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1.
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 ∈ ?. 相似文献
2.
Let R be a prime ring with center Z and L a noncommutative Lie ideal of R. Suppose that f is a right generalized β-derivation of R associated with a β-derivation δ such that f( x) n ∈ Z for all x ∈ L, where n is a fixed positive integer. Then f = 0 unless dim C RC = 4. 相似文献
3.
Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n( u, w) ≥1 such that ( F( uw) ? bwu) n = 0, then one of the following statements holds: F = 0 and b = 0; R ? M2(K), the ring of 2 × 2 matrices over a field K, b2 = 0, and F(x) = ?bx, for all x ∈ R. 相似文献
4.
Given a prime ring R, a skew g-derivation for g : R → R is an additive map f : R → R such that f( xy) = f( x) g( y) + xf( y) = f( x) y + g( x) f( y) and f( g( x)) = g( f( x)) for all x, y ∈ R. We generalize some properties of prime rings with derivations to the class of prime rings with skew derivations. 相似文献
5.
Let R be a semiprime ring, RF be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of RE. Furthermore, if there exists a fixed positive integer n such that μ(x)^n = 0 for all x∈I, then μ=0. 相似文献
6.
This article gives characterizations of generalized derivations with skew nilpotent values on noncommutative Lie ideals of a prime ring. The results simultaneously generalize the ones of Herstein, Lee and Carini et al. 相似文献
9.
Let R be a prime ring, L a noncentral Lie ideal of R, and a ∈ R. Set [ x, y] 1 = [ x, y] = xy ? yx for x, y ∈ R and inductively [ x, y] k = [[ x, y] k?1, y] for k > 1. Suppose that δ is a nonzero σ-derivation of R such that a[δ( x), x] k = 0 for all x ∈ L, where σ is an automorphism of R and k is a fixed positive integer. Then a = 0 except when char R = 2 and R ? M2( F), the 2 × 2 matrix ring over a field F. 相似文献
10.
Let R be a ring with unity, g a generalized derivation on R and f( X 1,…, X k ) a multilinear polynomial. In this article we describe the structure of R provided that g( f( x 1,…, x k )) is either invertible or nilpotent for every x 1,…, x k in some nonzero ideal of R. 相似文献
11.
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. 相似文献
12.
设R是一个半素环, RF(resp.Q)是它的左Martindale商环(对称Martindale 商环),K是R的一个本质理想,则K上的每一个广义斜导子μ能被唯一地扩展到RF和Q 上.设R是一个素环,K是R的一个本质理想,μ是K上的一个广义斜导子且α为其伴随自同构,d为其伴随斜导子,如果存在n≥0,使得对任意的x∈K都有μ(x)n=0,那么μ=0. 相似文献
13.
Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f( x1,…, xn) 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,…, rn) ∈ Rn, then one of the following holds: 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,…, xn)2 is central valued on R; There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R; 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; 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; 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,…, xn)2 is central valued on R. 相似文献
14.
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( xm) xn + xng( xm), xr] 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]]. 相似文献
15.
设R是中心为Z的素环.本文证明了:(1)设R的特征>n,n为自然数,D是R上的导子,若R是交换的并且Dn(R)=(0),则D(R)=(0);若R不是交换的并且Dn(R)Z,则D(Z)=(0).(2)设R的特征≠2,D1,D2是R上的两个导子,若[D1(R),D2(R)]Z,则D1=(0),或者D2=(0),或者R是交换的. 相似文献
16.
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. 相似文献
17.
Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f( x 1,…, x n ) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f( R) the set of all evaluations of the polynomial f( x 1,…, x n ) in R. If [ G( u) u, G( v) v] = 0, for any u, v ∈ f( R), we prove that there exists c ∈ U such that G( x) = cx, for all x ∈ R and one of the following holds: 1. f( x 1,…, x n ) 2 is central valued on R; 2. R satisfies s 4, the standard identity of degree 4. 相似文献
19.
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. 相似文献
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