A Note on Range-Inclusive Maps in Rings with Idempotents

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. Recently, Bre?ar proved that if A is a unital ring and M a unital A-bimodule such that A contains 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. Our main purpose is to remove the assumption of unitality in the above result.