Central idempotents in group-graded rings

TL;DR

This paper investigates the structure of central idempotents in group-graded rings, establishing conditions under which they have finite support and lie in the principal component, generalizing classical results.

Contribution

It extends classical results on group rings to non-commutative, possibly non-unital, group-graded rings under broad conditions.

Findings

Central idempotents have finite support in abelian or certain non-abelian group-graded rings.

If the group is torsion-free, all central idempotents are in the principal component.

Results apply to various algebraic structures like Leavitt path rings and Cuntz-Pimsner rings.

Abstract

Let $G$ be a group and let $R$ be a $G$-graded ring. We show that a nonzero central idempotent in $R$ has finite support group in two broad settings: when $G$ is abelian, and when $G$ is arbitrary but the grading satisfies a certain one-sided non-annihilation condition on nonzero homogeneous elements. In particular, under the respective hypotheses, if $G$ is torsion-free, then every central idempotent lies in the principal component of the grading. Our results generalize those of H. Bass and R. G. Burns from group rings to non-commutative, possibly non-unital, group-graded rings. We demonstrate the utility of our results by applying them to semigroup-graded rings, Leavitt path rings, fractional skew monoid rings, partial skew group rings, and algebraic Cuntz-Pimsner rings.