On derivations in Banach algebras

Abstract

Our main goal is to show that if there exist Jordanderivations D and G on a noncommutative (n+1)!-torsion freeprime ring R such that D(x)x^{n}-x^{n}G(x) in C(R) forall x in R, then we have D=0 and G=0. We also prove that ifthere exists a derivation D on a noncommutative 2-torsion freeprime ring R such that the mapping x mapsto [aD(x),x] iscommuting on R, then we have either a=0 or D=0.