Central idempotents in group-graded rings

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.