Lifting idempotents and exchange rings

Source abstract

Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given x ∈ R x \in R with x − x 2 ∈ L x - {x^2} \in L , there exists an idempotent e ∈ R e \in R such that e − x ∈ L e - x \in 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 P = N + M where N and M are submodules, there is a decomposition P = A ⊕ B P = A \oplus B with A ⊆ N A \subseteq N and B ⊆ M B \subseteq M .