Noncrossed Product Matrix Subrings and Ideals of Graded Rings

TL;DR

This paper investigates the structure of ideals and centers in groupoid graded rings, establishing conditions for maximal commutativity and nontrivial intersections of ideals, and providing examples of non-crossed product graded rings.

Contribution

It introduces new criteria for ideal intersections and maximal commutativity in groupoid graded rings, and constructs examples of non-crossed product graded rings for finite groupoids.

Findings

Nonzero ideal intersection with the commutant under certain conditions.

Characterization of maximal commutative principal components.

Existence of strongly graded rings that are not crossed products.

Abstract

We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component of the ring is nonzero. Furthermore, we show that for a skew groupoid ring with commutative principal component, the principal component is maximal commutative if and only if it is intersected nontrivially by each nonzero ideal of the skew groupoid ring. We also determine the center of strongly groupoid graded rings in terms of an action on the ring induced by the grading. In the end of the article, we show that, given a finite groupoid $G$, which has a nonidentity morphism, there is a ring, strongly graded by $G$, which is not a crossed product over $G$.