Simple Semigroup Graded Rings

TL;DR

This paper characterizes when a semigroup graded ring is simple, extending previous results to non-unital rings and partial skew group rings, with conditions involving graded simplicity and the center of a corner subring.

Contribution

It generalizes Jespers' simplicity criterion to non-unital rings graded by semigroups with specific properties and applies this to partial skew group rings.

Findings

Provides necessary and sufficient conditions for simplicity of graded rings.

Extends Jespers' result to non-unital rings and partial skew group rings.

Connects simplicity criteria with properties of the center of a corner subring.

Abstract

We show that if $R$ is a, not necessarily unital, ring graded by a semigroup $G$ equipped with an idempotent $e$ such that $G$ is cancellative at $e$, the non-zero elements of $eGe$ form a hypercentral group and $R_e$ has a non-zero idempotent $f$, then $R$ is simple if and only if it is graded simple and the center of the corner subring $f R_{eGe} f$ is a field. This is a generalization of a result of E. Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of D. Gon\c{c}alves'. We also point out how E. Jespers' result immediately implies a generalization of a simplicity result, recently obtained by A. Baraviera, W. Cortes and M. Soares, for crossed products by twisted partial actions.