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

A ring idempotent e∈R is said to be quarter-central (or q-central for short) if e R (1−e) R e=0. If all idempotents in a ring R are q-central, R is said to be a quarter-abelian ring (or more simply, a q-abelian ring). In this paper, we show that such a ring R is characterized by the property that [ e,R ] [e,R ]=0 for all idempotents e∈R, where [x,y] denotes the additive commutator xy−yx. For any nonzero ring S and any integer n≥3, we show that the ring Tn(S) of n×n upper triangular matrices over S is not q-abelian. On the other hand, T2(S) is q-abelian iff S is abelian (in the classical sense that all idempotents are central in S). 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.