Commutativity results for semiprime rings with derivations

Abstract

We extend a result of Herstein concerning a derivationdon a prime ringRsatisfying[d(x),d(y)]=0for allx,y∈R, to the case of semiprime rings. An extension of this result is proved for a two-sided ideal but is shown to be not true for a one-sided ideal. Some of our recent results dealing withU*- andU**- derivations on a prime ring are extended to semiprime rings. Finally, we obtain a result on semiprime rings for whichd(xy)=d(yx)for allx,yin some idealU.