Vanishing Derivations and Co-Centralizing Generalized Derivations 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, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

Identities with Generalized Derivations in Prime Rings

In this paper we investigate identities with two generalized derivations in prime rings. We prove, for example, the following result. Let R be a prime ring of characteristic different from two and let F ₁, F ₂ : R → R be generalized derivations satisfying the relation F ₁(x)F ₂(x) + F ₂(x)F ₁(x) = 0 for all ${x \in R}$ . In this case either F ₁ = 0 or F ₂ = 0.     

Engel-Type Conditions Involving Two Generalized Skew Derivations in Prime Rings

Let ? be a prime ring of characteristic different from 2, 𝒬_r the right Martindale quotient ring of ?, 𝒞 the extended centroid of ?, F, G two generalized skew derivations of ?, and k ≥ 1 be a fixed integer. If [F(r), r]_kr ? r[G(r), r]_k = 0 for all r ∈ ?, then there exist a ∈ 𝒬_r and λ ∈ 𝒞 such that F(x) = xa and G(x) = (a + λ)x, for all x ∈ ?.     

带有广义导子的素环

The concept of derivations and generalized inner derivations has been generalized as an additive function δ: R→ R satisfying δ(xy) = δ(x)y xd(y) for all x,y∈R,where d is a derivation on R.Such a function δis called a generalized derivation.Suppose that U is a Lie ideal of R such that u2 ∈ U for all u ∈U.In this paper,we prove that U(C)Z(R) when one of the following holds:(1)δ([u,v]) = uov (2)δ([u,v]) uov=O(3)δ(uov) =[u,v](4)δ(uov) [u,v]= O for all u,v ∈U.     

A Quadratic Differential Identity with Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C, and let ${f(x_1, \ldots, x_n)}$ be a multilinear polynomial over C, not central valued on R. Suppose that d is a derivation of R and G is a generalized derivation of R such that $$G(f(r_1, \ldots, r_n))d(f(r_1, \ldots, r_n)) + d(f(r_1, \ldots, r_n))G(f(r_1, \ldots, r_n)) = 0$$ for all ${r_1, \ldots, r_n \in R}$ . Then either d =  0 or G =  0, unless when d is an inner derivation of R, there exists ${\lambda \in C}$ such that G(x) =  λ x, for all ${x \in R}$ and ${f(x_1, \ldots, x_n)^2}$ is central valued on R.     

素环中的协中心化广义导子

Let R be a prime ring with center Z and S (?) R. Two mappings D and G of R into itself are called cocentralizing on S if D(x)x-xG(x) ∈ Z for all x ∈ S. The main purpose of this paper is to describe the structure of generalized derivations which are cocentralizing on ideals, left ideals and Lie ideals of a prime ring, respectively. The semiprime case is also considered.     

Power-Commuting Generalized Skew Derivations in Prime Rings

Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x] ⁿ  = 0, for all x ∈ R. Then there exists an element λ ∈ C such that F(x) = λx, for all x ∈ R.     

环上带有广义导子和自同构的恒等式

设R是一个素环,RF是它的左Martindale商环.如果φ(xigik△ij)是环R的某个本质理想I的一个多重线性既约且带有自同构的广义微分恒等式,那么φ(zikj)是环RF的一个广义多项式恒等式.设R是一个具有特征P≥0的半素环,RF是它的左Martindale商环.如果φ(xigik△ijfik)是环R的一个多重线性既约且带有自同构的广义微分恒等式,那么φ(zikjfike(△ij)是环RF的一个广义多项式恒等式,这里fik和e(△ij)是RF中的幂等元.     

A Characterization of Generalized Derivations on Prime Rings

Let R be a noncommutative prime ring, U be the left Utumi quotient ring of R, and k, m, n, r be fixed positive integers. If there exist a generalized derivation G and a derivation g (which is independent of G) of R such that [G(x^m)xⁿ + xⁿg(x^m), x^r]_k = 0, for all x ∈ R, then there exists a ∈ U such that G(x) = ax, for all x ∈ R. As a consequence of the result in the present article, one may obtain Theorem 1 in Demir and Argaç [10 Demir, Ç., Argaç, N. (2010). A result on generalized derivations with Engel conditions on one-sided ideals. J. Korean Math. Soc. 47(3):483–494.[Crossref], [Web of Science ®] , [Google Scholar]].     

Generalized Skew Derivations with Nilpotent Values in Prime Rings

Let ? be a prime ring, 𝒞 the extended centroid of ?, ? a Lie ideal of ?, F be a nonzero generalized skew derivation of ? with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) ? yx) ⁿ  = 0 for all x, y ∈ ?, then ? is commutative and one of the following statements holds:

(1) Either ? is central;

(2) Or ? ? M ₂(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2.     

关于素环广义导子和对称双导的几点备注

设R是素环,I是R的非零理想,如果R容许一个非单位映射的左乘子使得对所有x,y∈I满足δ(x°y)=x°y或δ(x°y) x°y=0,那么R可交换.此外,如果R是2-扭自由的素环,U是平方封闭的李理想,γ是伴随导子非零的广义导子,B:R×R→R是迹函数为g(x)=B(x,x)的对称双导,当下列条件之一成立时U为中心李理想(1)γ同态作用于U(2)2[x,y]-g(xy) g(yx)∈Z(R)(3)2[x,y] g(xy)-g(yx)∈Z(R)(4)2(x°y)=g(x)-g(y)(5)2(x°y)=g(y)-g(x)对所有的x,y∈U.     

q-Skew Derivations and Polynomial Identities

Let R be a prime ring and its left Martindale quotient ring. Assume that a q-skew -derivation of R satisfies the identity relationfor all x R, where the subring of constants of on R. It is proved that if R⁽⁾ satisfies nontrivial polynomial identities, then so does R. This answers affirmatively a problem raised in Bergen and Grzeszczuk [2] by removing the assumption on the algebraicity of .Mathematics Subject Classification (2000): 16W20, 16W25, 16W55Members of Mathematics Division, National Center for Theoretical Sciences at Taipei.Acknowledgement The authors are thankful to the referee for her/his useful suggestions and comments. This research was supported by the National Science Council of Taiwan.     

质环上的微商及环的结构

本文证明了如下定理:R为质环,char R≠2,d为R上非零微商，R中无非零诣零元，(?)则R为交换环，或R可嵌入体中.     

Power Values of Generalized Derivations with Annihilator Conditions in Prime Rings

Let R be a prime ring, H a nonzero generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that there exists ${0 \neq a \in R}$ such that a(u ^s H(u)u ^t ) ⁿ = 0 for all ${u \in L}$ , where s ≥ 0, t ≥ 0, n ≥ 1 are fixed integers. Then s = 0, H(x) = bx for all ${x \in R}$ with ab = 0, unless R satisfies s ₄, the standard identity in four variables. We also describe completely this last case.     

Derivations and Skew Polynomial Rings

Soient D un corps non nécessairement commutatif et L un sous-corps de D. On établit une condition nécessaire et suffisante pour que le groupe multiplicatif L de L soit d'indice fini dans son normalisateur N dans D. Lorsque la dimension à gauche [D : L]_g est un nombre premier, on précise le groupe N/L et la structure de D.