The Ideal Intersection Property for Groupoid Graded Rings

TL;DR

This paper investigates the ideal intersection property in groupoid graded rings, establishing conditions under which the commutant of the center of the principal component intersects nontrivially with all nonzero ideals, especially in skew groupoid algebras.

Contribution

It introduces new criteria linking the ideal intersection property to the maximal commutativity of the principal component in groupoid graded rings.

Findings

The commutant of the center intersects all nonzero ideals under certain ideal properties.

In skew groupoid algebras with commutative principal component, maximal commutativity is equivalent to the ideal intersection property.

Abstract

We show that if a groupoid graded ring has a certain nonzero ideal property, then the commutant of the center of the principal component of the ring has the ideal intersection property, that is it intersects nontrivially every nonzero ideal of the ring. Furthermore, we show that for skew groupoid algebras with commutative principal component, the principal component is maximal commutative if and only if it has the ideal intersection property.