Derivations with nilpotent values on Lie ideals

Abstract

Let R be a ring containing no nonzero nil right ideal and let LU be a Lie ideal of R. If d is a derivation of R so that d(u) is a nilpotent element for each it E U, then d = 0 when R is a prime ring and Uis not commutative. The main result shows that in general, d(I) = 0 for I the ideal R generated by [U-, U] and that R is the subdirect sum of two images so that d induces the zero derivation on one, and the image of U in the other is commutative. This paper continues a line of investigation in the literature concerning derivations having nilpotent values. The first such result, which inspired all those which followed, is due to I. N. Herstein [6] who proved that if R is a prime ring and d is an inner derivation of R satisfying = 0 for all x E R and n a fixed integer, then d = 0. This result was extended to arbitrary derivations in semiprime rings in [3]. In [2], the authors considered derivations satisfying d(x) n= 0 for all x E I, an ideal of R, and proved that d(I) = 0 when R has no nil right ideal. Using the general approach in [2], Carini and Giambruno [1] study the situation when d(x)f(X) = 0 for all x E U, a Lie ideal of R. They prove first that d(U) = 0 when R is a prime ring, charR 7& 2, and R contains no nil right ideal, and then obtain the same conclusion when n(x) = n is fixed and R is a 2-torsion free semiprime ring. Our purpose here is to remove both the bound on the indices of nilpotence and the characteristic assumptions on R. We rely on the ideas in [1] to handle the semisimple case, and the methods in [2] to complete the proof in general. Throughout this paper let R denote a ring containing no nonzero nil right ideal, U a Lie ideal of R, Z the center of R, and d a derivation of R. For any nonempty subset A of R, call d nil on A if d(a)n(a) = 0 for each a E A. It is clear that to conclude that d = 0, or that d(U) = 0, when d is nil on U, one must assume that R does not contain a nil ideal I, since otherwise for any y E I, [r,y] = d,(r) E I is nilpotent and so d,, is nil on R. The reason why we must assume that R contains no nil right ideal is given in [2]: If d is nil Received by the editors September 23, 1988 and, in revised form, February 23, 1989. 1980 Alathematics Subject Classification (1985 Revtision). Primary 16A72; Secondary 16A68, 1 6A 1 2 ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $ 25 per page