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)→c(t), there is a
strongly G-graded ring R with the properties that each Rs, for s∈G, is nonzero and Rt is a nonfree left Rc(t)-module.Comment: This article is an improvement of, and hereby a replacement for,
version 1 (arXiv:1001.1459v1) entitled "Commutants in Strongly Groupoid
Graded Rings