When is a Multiplicative Derivation Additive?

Abstract

Our main objective in this note is to prove the following. Suppose R is a ring having an idempotent element e (e≠0, e≠l) which satisfies: (M1) xR=0 implies x=0. (M2) eRx=0 implies x=0 (and hence Rx=0 implies exe=0). (M3) exeR(l-e)=0 implies exe=0. If d is any multiplicative derivation of R, then d is additive. © 1991, Hindawi Publishing Corporation. All rights reserved.