A note on generalized Lie derivations of prime rings

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D, K: A-R be additive maps such that F([x, y]) = F(x)_y-yK(x)-T(y)_{x + x}D(y) for all x, yEA. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R) > 3 and also in the case A is a noncentral Lie ideal and deg(R) > 9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

Vanishing and cocentralizing generalized derivations on Lie ideals

Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and L a not central Lie ideal of R. Suppose that F, G and H are generalized derivations of R, with F≠0, such that F(G(x)x?xH(x)) = 0, for any x∈L. In this paper we describe all possible forms of F, G and H.     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

Commuting and centralizing generalized derivations on lie ideals in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] _k x ? x[G(x), x] _k = 0 for any x ∈ L, then one of the following holds:

1. either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R
2. or R satisfies the standard identity s ₄(x ₁, …, x ₄) and one of the following conclusions occurs

1. there exist a, b, c, q ∈ U, such that a ?b + c ?q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R
2. there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc?d(x) for all x ∈ R, with a + b ? c ∈ C.

     

Multilinear polynomials and cocentralizing conditions in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x ₁,..., x _n ) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x ₁, …, xn) on R. If F and G are generalized derivations of R such that [[F(x), x], [G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds:

Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings

Let R be a prime ring of characteristic different from 2 and 3, Q_r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y ∈ R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y ∈ R.     

On derivations and commutativity in prime rings

Supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A3961.     

Strong commutativity-preserving generalized derivations on semiprime rings

Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving （scp） on S, if [f（x）, f（y）] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.     

Generalized derivations acting on multilinear polynomials in prime rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x₁,..., x _n ) a multilinear polynomial over C which is not central valued on R. If

$$F(f(r))G(f(r)) = H(f(r)^2 )$$

for all r = (r₁,..., r _n ) ∈ I ⁿ , then one of the following conditions holds:

1. (1)
   there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R
    
2. (2)
   there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C
    
3. (3)
   there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R
    
4. (4)
   f(x₁,..., x _n )² is central valued on R and one of the following conditions holds

   1. (a)
      there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’
       
   2. (b)
      there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.
       
    

     

Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2 and extended centroid C and let f(x1,..., x _n ) be a multilinear polynomial over C not central-valued on R, while δ is a nonzero derivation of R. Suppose that d and g are derivations of R such that

$\delta (d(f(r_1 , \ldots ,r_n ))f(r_1 , \ldots ,r_n ) - f(r_1 , \ldots ,r_n )g(f(r_1 , \ldots ,r_n ))) = 0$

for all r1,..., r _n ∈ R. Then d and g are both inner derivations on R and one of the following holds: (1) d = g = 0; (2) d = ?g and f(x ₁,..., x _n )² is central-valued on R.

     

Primitive algebraic relations of skew derivations of prime rings

In [1], the question was posed as to whether or not all algebraic relations of skew derivations of prime rings follow from primitive algebraic relations. Here we argue to obtain a negative answer to a milder question, and namely, an example is constructed in which a pointed Hopf algebra H (generated as an algebra with unity by its relatively primitive elements) acts trivially on the generalized centroid C of a prime ring R, but not all algebraic relations of skew derivations (corresponding to relatively primitive elements in H) follow from primitive algebraic ones. The R in the counterexample is a free associative C-algebra. Supported by ISF grant No. RPS300 and by RFFR grant No. 95-01-01356a. Translated from Algebra i Logika, Vol. 36, No. 4, pp. 407–421, July–August, 1997.