无穷矩阵环上的导子

讨论无穷矩阵环上的导子,证明了环R上有限个元素不为零的无穷矩阵坏的每个导子均可表示为两个特殊导子之和。     

交换环上反对称矩阵李代数的局部导子和2-局部导子

令R是有单位元1的2-挠自由交换环, L_n(R)是由R上所有n阶反对称矩阵构成的李代数.本文研究了L_n(R)(n≥3)上局部导子和2-局部导子的性质.利用L_n(R)作为李代数的完备性和矩阵计算技巧,证明了L_n(R)上的每个局部导子和2-局部导子都是导子.推广了L_n(R)上关于导子的主要结果.     

交换环上四阶反对称矩阵李代数的BZ导子

令R为有单位元1的2-挠自由的交换环.本文给出R上四阶反对称矩阵的李代数L4(R)的任意BZ导子的分解,及BZ导子成为内导子的一个充要条件.     

矩阵环的半交换子环

称环R是半交换的,如果对任意a∈R,rR(a)是R的理想．若n≥2,则任意具有单位元的环R上的n阶上三角矩阵环不是半交换环．我们证明了reduced环上的上三角矩阵环的一类特殊子环是半交换环．     

M_2(R)上的非线性Lie导子

设R是一个含有单位元的2无扰的交换环,M_2(R)是定义在R上的全矩阵代数,证明了M_2(R)上的每一个非线性Lie导子都可以表示成一个内导子,一个可加诱导导子和一个映所有二次换位子为零的中心映射的和.     

交换环上的严格上三角矩阵代数上的Lie导子

设R是任意含单位元的交换环,N(R)为R上(n+1)×(n+1)严格上三角矩阵构成的代数．本文证明了当n≥3且2是R的单位时,N(R)上任意Lie导子D可以唯一的表示为D=D_d+D_b+D_c+D_x,其中D_d,D_b,D_c,D_x分别是N(R)上的对角,极端,中心和内Lie导子,在n=2的情况,我们也证明了N(R)上任意Lie导子D可以表示为对角,极端,内Lie导子的和。     

稳定秩1环上三个三角矩阵的积

证明了环R为稳定秩 1环当且仅当R上的每个 2× 2可逆矩阵均可以表成乘积1　 0x　 11　y0　 1u　 0z　v ,其中x ,y ,z∈R ,u ,v∈GL1(R) ;这证明了 [1]中定理 1的逆命题也成立 ;并把 [2 ]中的主要结果推广到了非交换环上 .     

形式三角矩阵环的导子和自同构

本文研究了形式上三角矩阵环Tri(A,M,B)的导子和自同构,利用与单位元相乘的方法,获得了形式上三角矩阵环Tri(A,M,B)的导子和自同构的结构形式.     

半质环的若干交换性条件

设R表示结合环(可以没有单位元)，Z(R)为环R的中心，对任意x·y∈R，[x，y]=xy-yx，郭元春证明了满足(xy)^2-xy^2x∈Z(R)的半质环是交换环，魏宗宣用类似的方法证明了满足(xy)^2-yx^2y∈Z(R)的半质环是交换环，我们推广上述结果，证明了下面的定理。     

主理想整环上保对合矩阵的线性映射

设R是特征不为2的交换主理想整环,Mn（R）表示R上n阶全矩阵模,本文基底生成元的方法刻划Mn（R）上保对合矩阵的R-线性映射的形式。     

Group of units of group algebras

By a right (left resp.) S_2n-polynomial we mean a multilinear polynomial f(X₁,…, X_t) over the ring of integers with noncommuting in-determinates X_isuch that for any prime ring R if f( X₁,…, X _t) is a PI of some nonzero right (left resp.) ideal of R, then R satisfies S_2nthe standard identity of degree 2n. In this paper we prove the theorem:Let R be a prime ring, d a nonzero derivation of R, L a noncommutative Lie ideal of R and f(X₁,…, X_t) a right or left S_2n-polynomial. Suppose that f(d( u₁)ⁿ¹,…,d(u_t)^nt)=0 for all u_iu,_i[d] L, where n₁,…,n_tare fixed positive integers. Then R satisfies S_2n+2. Also, the one-sided version of the theorem is given.     

Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X₁,...,X_t) a multilinear polynomial not central-valued on R. Suppose d(f(x₁,...,x_t)) is either invertible or nilpotent for all x₁,...,x_t in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier.1991 Mathematics Subject Classification: primary 16W25, secondary 16R50, 16N60, 16U80     

A Family of Complex Irreducible Characters Possessed by Unitary and Special Unitary Groups Defined over Local Rings

Abstract

Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 ^l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U _n (R) be the subgroup of GL(V) that preserves ( , ). Let SU _n (R) be the subgroup of all g ∈ U _n (R) whose determinant is equal to one. Let Ψ be the Weil character of U _n (R).

All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU _n (R).     

Derivations with engel conditions in prime and semiprime rings

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y]) ^m = [x, y] _n for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)] _m = [x, y] ⁿ for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring.     

Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L _? be the ?-span of a Chevalley basis of L, and L _R  = R ?_? L _? be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L _R spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).     

A Note on Comaximal Graph of Non-commutative Rings

Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra?+?Rb?=?R. Let Γ₂(R) be the subgraph of Γ(R) induced by the non-unit elements of R. Let R be a commutative ring with unity and let J(R) denote the Jacobson radical of R. If R is not a local ring, then it was proved that:

1. If $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph, then n?=?2.
2. If there exists a vertex of $\Gamma_2(R)\backslash J(R)$ which is adjacent to every vertex, then R????₂×F, where F is a field.

In this note we generalize the above results to non-commutative rings and characterize all non-local ring R (not necessarily commutative) whose $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph.     

General symbolic powers: A homological point of view

Let I be an ideal of a Noetherian ring R and let S be a multiplicatively closed subset of R. We define the n-th (S)-symbolic power of 7 as S(Iⁿ) = IⁿR_s ∩R. The purpose of this paper is to compare the topologies defined by the adic {I_n}_n≤0 and the (S)-symbolic filtration {S(Iⁿ)}n≥o using the direct system {Extⁱ _R(R/Iⁿ,R)}_n≥0     

Generalized Skew Derivations Implemented by Elementary Operators

Let R be a semiprime ring with the maximal right ring of quotients Q _mr . An additive map d: R →Q _mr is called a generalized skew derivation if there exists a ring endomorphism σ:R →R and a map \(\d:R \to Q_{mr}\) such that \(d(xy)=\d(x)y+\sigma(x)d(y)\) for all x,y?∈?R. If σ is surjective, we determine the structure of generalized skew derivations for which there exists a finite number of elements a _i ,b _i ?∈?Q _mr such that d(x)?=?a ₁ xb ₁?+???+?a _n xb _n for all x?∈?R.     

Lie triple derivation of the Lie algebra of strictly upper triangular matrix over a commutative ring

Let N(n,R) be the nilpotent Lie algebra consisting of all strictly upper triangular n×n matrices over a 2-torsionfree commutative ring R with identity 1. In this paper, we prove that any Lie triple derivation of N(n,R) can be uniquely decomposited as a sum of an inner triple derivation, diagonal triple derivation, central triple derivation and extremal triple derivation for n≧6. In the cases 1≦n≦5, the results are trivial.