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 ∈ R, wheren (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 σ([x,y])n=σ([x,y]), where F and σ are generalized derivation and automorphism of a prime ring R, respectively and n>1a fixed integer.