A Class of Exchange Rings

Abstract

Abstract.It is well known that a ring R is an exchange ring iff, for any a ∈ R, a − e ∈ (a 2 − a)R for some e 2 = e ∈ R iff, for any a ∈ R, a − e ∈ R(a 2 − a) for some e 2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a − e ∈ (a 2 − a)R for a unique e 2 = 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), 227-236) 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.1991 Mathematics Subject Classification. Primary 16U99. Introduction.Rings R are associative with unity. We write J(R) and U(R) for the Jacobson radical and the group of units of R. In [13], Nicholson observed an interesting relation between elements in a ring R,

A Class of Quasipolar Rings

“…The following are equivalent for a ring R Recall that a ring R is local if for any a ∈ R, either a or 1 − a is invertible in R (see [1]). Similar to [11,Proposition 10], we have the following result.…”

A class of uniquely (strongly) clean rings

“…Note that there exists a ring R with R/δ r is Boolean but such that idempotents do not lift modulo δ r . There is a ring R with R/J(R) Boolean but such that idempotents do not lift modulo J(R) (see [13,Example 15] …”

“…Note that there exists a ring R with R/δ r is Boolean but such that idempotents do not lift modulo δ r . There is a ring R with R/J(R) Boolean but such that idempotents do not lift modulo J(R) (see [13,Example 15]). In this ring, idempotents do not lift modulo δ r , for, if they did, then R would be δ r -clean and therefore exchange, by Theorem 2.2 below.…”

“…2) R is exchange by Theorem 2.2. If we show that every nonzero idempotent of R is not the sum of 2 units, then by [13,Theorem 13], R/J will be Boolean. Let e be a nonzero idempotent in R .…”

Exchange Ideals with All Idempotents Central

“…As is well known, an ideal I of a ring R is an exchange ideal if and only if for any x ∈ I, there exists an idempotent e ∈ xR such that 1 − e ∈ (1 − x)R (cf. [5], [6] and [7]). We say that R is an exchange ring provided that R itself is an exchange ideal of R. For general theory of exchange rings, refer to [3] and [10].…”