Lie ideals and derivations of prime rings

Abstract

In this paper, we shall consider the relationship between the derivations and Lie ideals of a prime ring. Some of the results we obtain have been obtained earlier, even for rings more general than prime rings, in the case of inner derivations. We shall also look at the action of derivations on Lie ideals; the results we obtain extend some that had been proved earlier only for the action of derivations on the ring itself. Let R be a ring and d # 0 a derivation of R. If U is a Lie ideal of R, we shall be concerned about the size of d(C/). How does one measure this size? One way is to look at the centralizer of d(U) in R; the bigger d(U), the smaller this centralizer should be. This explains our interest in the centralizer of d(U). The result we obtain generalizes the principal theorem of [ 11. We may also measure the size of d(U) by looking at how large d(U), the subring generated by d(U), turns out to be. We view d(U) as large if it contains a non-zero ideal of R. For our special setting, we will obtain a result which generalizes one in [2]. Finally, a well-known and often used result states that if d is a derivation of R, which is semi-prime and 2-torsion-free, such that dZ = 0 then d = 0 (see the proof of Lemma 1.1.9 in [3]). If R is prime, of characteristic not 2, and d*(I) = 0 for a non-zero ideal, 1, of R, it also follows that d = 0. What can one say if d’(U) = 0 for some non-central Lie ideal of R? For inner derivations this was studied and answered in [4]. For prime rings and for any derivation d # 0 we answer the question of when d*(U) = 0 completely in our Theorem 1. We shall be working in the context of prime rings of characteristic not 2 in all that we do here. However, many of the results have some suitable analog for semi-prime, 2-torsion-free rings. In the presence of 2-torsion most of our results are not valid as stated, but something non-trivial can be said in