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.

In this paper we introduce some covering properties related with paracompactness, which allow us to characterize paracompact, compact, pseudocompact, lightly compact, collectionwise normal with respect to paracompact and closed sets, and countably compact spaces.

This paper provides a careful and accessible exposition of the theory of analytic semigroups which may have weak singularity at the origin, with emphasis on its applications to a class of subelliptic initial-boundary value problems for linear parabolic differential equations of second order.