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2.
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
3.
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
4.
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
5.
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 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. 相似文献
8.
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
9.
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
10.
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
11.
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
12.
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
13.
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
15.
Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f( x 1,…, x n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r 1,…, r n ∈ R, a[ F 2( f( r 1,…, r n )), f( r 1,…, r n )] = 0, then one of the following statements hold: 1. a = 0; 2. There exists λ ∈ C such that F( x) = λ x, for all x ∈ R; 3. There exists c ∈ U such that F( x) = cx, for all x ∈ R, with c 2 ∈ C; 4. There exists c ∈ U such that F( x) = xc, for all x ∈ R, with c 2 ∈ C. 相似文献
17.
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. 相似文献
18.
Let R be a noncommutative prime ring and d, δ two nonzero derivations of R. If δ([ d( x), x] n ) = 0 for all x ∈ R, then char R = 2, d 2 = 0, and δ = α d, where α is in the extended centroid of R. As an application, if char R ≠ 2, then the centralizer of the set {[ d( x), x] n | x ∈ R} in R coincides with the center of R. 相似文献
20.
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
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