共查询到10条相似文献,搜索用时 52 毫秒
1.
Let \(\mathcal {R}\) be a prime ring, \(\mathcal {Z(R)}\) its center, \(\mathcal {C}\) its extended centroid, \(\mathcal {L}\) a Lie ideal of \(\mathcal {R}, \mathcal {F}\) a generalized skew derivation associated with a skew derivation d and automorphism \(\alpha \). Assume that there exist \(t\ge 1\) and \(m,n\ge 0\) fixed integers such that \( vu = u^m\mathcal {F}(uv)^tu^n\) for all \(u,v \in \mathcal {L}\). Then it is shown that either \(\mathcal {L}\) is central or \(\mathrm{char}(\mathcal {R})=2, \mathcal {R}\subseteq \mathcal {M}_2(\mathcal {C})\), the ring of \(2\times 2\) matrices over \(\mathcal {C}, \mathcal {L}\) is commutative and \(u^2\in \mathcal {Z(R)}\), for all \(u\in \mathcal {L}\). In particular, if \(\mathcal {L}=[\mathcal {R,R}]\), then \(\mathcal {R}\) is commutative. 相似文献
2.
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{xin L}, where n is a fixed positive integer. Then f = 0. 相似文献
3.
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
4.
5.
ABSTRACTLet n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)n = 0 for all x,y∈L, then char(R) = 2 and R?M2(C), the ring of 2×2 matrices over C. 相似文献
6.
Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a ∈ R and [a, f(U)] = 0 then a ∈ Z or d(a) = 0 or U ? Z; (ii) If f 2(U) = 0 then U ? Z; (iii) If u 2 ∈ U for all u ∈ U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U ? Z. 相似文献
7.
Let m, n be two fixed positive integers and let R be a 2-torsion free prime ring, with Utumi quotient ring U and extended centroid C. We study the identity F(x
m+n+1) = F(x)x
m+n
+ x
m
D(x)x
n
for x in a non-central Lie ideal of R, where both F and D are generalized derivations of R and then determine the relationship between the form of F and that of D. In particular the conclusions of the main theorem say that if D is the non-zero map in R, then R satisfies the standard identity s
4(x
1, . . . , x
4) and D is a usual derivation of R. 相似文献
8.
Let R be a prime ring with char R ≠ 2, L a non-central Lie ideal of R, d, g non-zero derivations of R, n ≥ 1 a fixed integer. We prove that if (d(x)x − xg(x))
n
= 0 for all x ∈ L, then either d = g = 0 or R satisfies the standard identity s
4 and d, g are inner derivations, induced respectively by the elements a and b such that a + b ∈ Z(R). 相似文献
9.
LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA
c
and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let
be the factor Lie algebra and let δ:
be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt
14, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A
c
such that
for allx∈R. Our result solves an old problem of Herstein. 相似文献
10.
Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and L a not central Lie ideal of R. Suppose that F, G and H are generalized derivations of R, with F≠0, such that F(G(x)x?xH(x)) = 0, for any x∈L. In this paper we describe all possible forms of F, G and H. 相似文献