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.     

Notes on Automorphisms of Prime Rings

Let R be a prime ring, L a noncentral Lie ideal and σ a nontrivial automorphism of R such that usσ(u)ut= 0 for all u ∈ L, where s, t are fixed non-negative integers. If either char R s + t or char R = 0, then R satisfies s4, the standard identity in four variables. We also examine the identity(σ([x, y])-[x, y])n=0 for all x, y ∈ I, where I is a nonzero ideal of R and n is a fixed positive integer. If either char R n or char R = 0, then R is commutative.     

Derivations And Homomorphisms in Prime Rings

Let R be an associative ring, recall that an additive mapping d of R into itself is a derivation if for all x,y in R: d(xy)=d(x)y+xd(y) It is shown that structure of a ring is very tightly determined by the imposition of a special behavior on one of its derivations. analogously,we shall consider the problem: Suppose that R is a prime ring with nonzero derivation d such that the derivation d is a homomorphism (anti-     

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive(generalized)ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive(generalized)ξ-Lie derivation with ξ = 1 if and only if it is an additive(generalized)derivation satisfying L(ξA)= ξL(A)for all A. These results are then used to characterize additive(generalized)ξ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.     

A Note on Derivations in Semiprime Rings

Throughout,R will be semiprime ring,Qmr=Qmr(R) the maximal right quotient ringof R,and C=Z(Qmr) the extended centroid of R(see[1 ,Chapter2 ] for details) .　 In [2 ] ,Bresar,M.discussed the identity of derivations in a prime ring,and provedthat if nonzero derivations d and g of R satisfy d(x) g(y) =g(x) d(y) for all x,y∈R,then there exists aλ in C such thatg(x) =λd(x) for all x∈ R.Itis natural to ask whatresults can be obtained if d(x) g(x) =g(x) d(x) for all x∈ R.In[3 ] ,Bresa…     

Strong commutativity-preserving generalized derivations on semiprime rings

Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving （scp） on S, if [f（x）, f（y）] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.     

Generalized Derivations with Nilpotent Values on Semiprime Rings

Let R be a semiprime ring, RF be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of RE. Furthermore, if there exists a fixed positive integer n such that μ(x)^n = 0 for all x∈I, then μ=0.     

EXISTENCE OF SOLUTIONS FOR NON-PERIODIC SUPERLINEAR SCHRODINGER EQUATIONS WITHOUT （AR） CONDITION

The existence of solutions is obtained for a class of the non-periodic Schrdinger equation -Δu + V (x)u = f (x, u), x ∈ R N , by the generalized mountain pass theorem, where V is large at infinity and f is superlinear as |u| →∞.     

交换环上上三角矩阵的李三导子

Let T(n,R) be the Lie algebra consisting of all n × n upper triangular matrices over a commutative ring R with identity 1 and M be a 2-torsion free unital T(n,R)-bimodule.In this paper,we prove that every Lie triple derivation d : T(n,R) → M is the sum of a Jordan derivation and a central Lie triple derivation.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

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.     

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）.     

非交换环的拟零因子图

In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Γ_*(R), is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if x Ry = 0. We show that the following three conditions on an FIC ring R are equivalent:(1) χ(R) is finite;(2) ω(R) is finite;(3)Nil_*R is finite where Nil_*R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Γ_*(R).     

交换环上矩阵代数的局部化导子

Let R be a commutative ring with identity, Nn（R） the matrix algebra consisting of all n × n strictly upper triangular matrices over R. Several types of proper local derivations of Nn（R） （n ≤ 4） are constructed, based on which all local derivations of Nn（R） （n ≤ 4） are characterized when R is a domain.     

拟-中心半交换环

环$R$称为拟-中心半交换的(简称QCS环)如果对$a,b\in R$, $ab=0$蕴含$aRb\subseteq Q(R)$, 其中$Q(R)$为$R$的拟中心.证明了如果$R$ 为QCS环, 那么$R$的幂零元集恰好是它的Wedderburn根, 且对$n\geq 2$, 上三角矩阵环$R=T_n(S)$ 是QCS 环当且仅当$n=2$ 且$S$ 是duo 环, 而$T_{2k+2}^k$是QCS环如果$R$是约化的duo环.     

有界超格上的微分

在有界超格上引入微分,研究了有界超格上微分的一些性质.定义并研究了微分超格的微分超理想和微分超同余,并证明了如果$(L,d)$是一个有界强单微分超格并且$R$是$(L,d)$的一个强微分同余,则$(L/R,g)$仍是一个强单微分超格,其中$g$是由$d$诱导的商超格上的单强微分.     

广义IP-内射环

对环R,令ip(R_R)={a∈R：任意一个从R的右理想到R且象为aR的模同态能开拓到R}。众所周知,R为右IP-内射环当且仅当R=ip(R_R),R为右单-内射环当且仅当{a∈R：aR is simple)(?)ip(R_R)。对环R的一个子集S,我们引进了S-IP-内射环的概念,即满足S(?)ip(R_R)的环。并得到了这种环的一些性质。     

${\mbox{\boldmath $R$}}^N$上奇异非线性多调和方程的正整体解

本文研究形如△((△nu)(p-1) )=f(|x|,u,|(?)u|)u-β,x∈RN的奇异非线性多调和方程在RN上的正整体解,此处P>1,β≥0是常数,n是自然数,f:R × R ×R →R 是一个连续函数, ξδ*:=sign(ξ)·|ξ|δ,,ξ∈R,δ>0,给出了该类方程具有无穷多个其渐进阶刚好为|x|2n的正整体解的充分条件与必要条件.这些结论可以推广到更一般的方程.     

半交换环的一个推广研究

In this paper, a generalization of the class of semicommutative rings is investigated.A ring R is called left GWZI if for any a ∈ R, l(a) is a GW-ideal of R. We prove that a ring R is left GWZI if and only if S3(R) is left GWZI if and only if Vn(R) is left GWZI for any n ≥ 2.