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.     

环上矩阵环的导子

文[1]讨论了除环上2阶全矩阵环的导子的一些性质,本文继此讨论一般结合环R上的R阶全矩阵环R_n的导子的性质.环R的加群自同态(?)称为R的导子,若对x、y∈R,有d(xy)=xd(y) d(x)y.如下总假定R有单位元,且用R_n表示R上的n阶全矩阵环,E_ij表示(i,j)位置元素为R的单位元1其余元素为零的R_n的矩阵单位,xE饰表示对角线上元素为x的数量阵.     

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.     

可换环上一般线性李代数的李三导子

本文研究了含幺可换环上一般线性李代数的李三导子.通过构造特殊矩阵并利用这些矩阵进行运算,得到了任意含幺可换环上一般线性李代数的任意一个李三导子的具体形式,从而推广了导子的概念.     

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.     

可换环上一类矩阵代数的导子和自同构

决定了含幺可换环上一类矩阵代数的所有导子和所有自同构.     

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.     

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.     

可换环上严格上三角矩阵李代数的BZ导子

本文研究了严格上三角矩阵李代数的BZ导子.利用BZ导子在其基上的作用,获得了严格上三角矩阵李代数的任意一个BZ导子的具体形式.对导子的概念进行了推广.     

交换环上一个线性李超代数的导子

本文研究了交换环上一个李超代数的导子.利用构造几类特殊的导子,获得了此李超代数的任意导子是几类特殊导子的和.推广了交换环上李代数的导子.     

Positive derivations on archimedean lattice-ordered rings

It is known that the only positive derivation on a reduced archimedean f-ring is the zero derivation. We investigate derivations on general archimedean lattice-ordered rings. First, we consider semigroup rings over cyclic semigroups and show that, in the finite case, the only derivation that is zero on the underlying ring is the zero derivation and that, in the infinite case, such derivations are always based on the derivative. Turning our attention to lattice-ordered rings, we show that, on many algebraic extensions of totally ordered rings, the only positive derivation is the zero derivation and that, for transcendental extensions, derivations that are lattice homomorphisms are always translations of the usual derivative and derivations that are orthomorphisms are always dilations of the usual derivative. We also show that the only positive derivation on a lattice-ordered matrix ring over a subfield of the real numbers is the zero derivation, and we prove a similar result for certain lattice-ordered rings with positive squares. The second author thanks Hamilton College for its support of his visits to the first author in Houston. He also thanks John Miller for his friendship and hospitality over the last thirty years.     

Derivations of Rings of Infinite Matrices

For any associative unital ring R we investigate two rings – the ring of all infinite upper triangular matrices and the ring of all infinite matrices with the finite number of nonzero entries in each column. We describe derivations of these rings. We prove that every derivation of any of them is a sum of an inner derivation and a derivation which is induced by some derivation of R.     

Morita equivalence for infinite matrix rings

In this paper, weak distinguished subcategory and distinguished subcategory of modules are introduced. Left(right) local unital rings are particularly considered. Also, representable equivalent functors between categories. By using the replacement techniques of modules, a general theory of Morita equivalence for infinite matrix rings is established. This theory not only extends the classical Morita theory of equivalence from finite matrix rings to infinite matrix rings and also contains some new results which are useful in studying the algebraic structures for infinite matrix rings. Some results of classical Morita theory are included as its special cases.     

Infinite Matrix Rings on Spaces with Signed-Weakly Gliding Hump Property

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On the Behrens radical of matrix rings and polynomial rings

It is shown that the Behrens radical of a polynomial ring, in either commuting or non-commuting indeterminates, has the form of “polynomials over an ideal”. Moreover, in the case of non-commuting indeterminates, for a given coefficient ring, the ideal does not depend on the cardinality of the set of indeterminates. However, in contrast to the Brown-McCoy radical, it can happen that the polynomial ring R[X] in an infinite set X of commuting indeterminates over a ring R is Behrens radical while the polynomial ring R〈X〉 in an infinite set Y of non-commuting indeterminates over R is not Behrens radical. This is connected with the fact that the matrix rings over Behrens radical rings need not be Behrens radical. The class of Behrens radical rings, which is closed under taking matrix rings, is described.     

Commutativity of rings and near-rings with generalized derivations

In this paper we prove some theorems of commutativity for prime rings or 3-prime near-rings with a suitably constrained generalized derivation. As a consequence of the results obtained, we prove several commutativity theorems for 3-prime near-rings and prime rings. Also, we prove some other results about the generalized derivation either on a near-ring or on a ring.     

McCoy rings and zero-divisors

We investigate relations between the McCoy property and other standard ring theoretic properties. For example, we prove that the McCoy property does not pass to power series rings. We also classify how the McCoy property behaves under direct products and direct sums. We prove that McCoy rings with 1 are Dedekind finite, but not necessarily Abelian. In the other direction, we prove that duo rings, and many semi-commutative rings, are McCoy. Degree variations are defined, studied, and classified. The McCoy property is shown to behave poorly with respect to Morita equivalence and (infinite) matrix constructions.     

Nonlinear Jordan Higher Derivations of Triangular Algebras

In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algeb...     

Valuations with preassigned proximity relations

We characterize the infinite upper triangular matrices (which we call formal proximity matrices) that can arise as proximity matrices associated with zero-dimensional valuations dominating regular noetherian local rings. In particular, for every regular noetherian local ring R of the appropriate dimension, we give a sufficient condition for such a formal proximity matrix to be the proximity matrix associated with a real rank one valuation dominating R. Furthermore, we prove that in the special case of rational function fields, each formal proximity matrix arises as the proximity matrix of a valuation whose value group is computable from the formal proximity matrix. We also give an example to show that this is false for more general fields. Finally in the case of characteristic zero, our constructions can be seen as a particular case of a structure theorem for zero-dimensional valuations dominating equicharacteristic regular noetherian local rings.     

Stably just infinite rings

We study just infinite algebras which remain so upon extension of scalars by arbitrary field extensions. Such rings are called stably just infinite. We show that just infinite rings over algebraically closed fields are stably just infinite provided that the ring is either right noetherian (4.2) or countably generated over a large field (6.4). We give examples to show that, over countable fields, a just infinite algebra which is either affine or non-noetherian need not remain just infinite under extension of scalars. We also give a concrete classification of PI stably just infinite rings (5.5) and give two characterizations of non-PI stably just infinite rings in terms of Martindale's extended center (3.4), (3.5).