Vanishing power values of commutators with derivations

Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x ⁿ , d(y)] _m , [y, x] _s ] ^t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let 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.     

素环上导子的线性组合

In this paper we discuss the linear combination of derivations in prime rings which was initiated by Niu Fengwen.Our aim is to improve Niu‘s result.For the proof of the main theorem we generalize a result of Bresar and obtain a lemma that is of independent interest.     

导子交换子幂值的零化子条件

Let R be a prime ring with center Z(R), I a nonzero ideal of R, d a nonzero derivation of R and 0≠a∈ R. In the present paper, our object is to study the situation a[d(xk), xk]n∈ Z(R) for all x ∈ I under certain conditions, where n(≥ 1), k(≥ 1) are fixed integers.     

Derivations with power central values on Lie ideals in prime rings

Let R be a prime ring of char R ≠ 2 with a nonzero derivation d and let U be its noncentral Lie ideal. If for some fixed integers n ₁ ⩾ 0, n ₂ ⩾ 0, n ₃ ⩾ 0, (u ⁿ¹ [d(u), u]u ⁿ²) ⁿ³ ∈ Z(R) for all u ∈ U, then R satisfies S ₄, the standard identity in four variables.     

Skew Derivations of Prime Rings

Given a prime ring R, a skew g-derivation for g : R → R is an additive map f : R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)) for all x, y ∈ R. We generalize some properties of prime rings with derivations to the class of prime rings with skew derivations.     

环的广义斜导子

设R是一个半素环, RF(resp．Q)是它的左Martindale商环(对称Martindale 商环),K是R的一个本质理想,则K上的每一个广义斜导子μ能被唯一地扩展到RF和Q 上．设R是一个素环,K是R的一个本质理想,μ是K上的一个广义斜导子且α为其伴随自同构,d为其伴随斜导子,如果存在n≥0,使得对任意的x∈K都有μ(x)n=0,那么μ=0．     

素环的李结构

设R是一个咎征非2的素环,U是R的一个平方封闭的李理想,d1,d2,d是R的导子,δ是R的广义导子．本文证明了U为中心李理想,如果以下条件之一成立：（1）d（x）od（y）=xoy;（2）d（x）οd（y）＋xοy=0;（3）d1（x）οd2（可）=0;（4）δ（[x,y]）=0;（5）δ（xοy）=0对所有的x,y∈U．     

A Note on Commutators with Power Central Values on Lie Ideals

This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identity in 4 variables. The case where R is a semiprime ring is also examined by the authors. The results of the note improve Carini and Filippis＇s results.