Miyashita Action in Strongly Groupoid Graded Rings

Details Miyashita Action in Strongly Groupoid Graded Rings Miyashita Action in Strongly Groupoid Graded Rings

2010-01-09

arxiv.org/abs/1001.1459v4

Mathematics

Rings and Algebras

This article is an improvement of, and hereby a replacement for, version 1 (arXiv:1001.1459v1) entitled "Commutants in Strongl

Scientific paper

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.

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