MIYASHITA ACTION IN STRONGLY GROUPOID GRADED RINGS

Authors : Johan Öinert, Patrik Lundström

Pages : 46-63

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Publication Date : 2012-06-01

Article Type : Research Paper

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 : dinsert ignore into journalissuearticles values(t); → cinsert ignore into journalissuearticles values(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 Rcinsert ignore into journalissuearticles values(t);-module.
Keywords : graded rings, commutants, groupoid actions, matrix algebras