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A ring idempotentis said to be  quarter-central  (or q-central for short) if. If all idempotents in a ringare q-central,is said to be a  quarter-abelian ring  (or more simply, a q-abelian ring). In this paper, we show that such a ringis characterized by the property thatfor all idempotents, wheredenotes the additive commutator. For any nonzero ringand any integer, we show that the ringofupper triangular matrices overis not q-abelian. On the other hand,is q-abelian iffis abelian (in the classical sense that all idempotents are central in). A final section of this paper relates q-central idempotents to the notion of exchange rings and the study of regular, unit-regular, and strongly regular elements in arbitrary rings. From the viewpoint of currently prevailing generalizations of abelian rings in the literature, q-abelian rings are situated between the class of semiabelian rings and the class of strongly IC rings.

An introduction to q-central idempotents and q-abelian rings