When in a multiplicative derivation additive?

Abstract

Our main objective in this note is to prove the following. SupposeRis a ring having an idempotent elemente(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.Ifdis any multiplicative derivation ofR, thendis additive.