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1.
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
2.
Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x
1,..., x
n
) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x
1, …, xn) on R. If F and G are generalized derivations of R such that [[F(x), x], [G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds:
(1) |
there exists α ∈ C such that F(x) = αx for all x ∈ R 相似文献
3.
Let H
1, H
2 be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T) in H
1 and taking values in H
2. In this article we prove the following results:
4.
Basudeb Dhara 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):401-410
Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x
1, ..., x
n
) a nonzero multilinear polynomial over C. Suppose that x
s
d(x)x
t
∈ Z(R) for all x ∈ {d(f(x
1, ..., x
n
))|x
1, ..., x
n
∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ
C = eRC for some idempotent e in the socle of RC and f(x
1, ..., x
n
)
N
is central-valued in eRCe, where N = s + t + 1.
相似文献
5.
F. G. Timmesfeld 《Archiv der Mathematik》2002,79(6):404-407
Let Φ be a root system of typeA
ℓ, ℓ ≧ 2,D
ℓ, ℓ ≧ 4 orE
ℓ, 6 ≧ ℓ ≧ 8 andG a group generated by nonidentity abelian subgroupsA
r,r∈Φ, satisfying:
6.
Let R be a ring with center Z(R), let n be a fixed positive integer, and let I be a nonzero ideal of R. A mapping h: R → R is called n-centralizing (n-commuting) on a subset S of R if [h(x),x
n
] ∈ Z(R) ([h(x),x
n
] = 0 respectively) for all x ∈ S. The following are proved:
|