When in 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≠1) which satisfies: (M1)   xR=0  implies  x=0.(M2)   eRx=0  implies  x=0  (and hence  Rx=0  implies  x=0).(M3)   exeR(1−e)=0  implies  exe=0. If d is any multiplicative derivation of R, then d is additive.