Lifting idempotents and exchange rings

Abstract

Abstract.Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given x E R with x -x2 6 L, there exists an idempotent e S R such that e -x e L. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever P = N + M where N and M are submodules, there is a decomposition P = A © B with A C. N and B Q M.In 1972 Warfield showed that if M is a module over an associative ring 7? then M has the finite exchange property if and only if end M has the exchange property as a module over itself. He called these latter rings exchange rings and showed (using a deep theorem of Crawley and Jónsson) that every projective module over an exchange ring is a direct sum of cyclic submodules.Let /(7?) denote the Jacobson radical of R. Warfield showed that, if R/J(R) is (von Neumann) regular and idempotents can be lifted modulo /(7?), then 7? is an exchange ring and so generalized theorems of Kaplansky and Müller.The main purpose of this paper is to prove the following theorem: A ring R is an exchange ring ¡S and only // idempotents can be USted modulo every leSt (respectively right) ideal. The properties of these rings are examined in the first section and the theorem is proved in the second section. The theorems of Warfield are then easily deduced and a new condition that a projective module have the finite exchange property is given.

g.x/-full clean rings

“…Clean rings were introduced by Nicholson [4]. A ring R is called clean if for every element a 2 R, there exist an idempotent e and a unit u in R such that a D e C u.…”

Maximal residuated lattices with lifting boolean center

“…Maximal rings are an important class of commutative rings with unit; we refer to [2] for a book treatment. The idea of lifting idempotents, due to Nicholson [16], turns out to be very useful in studying different classes of rings.…”

Two-Sided Properties of Elements in Exchange Rings

“…In [14], Nicholson proved that suit (R) = suit r (R) for any ring R. The rings R with the property that R = suit r (R) are called (right) suitable rings. In the seminal paper [13], Nicholson proved that these are precisely Warfield's exchange rings in [17]; namely, those rings R for which the right module R R satisfies the (finite) exchange property of Crawley and Jónsson. In our paper, we'll use the terms "suitable rings" and "exchange rings" interchangeably, noting that suitable elements are sometimes also called "exchange elements" in the literature.…”

“…Various characterizations for right suitable elements in rings were given in [13]. Among the most useful ones is the following "Goodearl-Nicholson characterization" (see [7] and [13]): a ∈ suit r (R) if and only if there exists an idempotent e ∈ aR such that 1 − e ∈ (1 − a) R. (In particular, R is an exchange ring if and only if every element a ∈ R has the above idempotent property.)…”