Maximal commutative subrings and simplicity of partial skew group rings

TL;DR

This paper characterizes when partial skew group rings are simple by linking simplicity to the maximal commutativity and G-simplicity of the base ring, extending previous results to non-unital cases.

Contribution

It provides necessary and sufficient conditions for simplicity of partial skew group rings, generalizing prior work to non-unital rings and partial actions.

Findings

Non-zero ideals intersect the base ring non-trivially iff the base ring is maximal commutative.

Partial skew group ring is simple iff the base ring is G-simple and maximal commutative.

Extends previous results from skew to partial skew group rings, including non-unital cases.

Abstract

In this article, we show that for a partial skew group ring R*G, where R is a commutative ring, each non-zero ideal of R*G intersects R non-trivially if and only if R is a maximal commutative subring of R*G. As a consequence, we obtain necessary and sufficient conditions for simplicity; the partial skew group ring R*G is simple if and only if R is a G-simple and maximal commutative subring of R*G. We thereby generalize our previous results for skew group rings, to partial skew group rings which are not necessarily unital.