共查询到20条相似文献,搜索用时 15 毫秒
2.
Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [ g( r k ), r k ] n = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g( x) = xc and I( c ? α) = 0 for a suitable α ∈ C. In particular we have that g( x) = α x, for all x ∈ I. 相似文献
3.
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
6.
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
7.
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
8.
Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f( X 1,…, X t ) a nonzero polynomial in noncommutative indeterminates X 1,…, X t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f( X 1,…, X t ) are characterized if the Engel identity is satisfied: [ D( f( x 1,…, x t )), f( x 1,…, x t )] k = 0 for all x 1,…, x t ∈ R. 相似文献
10.
Let ? be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent P. It is shown that, under some mild conditions, an additive map L on ? satisfies L([ A, B]) = [ L( A), B] + [ A, L( B)] whenever AB = 0 (resp., AB = P) if and only if it has the form L( A) = ?( A) + h( A) for all A ∈ ?, where ? is an additive derivation on ? and h is an additive map into its center. 相似文献
11.
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
12.
We prove that a Jordan (φ, θ)-derivation of a 2-torsion free semiprime ring, with θ an automorphism, must be a (φ, θ)-derivation. This is then used to show more generally that an additive mapping of a semiprime ring to itself that behaves like a generalized (φ, θ)-derivation on nth powers is, in fact, a generalized (φ, θ)-derivation of the ring. 相似文献
13.
Let R be a prime ring of characteristic different from 2, Q r be its right Martindale quotient ring and C be its extended centroid. Suppose that F, G are generalized skew derivations of R and \({f(x_1, \ldots, x_n)}\) is a non-central multilinear polynomial over C with n non-commuting variables. If F and G satisfy the following condition: $$F(f(r_1,\ldots, r_n))f(r_1, \ldots,r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))\in C$$ for all \({r_1, \ldots, r_n \in R}\), then we describe all possible forms of F and G. 相似文献
14.
For prime algebras, we describe a linear map which behaves like a left derivation on a fixed multilinear polynomial in noncommuting indeterminates and, in particular, we characterize left derivations by their action on mth powers. 相似文献
16.
Let K be a commutative ring with unity, R a prime K-algebra, Z( R) the center of R, d and δ nonzero derivations of R, and f( x 1,…, x n ) a multilinear polynomial over K. If [ d( f( r 1,…, r n )), δ ( f( r 1,…, r n ))] ? Z( R), for all r 1,…, r n ? R, then either f( x 1,…, x n ) is central valued on R or { d, δ} are linearly dependent over C, the extended centroid of R, except when char( R) = 2 and dim C RC = 4. 相似文献
17.
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
19.
令A与B是含单位元的环,M是(A,B)-双模,U=Tri(A,M,B)是三角环.在一些附加假设条件下,本文从几个不同的角度给出了U上可加左导子的结构性质.此外,也得到了满足一定条件的环上可加左导子的两个不同刻画. 相似文献
20.
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. 相似文献
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