1, $G$ an automorphism group of $B$ of order $n$ for some integer $n$, $B\ast G$ the skew group ring over $B$ with a free basis $\{g\mid gın G\}$, $B^G$ the set of elements in $B$ fixed under $G$, and $\overline G$ the inner automorphism group of $B\ast G$ induced by~$G$. It is shown that when the center $C$ of $B$ is a $G$-Galois algebra over $C^G$ with Galois group $G|_C\cong G$ or $B$ is a $G$-Galois extension of $B^G$ and $n^{-1}ın B$, then, $B\ast G$ is an Azumaya algebra if and only if so is $(B\ast G)^{\overline G}$, and some splitting rings of $B\ast G$, $(B\ast G)^{\overline G}$ and $B$ are shown to be the same.