Charles Lanski

An Engel condition with derivation

Let R be a prime ring, L a noncommutative Lie ideal of R, and D a nonzero derivation of R. If for each x E L, [D(x), xlk = [[ ... [D(x), x], x], ... , x] = 0 with k fixed, then char(R) = 2 and R C M2(F) for F a field. In a recent paper, Vukman [12] gives an elementary but lengthy calculation that yields extensions of a well-known theorem of Posner [10] on centralizing derivations of prime rings. The purpose of this note is to show that by using the theory of differential identities one can fairly quickly generalize the results of Vukman to higher commutators, eliminate his restriction on characteristic, and extend the results from prime rings to Lie ideals in prime rings. The theory of differential identities was formally initiated by Kharchenko [5]. Its use here will not be explicit and arises only in reference to results in subsequent work of Chuang [1] and the author [6]. The one related object we need to mention is the symmetric quotient ring, introduced in [5] and required in [1] and [6] (see [9] for some details). Henceforth, we let R denote a prime ring with extended centroid C and symmetric quotient ring Q. All that we need here about these objects is that R C Q, Q is a prime ring whose center is the field C, and that C is the centralizer of R in Q. By D we always mean a nonzero derivation of R, and D = ad(A) for A E Q implies that D(r) = [A, r] = ArrA. In [10] Posner proved that R must be commutative if [D(x), x] is central for any x E R. Set [y, x]1 = [y, x] = yx xy for any x, y E R, and for k > 1 let [y, X]k = [[y, X]k-1, x]. The results of Vukman [12] show that R is commutative if either [D(x), X]2 = 0 for all x E R and char(R) :

An Engel condition with derivation for left ideals

2 or if [D(x), X]2 is central for all x E R and char(R) :

On the primitivity of prime rings

2, 3. We consider the more general Engel condition when for a fixed k > 0, [D(x), X]k = 0 for all x E L, a noncommutative Lie ideal of R1. Our first theorem will give the result for ideals. It incorporates the arguments needed for Lie ideals but avoids some technical complications of that case. Two well-known observations are crucial to our arguments and for convenience we state them as lemmas. Received by the editors January 8, 1991 and, in revised form, November 11, 1991. 1991 Mathematics Subject Classification. Primary 16W10; Secondary 16W25, 16N60, 16U80. i 1993 American Mathematical Society 0002-9939/93

Pseudosimilarity and cancellation of modules

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Differential identities in prime rings with involution

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On the centralizers of ideals and nil derivations

We generalize a number of results in the literature by proving the following theorem: Let R be a semiprime ring, D a nonzero derivation of R, L a nonzero left ideal of R, and let [x, y] = xy yx. If for some positive integers tooth, ... , tn, and all x E L, the identity [[... [[D(xtO),xtl],xt2], . .],Xtn] -0 holds, then either D(L) = 0 or else the ideal of R generated by D(L) and D(R)L is in the center of R. In particular, when R is a prime ring, R is commutative. In this paper we prove a theorem generalizing several results, principally [20] and [9], which combine derivations with Engel type conditions. Before stating our theorem we discuss the relevant literature. If one defines [x, y]0 = x and [x, y]1 = [x, y] = xy yx, then an Engel condition is a polynomial [x, Y]n+l = [[X, Y]n, y] in noncommuting indeterminates. A commutative ring satisfies any such polynomial, and a nilpotent ring satisfies one if n is sufficiently large. The question of whether a ring is commutative, or nilpotent, if it satisfies an Engel condition goes back to the well known work of Engel on Lie algebras [15, Chapter 2], and has been considered, with various modifications, by many since then (e.g. [2] or [7]). The connection of Engel type conditions and derivations appeared in a well known paper of E. C. Posner [23] which showed that for a nonzero derivation D of a prime ring R, if [D(x),x] is central for all x C R, then R is commutative. This result has led to many others (see [19] for various references), and in particular to a result of J. Vukman [25] showing that if [D(x), X]2 is central for all x E R, a prime ring with char R + 2,3, then again R is commutative. We extended this result [20] by proving that if [D(x), X]n = 0 for all x c I, an ideal of the prime ring R, then R is commutative, and if instead, this Engel type condition holds for all x C U, a Lie ideal of R, then R embeds in M2(F) for F a field with char F 2. Recently, [9] proved that for a left ideal L of a semiprime ring R, either D(L) = 0 or R contains a nonzero central ideal if either: R is 6-torsion free and [D(x), x]2 is central for all x E L; or if [D(x), Xn] is central for all x C L and R is n!-torsion free. The first of these conditions generalized [1, Theorem 3, p. 99], which assumed that [D(x), x] is central for all x E L, with no restriction on torsion. The second, involving powers, is related to both [12], which showed that a prime ring R is commutative if D(Xk) = 0 for all x C R, and to [8], a significant extension of [12], showing that R is commutative if it contains no nonzero nil ideal and [D(xk(x)), Xk(x)]n = 0 on Received by the editors August 2, 1995. 1991 Mathematics Subject Classification. Primary 16W25; Secondary 16N60, 16U80. ?D1997 American Mathematical Society

The group of units of a simple ring II

In this paper we obtain some conditions which force prime rings to be primitive. Our main theorems are converses to well-known results on the primitivity of certain subrings of primitive rings. Applications are given to the case of primitive domains, and a tensor prod&t theorem is proved which answers a question of Herstein on the primitivity of E[x, ,..., x,J, for E the endomorphism ring of a vector space over a division ring. Throughout the paper, all modules are right (unital) modules and “primitive” will mean right primitive. When R has an identity, the existence of a faithful irreducible R module is equivalent to the existence in R of a proper right ideal T satisfying T + I = R for every nonzero ideal I of R [2, Theorem 1, p. 5081. We begin by stating a useful and well-known result, the proof of which is straightforward using the existence of a faithful irreducible module.

Generalized Derivations and nth Power Maps in Rings

Abstract Two square matrices A and B over a ring are pseudosimilar if there exist X , Y , and Z satisfying XAY = B , ZBX = A , and XYX = XZX = X . Hartwig and Hall showed this is equivalent to similarity over a field. This result is extended to rings where free modules satisfy a cancellation property. These include rings R with R /rad R artinian (or more generally rings with one in the stable range) and polynomial rings over Dedekind domains. Furthermore, it is shown for commutative rings that if A and B are pseudosimilar, then diag( A, O m ) and diag( B, O m ) are similar for some m .

The rank of a commutator

Let R be a prime ring with involution. Using work of V. K. Kharchenko it is shown that any generalized identity for R involving derivations of R and the involution of R is a consequence of the generalized identities with involution which R satisfies. In obtaining this result, a generalization, to rings satisfying a GPI, of the classical theorem characterizing inner derivations of finite-dimensional simple algebras is required. Consequences of the main theorem are that in characteristic zero no outer derivation of R can act algebraically on the set of symmetric elements of R, and if the images of the set of symmetric elements under the derivations of R satisfy a polynomial relation, then R must satisfy a generalized polynomial identity. This paper deals with differential identities of prime rings with involution, and was motivated by work of V. K. Kharchenko and of 1. N. Herstein. In [5], Kharchenko shows that the differential identities of prime rings are consequences of formal identities for endomorphisms, satisfied in any ring, and of the generalized polynomial identities satisfied by the prime ring under consideration. In [4], Herstein proves that a certain identity, namely D(s)D(t) D(t)D(s) = 0, where D is a derivation and both s and t are symmetric elements, cannot hold in a prime ring of characteristic different from two, unless the ring satisfies the standard identity of degree four. The extension of Kharchenkos theorem to differential identities involving involution would provide a more general context for the result of Herstein. Our goal is to provide a careful setting for the theory of differential identities (with involution) and to prove an extension of Kharchenkos theorem which shows that the differential identities of a prime ring with involution are consequences of the formal identities for endomorphisms and of the generalized polynomial identities with involution satisfied by the ring under consideration. The proof relies heavily on the result and techniques of Kharchenko, although we have attempted to make our exposition as self-contained as possible. In particular, in our Theorem 1, we adapt Kharchenkos argument [5, Lemma 2, p. 158] to our more general setting. Our main result also requires an extension to rings satisfying a generalized polynomial identity of the classical result characterizing inner derivations of finite-dimensional simple algebras. We use our main result to show, as in [5], that when the characteristic of the ring is zero, any derivation which is algebraic when restricted to the symmetric or skew-symmetric elements must be an inner derivation. We also show that a prime ring with involution, must satisfy a generalized polynomial identity if its symmetric, or skew-symmetric elements satisfy an identity of the form Received by the editors February 8, 1984. 1980 Mathematics Subject Classification. Primary 16A38; Secondary 16A28, 16A72, 16A12, 16A48. 1 Most of the research in this paper was done while the author was visiting the Department of Mathematics at VPI and SUo The hospitality of the department is gratefully acknowledged. 765 ©1985 American Mathematical Society 0002-9947/85

Ring elements as sums of units

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