欧氏环上某些矩阵特征性质的刻化

设Ｒ是一个欧氏环。本文首先给出Ｒ上矩阵多项式的秩的一个定理，然后用此定理刻化了Ｒ上某些矩阵的特征性质。     

关于p-除环上分块矩阵秩的一些恒等式

本文给出了p-除环上分块矩阵秩的一些恒等式，从而推广了文[1]的结果.     

关于除环上分块矩阵秩的等式

本文使用双矩阵分解方法研究除环上分块矩阵秩的等式,给出了Marsaglia-Styan公式一个新的证明并获得了一些新的秩等式的刻划.     

除环上矩阵的保逆线性算子

除环上矩阵的保逆线性算子曹重光（黑龙江大学数学系，哈尔滨１５００８０）关键词除环，逆矩阵，线性算子．分类号ＡＭＳ（１９９１）１５Ａ０４，１５Ａ３３／ＣＣＬＯ１５１．２１设Ｒ是一个除环，Ｆ为其中心域，又设ｃｈａｒＲ≠２，３；Ｗ＿ｎ（Ｒ）及Ｓ＿ｎ（Ｒ）分...     

关于幂等矩阵秩的一个等式

目的:将复数域上幂等矩阵秩的一个等式推广到除环上.方法:采用广义逆矩阵的理论.结果:得到了除环上的类似等式.结论:复幂等矩阵的某些等式是可以推广到除环上的,并且可以进行必要的改进.     

无限矩阵环和完备环

无限矩阵环和完备环郭善良（复旦大学）Ｓｈａｎｎｙ在１９７１证明了一个环Ｒ是半单Ａｒｔｉｎ环当且仅当Ｒ上的无限矩阵环是ＶｏｎＮｅｕｍａｎｎ正则环［１］。这也就是说一个环的无限矩阵环在一定程度上唯一确定了Ｒ本身。我们注意到若ＥｎｄＦ，为ＶｏｎＮｅｕｍｍａ...     

非交换主理想整环上三矩阵乘积的Moore-Penrose逆的倒换顺序律

本文研究非交换主理想整环R上三矩阵乘积M-P逆的倒序律成立的刻划问题文中阐述R上矩阵的秩的定义,并利用R上矩阵与 R所嵌入的商除环K上矩阵的秩之间的关系,把文[2]中复矩阵的主要结果直至推广到R上,得到了倒序律成立的若干个刻划.     

几乎弱稳定秩的环

主要是介绍了非交换环的几乎弱稳定秩的概念,并利用它来研究非交换环上的右Hermite环,右Bezout环及初等因子环之间的关系.证明了具有几乎弱稳定秩的满足条件V的右(左)Hermite环是初等因子环;还证明了具有几乎弱稳定秩的满足条件V的右Bezout环是右Hermite环;除此之外还得到了几乎的Exchang环具有几乎弱稳定秩.最后,给出了在具有几乎弱稳定秩且J(R)不为零的右(左)Hermite环上的任意矩阵都可以分解成LUM的乘积,其中L,M为下三角矩阵,U为上三角矩阵.     

p－除环上子空间的交与和

本文讨论ｐ－除环上子空间的交与和的关系，用Ａｂｅｌ范畴中裂正合序列的性质证明矩阵秩的恒等式和维数公式．     

Abel范畴中态射乘积诱导的短正合列及其应用

本文给出了Ａｂｅｌ范畴中态射乘积αβγ诱导的短正合列的立体交换图。其中共有７个３×３平面，３０条线。每一条线（两端的零对象省略）都是一个短正合列．这些短正合列用到ｐ－除环上的矩阵，可以导出一系列矩阵秩的恒等式，它不仅推广了域上矩阵秩的结果，还能得到一些新的关系式．     

The rank of the difference of matrices with prescribed invariant factors

We give a necessary and sufficient condition for the existence, over a principal ideal domain, of matrices A and B with prescribed invariant factors, such that rank(A-B) is equal to a fixed integer.     

Diagonalization of matrices over graded principal ideal domains

The main result of this paper is the analogue of the classical diagonal reduction of matrices over PIDs, for graded principal ideal domains. A method for diagonalizing graded matrices over a graded principal ideal domain is obtained. In Section 2 we emphasis on some applications. A procedure is given to decide whether or not a matrix defined over an ordinary Dedekind domain (i.e. nongraded), with cyclic class group, is diagonalizable. In case the answer is positive the diagonal form can be calculated. This can be done by taking a suitable graded PID which has the Dedekind domain as its part of degree zero. It turns out that, even in the case where diagonalization of a matrix over the part of degree zero is not possible, the diagonal representation over the graded ring contains useful information. The main reason for this is that the graded ring hasn't essentially more units than its part of degree zero. We illustrate this by considering the problem of von Neumann regularity of a matrix over a Gr-PID and to matrices over Dedekind domains with cyclic class group. These problems were the original motivation for studying diagonalization over graded rings.     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.     

A note on matrix inversion

We generalize a recent result of Thompson on inverses of block matrices over principal ideal domains to isomorphisms of direct sums of modules over an arbitrary ring.     

环上矩阵的广义Moore-Penrose逆

本文给出带有对合的有1的结合环上一类矩阵的广义Moore-Penrose逆存在的充要条件，而这类矩阵概括了左右主理想整环，单Artin环上所有矩阵。     

Similarity Classes of 3 × 3 Matrices Over a Local Principal Ideal Ring

In this article similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.     

Group representations and matrix spectral problems

We study a connectionvia group representation theory, between the problem of describing the invariant factors of a product of two matrices over a principal ideal domain and the problem of describing the spectrum of a sum of two Hermitian matrices.     

Determinants of matrices over commutative finite principal ideal rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring R of a fixed determinant a is determined for all $a\in R$ and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.     

Polynomes a valeurs entieres sur un anneau de pseudovaluation

We study the ring of integral valued polynomials over a pseudovaluation domain A. We entirely determine the set of prime ideals above the maximal ideal M of A: if M is a principal ideal in the valuation domain V associated with A and if its residue field is finite, then this set is in bijection with a topologically complete ring, as in the Noetherian case; if M is principal but of infinite residue field in V, then this set is finite; at last, if M is not principal, then the ring of integral valued polynomials is included in V[X] and has the same set of prime ideals above M.