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

Abstract

A ring idempotent is said to be quarter-central (or q-central for short) if . If all idempotents in a ring are 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 ring is characterized by the property that for all idempotents , where denotes the additive commutator . For any nonzero ring and any integer , we show that the ring of upper triangular matrices over is not q-abelian. On the other hand, is q-abelian iff is 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.