On Herstein's identity in prime rings

Abstract

A celebrated result of Herstein [10, Theorem 6] states that a ring \(R\) must be commutative if \([x,y]^{n(x,y)}=[x,y]\) for all \(x,y\in R,\) where \(n(x,y)>1\) is an integer. In this paper, we investigate the structure of a prime ring satisfies the identity \(F([x,y])^{n}=F([x,y])\) and \(\sigma([x,y])^{n}=\sigma([x,y]),\) where \(F\) and \(\sigma\) are generalized derivation and automorphism of a prime ring \(R\), respectively and \(n>1\) a fixed integer.