On some generalizations of abelian rings

Abstract

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings [Formula: see text] are always “J-abelian” in the sense that each idempotent of [Formula: see text] maps onto a central idempotent in [Formula: see text]. On the other hand, J-abelian rings turn out to be precisely the “strongly perspective rings” as well as the “strongly IC rings”, and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking [Formula: see text]-th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings.