Miyashita Action in Strongly Groupoid Graded Rings

TL;DR

This paper extends Miyashita's classical result on the commutant of homogeneous subrings from group graded rings to the more general strongly groupoid graded rings, providing explicit constructions and examples.

Contribution

It generalizes Miyashita's theorem to strongly groupoid graded rings and offers explicit examples demonstrating the new theoretical results.

Findings

The commutant of homogeneous subrings can be described via an induced action in strongly groupoid graded rings.

A construction of strongly G-graded rings with specific module properties related to nonidentity morphisms.

Extension of classical results from group graded rings to the broader context of groupoid graded rings.

Abstract

We determine the commutant of homogeneous subrings in strongly groupoid graded rings in terms of an action on the ring induced by the grading. Thereby we generalize a classical result of Miyashita from the group graded case to the groupoid graded situation. In the end of the article we exemplify this result. To this end, we show, by an explicit construction, that given a finite groupoid $G$, equipped with a nonidentity morphism $t : d(t) \to c(t)$, there is a strongly $G$-graded ring $R$ with the properties that each $R_s$, for $s \in G$, is nonzero and $R_t$ is a nonfree left $R_{c(t)}$-module.