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# Simplicity of Partial Skew Group Rings of Abelian Groups

Published:2014-04-05
Printed: Sep 2014
• Daniel Gonçalves,
Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, 88040-900, Brasil
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## 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. 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.
 Keywords: partial skew group rings, simple rings, partial actions, abelian groups
 MSC Classifications: 16S35 - Twisted and skew group rings, crossed products 37B05 - Transformations and group actions with special properties (minimality, distality, proximality, etc.)