An Engel condition with derivation for left ideals

Abstract

We generalize a number of results in the literature by proving the following theorem: Let R be a semiprime ring, D a nonzero derivation of R, L a nonzero left ideal of R, and let [x, y] = xy yx. If for some positive integers tooth, ... , tn, and all x E L, the identity [[... [[D(xtO),xtl],xt2], . .],Xtn] -0 holds, then either D(L) = 0 or else the ideal of R generated by D(L) and D(R)L is in the center of R. In particular, when R is a prime ring, R is commutative. In this paper we prove a theorem generalizing several results, principally [20] and [9], which combine derivations with Engel type conditions. Before stating our theorem we discuss the relevant literature. If one defines [x, y]0 = x and [x, y]1 = [x, y] = xy yx, then an Engel condition is a polynomial [x, Y]n+l = [[X, Y]n, y] in noncommuting indeterminates. A commutative ring satisfies any such polynomial, and a nilpotent ring satisfies one if n is sufficiently large. The question of whether a ring is commutative, or nilpotent, if it satisfies an Engel condition goes back to the well known work of Engel on Lie algebras [15, Chapter 2], and has been considered, with various modifications, by many since then (e.g. [2] or [7]). The connection of Engel type conditions and derivations appeared in a well known paper of E. C. Posner [23] which showed that for a nonzero derivation D of a prime ring R, if [D(x),x] is central for all x C R, then R is commutative. This result has led to many others (see [19] for various references), and in particular to a result of J. Vukman [25] showing that if [D(x), X]2 is central for all x E R, a prime ring with char R + 2,3, then again R is commutative. We extended this result [20] by proving that if [D(x), X]n = 0 for all x c I, an ideal of the prime ring R, then R is commutative, and if instead, this Engel type condition holds for all x C U, a Lie ideal of R, then R embeds in M2(F) for F a field with char F 2. Recently, [9] proved that for a left ideal L of a semiprime ring R, either D(L) = 0 or R contains a nonzero central ideal if either: R is 6-torsion free and [D(x), x]2 is central for all x E L; or if [D(x), Xn] is central for all x C L and R is n!-torsion free. The first of these conditions generalized [1, Theorem 3, p. 99], which assumed that [D(x), x] is central for all x E L, with no restriction on torsion. The second, involving powers, is related to both [12], which showed that a prime ring R is commutative if D(Xk) = 0 for all x C R, and to [8], a significant extension of [12], showing that R is commutative if it contains no nonzero nil ideal and [D(xk(x)), Xk(x)]n = 0 on Received by the editors August 2, 1995. 1991 Mathematics Subject Classification. Primary 16W25; Secondary 16N60, 16U80. ?D1997 American Mathematical Society