Simplicity of Skew Group Rings of Abelian Groups

Abstract

Necessary and sufficient conditions for simplicity of a general skew group ring A ⋊σ G are not known. In this article, we show that a skew group ring A ⋊σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ⋊σ G is simple.