Simplicity of partial skew group rings of abelian groups

TL;DR

This paper characterizes when partial skew group rings of abelian groups are simple, linking simplicity to the $ ext{G}$-simplicity of the base ring and the field nature of certain centers, with applications to topological and set-level actions.

Contribution

It provides a complete characterization of simplicity for partial skew group rings of abelian groups, extending understanding of their structure and applications.

Findings

$ ext{A} times_ ext{α} ext{G}$ is simple iff $ ext{A}$ is $ ext{G}$-simple and centers are fields.

The result applies to partial actions on compact sets and at the set level.

Provides criteria for simplicity in terms of centers and $ ext{G}$-simplicity.

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.