AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS

Abstract

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.