Simplicity of partial skew group rings of abelian groups

Abstract

Let $\A$ be a ring with local units, $\E$ a set of local units for $\A$, $\G$ an abelian group and $\alpha$ a partial action of $\G$ by ideals of $\A$ that contain local units and such that the partial skew group ring $\A\star_{\alpha} \G$ is associative. We show that $\A\star_{\alpha} \G$ is simple if and only if $\A$ is $\G$-simple and the center of the corner $e\delta_0 (\A\star_{\alpha} \G) e \delta_0$ is a field for all $e\in \E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.