Lifting idempotents and exchange rings

Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given x e R with x-x2 cL, there exists an idempotent e c 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 S N and B C M. In 1972 Warfield showed that if M is a module over an associative ring R 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 Jonsson) that every projective module over an exchange ring is a direct sum of cyclic submodules. Let J(R) denote the Jacobson radical of R. Warfield showed that, if R/J(R) is (von Neumann) regular and idempotents can be lifted modulo J(R), then R is an exchange ring and so generalized theorems of Kaplansky and Muller. The main purpose of this paper is to prove the following theorem: A ring R is an exchange ring if and only if idempotents can be lifted modulo every left (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. 1. Suitable rings. In this section, the rings of interest are defined, some of their properties are deduced, and several examples are given. All rings are assumed to be associative with identity and J(R) denotes the Jacobson radical of a ring R. 1.1. PROPOSITION. If R is a ring, the following conditions are equivalent for an element x of R. Received by the editors December 2, 1975. AMS (MOS) subject classifications (1970). Primary 16A32, 16A64; Secondary 16A30, 16A50.