A result on derivations

Abstract

Let R be a semiprime ring with a derivation d and let U be a Lie ideal of R, a Ef R. Suppose that ad(u)n = 0 for all u G U, where n is a fixed positive integer. Then ad(I) = 0 for I the ideal of R generated by [U, U] and if R is 2-torsion free, then ad(U) = 0. Furthermore, R is a subdirect sum of semiprime homomorphic images RI and R2 with derivations d1 and d2, induced canonically by d, respectively such that ad1(Rf) = 0 and the image of U in R2 is commutative (central if R is 2-torsion free), where -a denotes the image of a in R1. Moreover, if U = R, then ad(R) = 0. This gives Bregar's theorem without the (n 1)!-torsion free assumption on R. In [8] I. N. Herstein proved that if R is a prime ring and d is an inner derivation of R such that d(x)n = 0 for all x E R and n a fixed integer, then d = O. In [6] A. Giambruno and I. N. Herstein extended this result to arbitrary derivations in semiprime rings. In [2] L. Carini and A. Giambruno proved that if R is a prime ring with a derivation d such that d(x)n(x) = 0 for all x E U, a Lie ideal of R, then d(U) = 0 when R has no nonzero nil right ideals, charR 7f 2 and the same conclusion holds when n(x) = n is fixed and R is a 2-torsion free semiprime ring. Using the ideas in [2] and the methods in [5] C. Lanski [11] removed both the bound on the indices of nilpotence and the characteristic assumptions on R. In [1] M. Bresar gave a generalization of the result due to I. N. Herstein and A. Giambruno [6] in another direction. Explicitly, he proved the theorem: Let R be a semiprime ring with a derivation d, a E R. If ad(x)n = 0 for all x E R, where n is a fixed integer, then ad(R) = 0 when R is an (n 1)!-torsion free ring. The present paper is then motivated by Bresar's result and by Lanski's paper [11]. We prove Bresar's result without the assumption of (n 1)!-torsion free on R. In fact, we study the Lie ideal case as given in [11] and then obtain Bresar's result as the corollary to our main result. More precisely, we shall prove the following Main Theorem. Let R be a semiprime ring with a derivation d and let U be a Lie ideal of R, a E R. Suppose that ad(u)n = 0 for all u E U, where n is a fixed integer. Then ad(I) = 0 for I the ideal of R generated by [U, U] and if R is 2-torsion free, then ad(U) = 0. Furthermore, R is a subdirect sum of semiprime homomorphic images R1 and R2 with derivations d1 and d2, induced canonically by d, respectively such that Received by the editors March 28, 1994 and, in revised form, May 9, 1994 and December 9, 1994. 1991 Mathematics Subject Classification. Primary 16W25.