Every nonzero central idempotent in a G-graded ring has finite support group if G is abelian or the grading meets a one-sided non-annihilation condition on nonzero homogeneous elements.
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Central idempotents in group-graded rings
Every nonzero central idempotent in a G-graded ring has finite support group if G is abelian or the grading meets a one-sided non-annihilation condition on nonzero homogeneous elements.