Pjek-Hwee Lee

Derivations with Engel conditions on multilinear polynomials

Let R be a prime algebra over a commutative ring K with unity and let f(X1, . . . ,Xn) be a multilinear polynomial over K. Suppose that d is a nonzero derivation on R such that for all r1, . . . , rn in some nonzero ideal I of R, [ d ( f(r1, . . . , rn) ) , f(r1, . . . , rn) ] k = 0 with k fixed. Then f(X1, . . . , Xn) is central–valued on R except when char R = 2 and R satisfies the standard identity s4 in 4 variables. Throughout this note K will denote a commutative ring with unity and R will denote a prime K–algebra with center Z. By d we always mean a nonzero derivation onR. For x, y ∈ R, set [x, y]0 = x, [x, y]1 = [x, y] = xy−yx and, for k > 1, [x, y]k = [ [x, y]k−1, y ] . A well–known result proved by Posner [10] states that R must be commutative if [ d(x), x ] ∈ Z for all x ∈ R. In [7], the authors generalized Posner’s theorem by showing that a Lie ideal L of R must be contained in Z if char R 6= 2 and [ d(x), x ] ∈ Z for all x ∈ L. As to the case when char R = 2, Lanski [5] obtained the same conclusion except when R satisfies the standard identity s4 in 4 variables. On the other hand, Vukman [11] showed that R is commutative if char R 6= 2 and [ d(x), x ] 2 = 0 for all x ∈ R, or if char R 6= 2, 3 and [ d(x), x ] 2 ∈ Z for all x ∈ R. In a recent paper [6], a full generalization of these results was proved by Lanski. He showed that a Lie ideal L of R is in Z if for some fixed k > 0, [ d(x), x ] k = 0 for all x ∈ L, unless char R = 2 and R satisfies s4. Note that a noncentral Lie ideal of R contains all the commutators [x, y] for x, y in some nonzero ideal of R except when char R = 2 and R satisfies s4. It is natural to consider the situation when [ d(x), x ] k = 0 for all commutators x = [x1, x2], or more generally, when [ d(x), x ] k = 0 for all x = f(x1, . . . , xn) where f(X1, . . . , Xn) is a multilinear polynomial over K. In the present paper, we shall extend Lanski’s theorem by imposing the condition [ d ( f(x1, . . . , xn) ) , f(x1, . . . , xn) ] k = 0 on some nonzero ideal of R. First we dispose of the simplest case when R is the matrix ring Mm(F ) over a field F and d is an inner derivation on R. Received by the editors November 4, 1994 and, in revised form, March 1, 1995. 1991 Mathematics Subject Classification. Primary 16W25; Secondary 16N60, 16R50, 16U80.

On additive maps of prime rings satisfying the engel condition

Let A be a prime ring with nonzero right ideal R and f : R → A an additive map. Next, let k,n1, n2,…,nk be natural numbers. Suppose that […[[(x), xn1], xn2],…, xnk]=0 for all x ∈ R. Then it is proved in Theorem 1.1 that [f(x),x]=0 provided that either char(A)=0 or char (A)> n1+n2+ …+nk Theorem 1.1 is a simultaneous generalization of a number of results proved earlier.

Linear identities and commuting maps in rings with involution

Abstract. In this paper we extend Martindales result by showing that the symmetric elements or the skew elements of a prime ring with involution of characteristic not 2 do not satisfy certain linear generalized polynomial identities. As applications, we determine the centralizing additive maps and the commuting traces of biadditive maps on the symmetric elements of a prime ring.

Supercentralizing Maps in Prime Superalgebras

Let A be a prime superalgebra over a commutative ring F with and f:A → A a supercentralizing F-linear map on A. We show that there exist an element λ in the extended centroid C of A and an F-linear map μ:A → C such that f(x) = λ x + μ(x) for all x ∈ A. This gives a version of Brešars theorem for superalgebras. As a consequence, we show that a nontrivial prime superalgebra admitting a “nontrivial” supercentralizing F-linear map satisfies the standard identity of degree 4.

ON CERTAIN SUBGROUPS OF PRIME RINGS WITH DERIVATIONS

Let R be a noncommutative prime ring and let d be a nonzero derivation on R. A classical theorem of Posner asserts that the subset {[x d , x]|x ∈ R} is not contained in the center of R. Under the additional assumption that char R ≠ 2 and d 3 ≠ 0, we show that the additive subgroup of R generated by the subset {[x d , x] | x ∈ R} contains a noncentral Lie ideal of R.

A NOTE ON DERIVATIONS, ADDITIVE SUBGROUPS, AND LIE IDEALS OF PRIME RINGS

ABSTRACT Let be a prime ring of characteristic not or and a noncentral Lie ideal of . If is a nonzero derivation of , then the additive subgroup of generated by the subset contains a noncentral Lie ideal of .

Prime Lie Rings of Derivations of Commutative Rings II

Let R be a 2-torsionfree commutative ring, and D a Lie subring and an R-submodule of Der(R) such that R is D-prime. Then any ideal of the Lie ring D is itself a prime Lie ring.

On maps preserving products

Maps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view.

A Note on Polynomial Rings over Nil Rings

Let R be a nil ring with p R = 0 for some prime number p. We show that the polynomial ring R[x,y] in two commuting indeterminates x, y over R cannot be homomorphically mapped onto a ring with identity. This extends, in finite characteristic case, a result obtained by Smoktunowicz [8] and gives a new approximation, in that case, of a positive solution of Kothe’s problem.

A linear algebra approach to Koethe's problem and related questions

We discuss several problems on the structure of nil rings from the linear algebra point of view. Among others, a number of questions and results are presented concerning algebras of infinite matrices over nil algebras, and nil algebras of infinite matrices over fields, which are related to the famous Koethes problem. Some questions on radicals of tensor products of algebras related to Koethes problem are also discussed.