Simple Semigroup Graded Rings

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.