A note on annihilator conditions in prime rings

Abstract

Let R be a prime ring with char\((R)\ne 2\) and a be a nonzero element in R. If F is a generalized derivation associated with a nonzero derivation d of R and \(k>1\) is a fixed integer such that \(a \Big (F([x,y]_{k})- [x,y]\Big )=0\) for all \(x,y \in R\), then R is commutative. Moreover, we will also study an identity involving an automorphism in prime ring.