Simplicity of skew group rings of abelian groups

TL;DR

This paper characterizes when skew group rings of abelian groups are simple, linking simplicity to the properties of the centre and the G-simplicity of the base ring, with applications to transformation groups.

Contribution

It provides necessary and sufficient conditions for simplicity of skew group rings of abelian groups, extending understanding of their structure and applications.

Findings

Skew group ring is simple iff its centre is a field and A is G-simple.

For commutative A, simplicity requires injective σ and G-simplicity.

Application: minimal and faithful transformation groups correspond to simple skew group algebras.

Abstract

Given a group G, a (unital) ring A and a group homomorphism $\sigma : G \to \Aut(A)$, one can construct the skew group ring $A \rtimes_{\sigma} G$. We show that a skew group ring $A \rtimes_{\sigma} G$, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. If G is abelian and A is commutative, then $A \rtimes_{\sigma} G$ is shown to be simple if and only if \sigma is injective and A is G-simple. As an application we show that a transformation group (X,G), where X is a compact Hausdorff space and G is abelian, is minimal and faithful if and only if its associated skew group algebra $C(X) \rtimes_{\sigma} G$ is simple. We also provide an example of a skew group algebra, of an (non-abelian) ICC group, for which the above conclusions fail to hold.