Lie isomorphisms of prime rings

Abstract

for all x, y E S. Our interest and viewpoint toward the study of Lie isomorphisms of rings was originally (and still is) inspired by the work done by I. N. Herstein on generalizing classical theorems on the Lie structure of total matrix rings to results on the Lie structure of arbitrary simple rings. In our case the starting point was the realization that it should be possible to extend the following theorem of L. Hua [1]: every Lie automorphism of the ring R of all n x n matrices over a division ring, n >2, characteristic & 2, 3, is of the form a + -r, where a is either an automorphism or the negative of an antiautomorphism of R and X is an additive mapping of R into its center which maps commutators into zero. Indeed, using some of Hua's techniques and some valuable suggestions due to Nathan Jacobson, we were in [2], roughly speaking, able to obtain the same conclusion under the weaker assumptiop that R was merely a primitive ring possessing three orthogonal idempotents whose sum was 1. In a recent paper [3], while making the stronger assumption that R was simple, we were able to lower the number of idempotents from three to two. For the most part, the same techniques were used in this second paper, although a tensor product method due to Jacobson was to replace tedious calculations involving matrix units due to Hua, and some results of Herstein on Lie ideals of simple rings seemed necessary. Our goal in this paper is Theorem 11, in which we extend the above results to the situation where R is a prime ring with two orthogonal idempotents whose sum is 1. Primeness is a natural generalization of simplicity and primitivity, and, in the sense of keeping free of the radical and of (sub)direct sums of ideals, it is perhaps the strongest generalization one may make. Whether the assumption of idempotents is necessary or not is still a major open question. In all our work on the subject (including the present paper) our arguments rest heavily on the presence of a nontrivial idempotent. A successful removal of the assumption of idempotents would certainly require totally new methods; one would, for example, have to face the situation of an arbitrary division ring.