W. S. Martindale

Lie isomorphisms of prime rings

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 Huas 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.

On Herstein’s Lie map conjectures, I

First published in Transactions- American Mathematical Society in Vol.353, No.10, pp.4235-4260, published by the American Mathematical Society

Herstein's Lie theory revisited

An associative ring R becomes a Jordan ring R + under x 0 y = xy + yx and a Lie ring R under [x, y] = xy yx. In case R has an involution *, i.e., an antiautomorphism of period 1 or 2, then the set of symmetric elements S = {s E R 1 s* = S} is a Jordan subring of R + and the set of skew elements K= {kER 1 k*= -k} is a Lie subring of Rp. In the early 1950s Herstein initiated a study of the Jordan and Lie ideals of R, S, and K in case that R was a simple associative ring (either without or with an involution). In the ensuring years his work has been generalized in various directions, on the one hand, to the setting of prime and semiprime rings, and, on the other hand, to invariance conditions other than that given by ideals. Besides Herstein himself we mention Lanski as having been a major force in this program. Their influence is certainly felt in the present paper; we have taken the liberty of using various arguments (without making specific mention) from their papers, notably [6] and [S]. Other important contributions in this area have also been made by Baxter, Chacron, Erickson, Montgomery, Osborn, Streb, and others. Part of our motivation in writing this paper is to obtain the Lie ideal theory for semiprime rings with involution by a somewhat different approach from the self contained, elementary, very clever methods embodied in the original style of Herstein. Let us describe very briefly the spirit of our approach. For R a ring with involution * and 1 any *-ideal (i.e., an ideal invariant under *) [In K, K] is always a Lie ideal of K, which we shall call a standard Lie ideal of K. A

Prime rings with involution and generalized polynomial identities

Our main purpose in writing this paper is to prove that if R is a prime ring with involution whose symmetric elements satisfy a generalized polynomial identity over the extended centroid C of R, then the central closure A = RC + C must in fact be a primitive ring with a minimal right ideal eA such that eAe is a finite dimensional division algebra over C. This generalizes a previous theorem of ours [8], in which we obtained the above result for the case where R was assumed to be primitive. It also generalizes a recent result of Skinner [lo, Theorem 5.11, where the above result was obtained for the case where R was a prime Goldie ring. In the course of proving this theorem, and without digressing too much from our main goal, we rework and combine the techniques of Amitsur in [l] and [2] with those of ours in [7], so as to obtain the various structure theorems (due to Amitsur, Posner, Herstein, Kaplansky, and the author) on simple, primitive, and prime rings (with involution) whose (symmetric) elements satisfy a (generalized) polynomial identity. We hope this attempt to give a more or less unified approach to a group of theorems, the existing proofs of which are not for the most part too closely related, will be of general interest. In Section 2, we recall the notion of extended centroid of a prime ring and discuss some of its key properties (Theorems 2.1-2.4). Next, putting together some ideas of Amitsur, we give a particular way of embedding prime rings in primitive rings (Theorems 2.5-2.8). Finally, we show how information about primitive rings can be pulled back to prime rings (Theorems 2.9-2.10). In Section 3, we recall the notion of generalized multilinear identity (GMI) and show (Theorems 3.1-3.2) that GMI’s are carried over by the aforementioned embedding of prime rings in primitive rings. A fundamental result of Amitsur (Theorem 3.3) together with its corollary, Theorem 3.5, are applicable to primitive rings and say in effect that some nonzero linear

On functional identities in prime rings with involution II

Let A be a prime ring with involution *Kbe its set of skew element, and R be a noncentral Lie ideal of K. Further, let Q be the maximal right ring of quotients of A and be maps. We study functional identities of the form for all (where means etc). In case A does not satisfy the standard identity of degree ≤ 4(m + 1), definitive results are obtained.

Multiplicative Semiprimeness of Skew Lie Algebras

Abstract Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z [K, K] denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z [K, K] is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z [K, K] is prime, whenever [K, K]/Z [K, K] is prime. We will obtain similar results for the Lie algebra K/Z K whenever the base field has characteristic zero.

The normal closure of coproducts of domains

Let R, and R, be algebras with 1 over a common field F, with each (Rj: F) > 1, and let R = R, II R, denote the coproduct (often called the free product) of R, and R, over F. It is easy to show that R is always prime; in fact R is primitive in case at least one of (Ri: F) > 2 171. On the other hand, when (Ri: F) = 2, i= 1,2, it is clear from [ 1, p. 261 that R is prime PI but never primitive. If R, and R, are domains then we have Cohn’s result [2 i that R is again a domain. In studying certain automorphisms of a prime ring R one is led in a natural way to the normal closure of R, and we now briefly recall the meaning of this notion. If R, is the left quotient ring of R relative to the filter F of all nonzero two-sided ideals of R, the set N* of all units u of R-,such that uwlRu = R is called the set of normalizing elements [ 10, p. 5 ]. The automorphisms thus induced on R are just the X-inner automorphisms of Kharchenko [IO, p. 31. The subring RN of R generated by R and N = N* U (0) is called the normal closure of R. An important subset of N is the center C of R-, (called the extended center of R); it is known that C is a field. We now return to the coproduct R = R, II R, , where R L and R2 are arbitrary F-algebras. It is known in this situation that if at least one of (Ri: F) > 2, then C = F [8]; when both (R,: F) = 2, then C =F(tj G R [ 1 j. Now fix F-bases {xii u 1 for R, and ( ~7~) u 1 for R2 and call the various products of alternating xI(s and -yj’s basis monomials. The degree of a basis

Polynomial preserving maps on certain Jordan algebras

LetB andQ be associative algebras and letS be a Jordan subalgebra ofB. Letf(x1,…,xm) be a (noncommutative) multilinear polynomial such thatS is closed underf. Letα:S→Q be anf-homomorphism in the sense that it is a linear map preservingf. Under suitable conditions it is shown thatα is essentially given by a ring homomorphism. An analogous theorem forf-derivations is also proved. The proofs rest heavily on results concerning functional identities andd-freeness.

Extended centroids of power series rings

Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study. The extended centroid has been computed for many classes of rings, such as some group rings [2], skew polynomial rings of automorphism and derivation type [7], [10], coproducts of algebras, and free algebras [6]. The ring of quotients Q(R) and its various subrings have had numerous applications. As mentioned above, they played a key role in the solution of the GPI problem. In [4], Kharchenko developed a Galois theory of semiprime rings which makes extensive use of these ideas. We refer the reader to [8] for a simplified account of this work over prime rings. In this paper, we determine the extended centroid of the power series ring R[[x]] over a closed prime ring R. One motivation for this problem comes from commutative algebra. In this case, finding the extended centroid of a prime ring is equivalent to finding its field of quotients. If F is a field, then the field of quotients of F[[x]] is F((x)), the field of Laurent series over F. However, if D is an integral domain then the field of quotients of D[[x]] is not known in general. In particular, it is rarely the case that the field of quotients of D[[x]] is F((x)), where F is the field of quotients of D. We will provide an example where D = Z (the integers) and refer the reader to [1], [3] for more general examples. If R is a closed prime ring over C, we prove the extended centroid of /?[[*]] is C((x)). This theorem generalizes the corresponding result for F[[JC]]. AS a corollary, we show that if R is a closed prime ring then so is the ring of Laurent series R((x)).

Generalized rational identities and rings with involution

The main goal of this paper is to present the following generalization of a theorem of Desmarais: LetD be a fixed division ring and letE⊇D be a division ring with involution * and with infinite centerC such that (E:C)=∞. ItS is the set of all 2m-tuples of the form (a1,a2, ...,am,a1*,a2*, ...,am*),ai∈E, then any generalized rational identity (overD) vanishing onS (where defined) must in fact vanish onE2m (where defined). The result follows as a corollary to Bergmans generalization of Amitsurs basic result on rational identities, and for completeness we present Cohns account of Bergmans result.