共查询到20条相似文献,搜索用时 8 毫秒
1.
Cheng-Kai Liu 《Monatshefte für Mathematik》2016,180(4):833-852
We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and \(\delta \) a nonzero \(\sigma \)-derivation of A, where \(\sigma \) is an epimorphism of A. For \(x,y\in A\), we set \([x,y] = xy - yx\). If \([[\ldots [[\delta (x^{n_0}),x^{n_1}],x^{n_{2}}],\ldots ],x^{n_k}]=0\) for all \(x\in R\), where \(n_{0},n_{1},\ldots ,n_{k}\) are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) \(C\cong GF(2)\), the Galois field of two elements; (3) there exist \(b\in Q\) and \(\lambda \in C\) such that \(\delta (x)=\sigma (x)b-bx\) for all \(x\in A\), \((b-\lambda )R=0\) and \(\sigma (R)=0\). The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995). 相似文献
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Vincenzo De Filippis 《Israel Journal of Mathematics》2007,162(1):93-108
Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x
1,..., x
n) a multilinear polynomial over C, I a nonzero right ideal of R.
If [g(f(r
1,..., r
n)), f(r
1,..., r
n)] = 0, for all r
1, ..., r
n ∈ I, then either g(x) = ax, with (a − γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds:
Supported by a grant from M.I.U.R. 相似文献
(i) | f(x 1,..., x n) is central valued in eRCe |
(ii) | g(x) = cx + xb, where (c+b+α)e = 0, for α ∈ C, and f (x 1,..., x n)2 is central valued in eRCe |
(iii) | char(R) = 2 and s 4(x 1, x 2, x 3, x 4) is an identity for eRCe. |
6.
Vincenzo De Filippis 《Israel Journal of Mathematics》2009,171(1):325-348
Let R be a prime ring with extended centroid C, δ a nonzero generalized derivation of R, f(x
1, ..., x
n
) a nonzero multilinear polynomial over C, I a nonzero right ideal of R and k ≥ a fixed integer.
If [δ(f(r
1, ..., r
n
)), f(r
1, ..., r
n
)]
k
= 0, for all r
1, ..., r
n
∈ I, then either δ(x) = ax, with (a-γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds
(1) if char(R) = 0 then f(x
1, ..., x
n
) is central valued in eRCe
(2) if char(R) = p > 0 then is central valued in eRCe, for a suitable s ≥ 0, unless when char(R) = 2 and eRCe satisfies the standard identity s
4
(3) δ(x) = ax−xb, where (a+b+α)e = 0, for α ∈ C, and f(x
1, ..., x
n
)2 is central valued in eRCe. 相似文献
7.
Let R be a semiprime ring with a derivation D. The focus is on the two identities with Engel condition on ${D: [x^m, D(x^{n_1}),\ldots,D(x^{n_s})]_s=0}$ for all ${x\in R}$ and ${[x^m, D(x)^{n_1},\ldots,D(x)^{n_s}]_s=0}$ for all ${x\in R}$ , where s, m, n 1, . . . , n s are fixed positive integers. Our results are natural generalizations of Posner’s theorem on centralizing derivations, Herstein’s theorem on derivations with power-central values and a recent result by A. Fo?ner, M. Fo?ner and Vukman. 相似文献
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An Engel condition with derivation for left ideals 总被引:5,自引:0,他引:5
Charles Lanski 《Proceedings of the American Mathematical Society》1997,125(2):339-345
We generalize a number of results in the literature by proving the following theorem: Let be a semiprime ring, a nonzero derivation of , a nonzero left ideal of , and let . If for some positive integers , and all , the identity holds, then either or else the ideal of generated by and is in the center of . In particular, when is a prime ring, is commutative.
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Nihan Baydar Yarbil 《代数通讯》2018,46(1):205-216
12.
Wen-Kwei Shiue 《Rendiconti del Circolo Matematico di Palermo》2003,52(3):505-509
LetR a prime ring,L a noncentral Lie ideal ofR anda ∈R. Suppose thatd is a nonzero derivation ofR such thata[d(u),u]
k
=0 for allu ∈L, wherek is a fixed positive integer. Thena=0 except when charR=2 and dim
C
RC=4. 相似文献
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Jui-Chi Chang 《Monatshefte für Mathematik》2010,58(1):155-160
Let R be a prime ring 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
= 0 for all x ? L{x\in L}, where n is a fixed positive integer. Then f = 0. 相似文献
16.
Jui-Chi Chang 《Monatshefte für Mathematik》2010,161(2):155-160
Let R be a prime ring 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 = 0 for all ${x\in L}$ , where n is a fixed positive integer. Then f = 0. 相似文献
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In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several approaches to generalize Reed-Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed-Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank metric and derive families of Maximum Distance Separable and Maximum Rank Distance codes. 相似文献
19.
Giovanni Scudo 《Rendiconti del Circolo Matematico di Palermo》2012,61(3):343-353
Let $R$ be a non-commutative prime ring, with center $Z(R)$ , extended centroid $C$ and let $F$ be a non-zero generalized derivation of $R$ . Denote by $L$ a non-central Lie ideal of $R$ . If there exists $0\ne a\in R$ such that $a[F(x),x]_k\in Z(R)$ for all $x\in L$ , where $k$ is a fixed integer, then one of the followings holds: (1) either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$ , (2) or $R$ satisfies $s_4$ , the standard identity in $4$ variables, and $char(R)=2$ ; (3) or $R$ satisfies $s_4$ and there exist $q\in U, \gamma \in C$ such that $F(x)=qx+xq+\gamma x$ . 相似文献
20.
Let R be a prime ring of char R≠2, d a non-zero derivation of R and ρ a non-zero right ideal of R such that [[d(x),d(y)]n [y,x]m] = 0 for all x,y ∈ ρ or [[d(x),d(y)]n d[y,x]m] = 0 for all x,y ∈ ρ, n, m ≥ 0 are fixed integers. If [ρ,ρ]ρ ≠ 0, then d(ρ)ρ = 0. 相似文献