Monomial Derivations without Darboux Polynomials

In this paper, it is proved that a monomial derivation d of k[x, y, z] has no Darboux polynomials if and only if d is a strict derivation with a trivial ring of constants, and we give the specific conditions when it has no Darboux polynomials.     

Constants and Darboux Polynomials for Tensor Products of Polynomial Algebras with Derivations

Abstract

Let d ₁ : k[X] → k[X] and d ₂ : k[Y] → k[Y] be k-derivations, where k[X] ? k[x ₁,…,x _n ], k[Y] ? k[y ₁,…,y _m ] are polynomial algebras over a field k of characteristic zero. Denote by d ₁ ⊕ d ₂ the unique k-derivation of k[X, Y] such that d| _k[X] = d ₁ and d| _k[Y] = d ₂. We prove that if d ₁ and d ₂ are positively homogeneous and if d ₁ has no nontrivial Darboux polynomials, then every Darboux polynomial of d ₁ ⊕ d ₂ belongs to k[Y] and is a Darboux polynomial of d ₂. We prove a similar fact for the algebra of constants of d ₁ ⊕ d ₂ and present several applications of our results.     

Central Extensions and Derivations of the Lie Algebras of Skew Derivations for the Quantum Torus

ABSTRACT

In this article, we study the central extensions and derivations of the Lie algebra of skew derivations for the quantum torus. The results of the article generalize those obtained in Jiang and Meng (1998a Jiang , C. , Meng , D. ( 1998a ). The derivation algebra of the associative algebra C _q [X, Y, X ^?1, Y ^?1] . Comm. Algebra 26 : 1723 – 1736 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] b Jiang , C. , Meng , D. ( 1998b ). The automorphism group of the derivation algebra of the Virasoro-like algebra . Adv. Math. (China) 27 : 175 – 183 . [CSA]  [Google Scholar]) and Kirkman et al. (1994 Kirkman , E. , Procesi , C. , Small , L. ( 1994 ). A q-analog for the Virasoro algebra . Comm. Algebra 22 : 3755 – 3774 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

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]].     

Eigenvector p-Bases of Rings of Constants of Derivations

Let A be a UFD of characteristic p > 0, let 𝒵 be a set of some eigenvectors of a derivation of A. We prove, under some additional assumptions, a necessary and sufficient condition for 𝒵 to be a p-basis of the minimal ring of constants containing 𝒵. The main preparatory result is the unique decomposition theorem with respect to a factor from a given subalgebra containing A^p.     

Derivations with Engel conditions on multilinear polynomials

Let be a prime algebra over a commutative ring with unity and let be a multilinear polynomial over . Suppose that is a nonzero derivation on such that for all in some nonzero ideal of , with fixed. Then is central--valued on except when char and satisfies the standard identity in 4 variables.

     

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.

     

3-李代数的广义导子

给出了3-李代数的广义导子、拟导子、拟型心的定义,研究了他们之间的结构关系,并对具有极大对角环面的3-李代数的拟导子和拟型心结构进行了系统的研究.证明了(1)广义导代数GDer(A)可以分解成拟导子代数QDer(A)和拟型心QΓ(A)的直和;(2)3-李代数A的拟导子可以扩张成一个具有较大维数的3-李代数的导子;(3)拟导子代数QDer(A)包含在拟型心的正规化子中,表示为[QDer(A),QΓ(A)]?QΓ(A);(4)如果A包含极大对角环面T,那么QDer(A)和Qr(A)是T的对角模,也就是(T,T)半单地作用在QDer(A)和QΓ(A)上.     

von Neumann代数中套子代数上的Lie导子

本文对因子von Neumann代数中套子代数上的线性映射L:alg_Mβ→M满足L(AB—BA)=L(A)B-BL(A)+AL(B)-L(B)A( A,B∈alg_Mβ)进行了刻划,证明了存在线性函数h:alg_Mβ→C;且对任意A,B∈alg_Mβ,有h(AB—BA)=0和算子T∈M,使得对任意X∈alg_Mβ,都有L(X)=XT-TX+h(X)I.     

微商共同作用在半素环的Lie理想上的结果

本文讨论了微商共同作用在半素环的某个Ｌｉｅ理想上的问题。给出了如下结果：设Ｒ是带有中心Ｚ（Ｒ）的半素环,Ｑｍｒ是Ｒ的极大右商环,Ｌ是Ｒ的非交换Ｌｉｅ理想,ｄ和δ是Ｒ的微商,假设ｒＲ（「Ｌ,Ｌ」）＝０且ｄ（ｘ）ｘ－ｘδ（ｘ）∈Ｚ（Ｒ）对任意ｘ∈Ｌ成立,则在Ｒ的扩张形心Ｃ中存在一个幂等元ｅ使得ｄ（１－ｅ）Ｑｍｒ＝０和δ（１－ｅ）Ｑｍｒ）＝０并且ｅＱｍｒ满足Ｓ４。另外给出微商共同作用在半素环上多项式的结     

AF C~*-代数中的子代数上的保幂等映射和局部导子

证明了从AFC-代数E中的子代数A到任意赋范代数B上的范数连续保幂等映射是Jordan同态,以及从A到任意赋范E-双模M上的局部导子是导子,从而推广了Crist关于局部导子的结果．     

A Generalization of Engel Conditions with Derivations in Rings

Let R be a prime ring of characteristic different from 2 with Z the center of R and d a nonzero derivation of R. Let k, m, n be fixed positive integers. If ([d(x ^k ), x ^k ] _n ) ^m  ∈ Z for all x ∈ R, then R satisfies S ₄, the standard identity in 4 variables.     

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.     

Skew Derivations and Engel Conditions

It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x ^{k ₀} ), x ^{k ₁} ], x ^{k ₂} ],…], x ^{k _n} ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result in two variables is obtained for d a (θ, ?)-derivation.     

Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X₁,...,X_t) a multilinear polynomial not central-valued on R. Suppose d(f(x₁,...,x_t)) is either invertible or nilpotent for all x₁,...,x_t in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier.1991 Mathematics Subject Classification: primary 16W25, secondary 16R50, 16N60, 16U80