A class of exchange rings

Abstract

It is well known that a ring R is an exchange ring iff, for any a R, ae (a2a)R for some e2 = e R iff, for any a R, ae R(a 2a) for some e2 = e R. The paper is devoted to a study of the rings R satisfying the condition that for each a R, ae (a2a)R for a unique e2 = e R. This condition is not leftright symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given. © 2008 Glasgow Mathematical Journal Trust.