The center of crossed products over simple rings

Abstract

Let $R*G$ be the crossed product of an arbitrary group $G$ over a simple ring $R$. Since $G$ acts on $Z(R)$ and $R$ is simple, $Z(R)$ is a $G$-field and the fixed field $Z(R)^{G}$ of $G$ is contained in $Z(R*G)$. The main result of this paper exhibits a distinguished basis for $Z(R*G)$ over the field $Z(R)^{G}$. A number of applications is also provided. Our method is based on the theory of similinear monomial representations. In this way we obtain conceptual proofs of results which otherwise require lengthy computations and ad hoc arguments.