MIYASHITA ACTION IN STRONGLY GROUPOID GRADED RINGS

Year 2012, Volume: 11 Issue: 11, 46 - 63, 01.06.2012

Johan Öinert Patrik Lundström

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) → 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.

Keywords

graded rings, commutants, groupoid actions, matrix algebras