Generalized derivations as Jordan homomorphisms on lie ideals and right ideals

Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.     

Orthogonality of Generalized (θ,φ)-Derivations on Ideals

In this paper,necessary and sufficient conditions concerning the orthogonality and the composition of a couple of generalized (θ,φ)-derivations on a nonzero ideal of a semiprime ring are presented.These results are generalizations of several results of Breˇsar and Vukman,which are related to a theorem of Posner on the product of two derivations on a prime ring.     

理想上的广义$(\theta,\phi)$-导子的正交性

In this paper,necessary and sufficient conditions concerning the orthogonality and the composition of a couple of generalized （θ,φ）-derivations on a nonzero ideal of a semiprime ring are presented.These results are generalizations of several results of Breˇsar and Vukman,which are related to a theorem of Posner on the product of two derivations on a prime ring.     

素环中的协中心化广义导子

Let R be a prime ring with center Z and S (?) R. Two mappings D and G of R into itself are called cocentralizing on S if D(x)x-xG(x) ∈ Z for all x ∈ S. The main purpose of this paper is to describe the structure of generalized derivations which are cocentralizing on ideals, left ideals and Lie ideals of a prime ring, respectively. The semiprime case is also considered.     

素环上的导子在Lie理想上的协交换子的零化子

Let R be a prime ring of characteristic different from 2,d and g twoderivations of R at least one of which is nonzero,L a non-central Lie ideal of R,anda∈R.We prove that if a(d(u)u-ug(u))=0 for any u∈L,then either a=0,or R is an s_4-ring,d(x)=[p,x],and g(x)=-d(x)for some p in the Martindalequotient ring of R.     

关于素环平方封闭的李理想的两个结果

Let R be a 2-torsion free prime ring, d1 a nonzero derivation, -γ a generalized derivation associated with a nonzero derivation d2, U a square closed Lie ideal of R. In the present paper,we prove that if [di^2（u）, u] ∈ Z（R） or γ acts as a homomorphism （or an antihomomorphism） on U, then U Z（R）.     

Generalized Derivations with Skew Nilpotent Values on Lie Ideals

This article gives characterizations of generalized derivations with skew nilpotent values on noncommutative Lie ideals of a prime ring. The results simultaneously generalize the ones of Herstein, Lee and Carini et al.     

Generalized Differential Identities of （Semi-）Prime Rings

Let R be a semiprime ring with characteristic p≥0 and RF be its left Martindale quotient ring. If ф（Xi^△j） is a reduced generalized differential identity for an essential ideal of R, then ф（Zije（△j ）） is a generalized polynomial identity for RF, where e（△j） are idempotents in the extended centroid of R determined by △j. Let R be a prime ring and Q be its symmetric Martindale quotient ring. If ф（Xi△j） is a reduced generalized differential identity for a noncommutative Lie ideal of R, then ф（Zij） is a generalized polynomial identity for [R, R]. Moreover, if ф（Xi△j） is a reduced generalized differential identity, with coefficients in Q, for a large right ideal of R, then ф（Zij） is a generalized polynomial identity for Q.     

LIE IDEALS, MORITA CONTEXT AND GENERALIZED （α β）-DERIVATIONS

A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson＇s famous result, several tech- niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized （α β）-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.     

特征为2的素环的Lie理想上中心化自同构

Let R be a prime ring with an automorphism σ≠1, an identity map. Let L be a noncentral Lie ideal of R such that [x^σ,x] E Z for all x ∈ L, where Z is the center of R. Then L is contained in the center of R, unless char(R)=2 and dimc RC = 4.     

微商共同作用在半素环的Lie理想上的结果

本文讨论了微商共同作用在半素环的某个Ｌｉｅ理想上的问题。给出了如下结果：设Ｒ是带有中心Ｚ（Ｒ）的半素环,Ｑｍｒ是Ｒ的极大右商环,Ｌ是Ｒ的非交换Ｌｉｅ理想,ｄ和δ是Ｒ的微商,假设ｒＲ（「Ｌ,Ｌ」）＝０且ｄ（ｘ）ｘ－ｘδ（ｘ）∈Ｚ（Ｒ）对任意ｘ∈Ｌ成立,则在Ｒ的扩张形心Ｃ中存在一个幂等元ｅ使得ｄ（１－ｅ）Ｑｍｒ＝０和δ（１－ｅ）Ｑｍｒ）＝０并且ｅＱｍｒ满足Ｓ４。另外给出微商共同作用在半素环上多项式的结     

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．     

交换环上严格上三角矩阵的广义李三导子(英文)

本文研究了交换环R上所有n×n严格上三角矩阵构成的李代数N(n,R)(n≥5)上广义李三导子.利用矩阵技巧,证明了N(n,R)(n≥5)上任意广义李三导子为一李三导子与一位似映射的和.对于N(n,R)(n≥3)上广义李导子,得出类似结果.     

Generalized Jordan derivations on Lie ideals associate with Hochschild 2-cocycles of rings

Let R be a 2-torsion free ring and L be a Lie ideal of R. In this paper we initiate the study of generalized derivations on Lie ideals associate with Hochschild 2-cocycles and prove that every generalized derivation associate with a Hochschild 2-cocycle on L is a generalized derivation on L under certain conditions.     

关于素环广义导子和对称双导的几点备注

设R是素环,I是R的非零理想,如果R容许一个非单位映射的左乘子使得对所有x,y∈I满足δ(x°y)=x°y或δ(x°y) x°y=0,那么R可交换.此外,如果R是2-扭自由的素环,U是平方封闭的李理想,γ是伴随导子非零的广义导子,B:R×R→R是迹函数为g(x)=B(x,x)的对称双导,当下列条件之一成立时U为中心李理想(1)γ同态作用于U(2)2[x,y]-g(xy) g(yx)∈Z(R)(3)2[x,y] g(xy)-g(yx)∈Z(R)(4)2(x°y)=g(x)-g(y)(5)2(x°y)=g(y)-g(x)对所有的x,y∈U.     

On Lie Ideals and Automorphisms in Prime Rings

Mathematical Notes - Let R be a prime ring of characteristic different from 2 with center Z and extended centroid C, and let L be a Lie ideal of R. Consider two nontrivial automorphisms α and...     

On the composition of derivations

Posner ([9]) has shown that for any prime ringR of characteristic different from 2 the composition of any two non-zero derivations is not a derivation. On the other hand, it is well known ([4]) that if charR=n for a prime numbern andd is a derivation ofR, thend ⁿ is also a derivation. Our main objective is to extend the above mentioned result of Posner in the case of commutative domains, and to apply this results to the investigation of connections either between derivations and a center, or between derivations and a generalized centroid of a prime ring. For this purpose, we are first going to introduce a method of notation for the composition of derivations which, we hope, will also be useful in other situations.