Central idempotents in group-graded rings

Abstract

Let $G$ be a group and let $R$ be a $G$-graded ring. We show that every 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 earlier results by H. Bass, R. G. Burns, and A. A. Bovdi--S. V. Mihovski, from group rings and crossed products, 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.

11 pages. Minor improvements of the presentation from v1 with the addition of references, Example 3.2 and Remark 3.7