Range-inclusive maps in rings with idempotents

Abstract

Let A be a ring, let M be an A-bimodule, and let C be the center of M. A map F:A → M is said to be range-inclusive if [F(x), A] ⊆[x, M] for every x ∈ A. We show that if A contains idempotents satisfying certain technical conditions (which we call wide idempotents), then every range-inclusive additive map F:A → M is of the form F(x)= λx + μ(x) for some λ ∈ C and μ : A → C. As a corollary we show that if A is a prime ring containing an idempotent different from 0 and 1, then every range-inclusive additive map from A into itself is commuting (i.e., [F(x), x]=0 for every x ∈ A).