Matrix product codes with rosenbloom-tsfasman metric

In this article, the Rosenbloom-Tsfasman metric of matrix product codes over finite commutative rings is studied and the lower bounds for the minimal Rosenbloom-Tsfasman distances of the matrix product codes are obtained. The lower bounds of the dual codes of matrix product codes over finite commutative Frobenius rings are also given.     

保持交换环上两类矩阵运算的映射

主要针对交换环上两类矩阵的保持问题进行展开:(1)刻画了交换环上全矩阵空间和上三角形矩阵空间的保持反对合矩阵映射的形式.(2)研究了交换环上n阶上三角形矩阵空间的保持伴随矩阵映射的形式.     

可换环上辛代数与一般线性李代数之间的中间李代数

本文研究了含幺可换环上一般线性李代数的子代数结构.通过构造特殊矩阵并利用这些矩阵进行计算, 得到了任意含幺可换环上辛代数与一般线性李代数之间的所有中间李代数的形式.并且有利于研究可换环上相应的典型群的子群结构.     

交换环上三角矩阵模间的线性保持映射(英文)

本文研究了交换环上三角矩阵模间的线性保持问题.利用矩阵计算技巧和局部化技巧,刻画了上三角矩阵T_n(R)上分别保持立方幂等,{1}逆,{1,2}逆和群逆的所有R模自同构集合中的元素,其中R是交换环.     

Rings in which elements are uniquely the sum of an idempotent and a unit that commute

A ring is called uniquely clean if every element is uniquely the sum of an idempotent and a unit. The rings described by the title include uniquely clean rings, and they arise as triangular matrix rings over commutative uniquely clean rings. Various basic properties of these rings are proved and many examples are given.     

Notes on linear codes over finite commutative chain rings

The properties of the generator matrix are given for linear codes over finite commutative chain rings,and the so-called almost-MDS (AMDS) codes are studied.     

Matrix product codes over finite commutative Frobenius rings

Properties of matrix product codes over finite commutative Frobenius rings are investigated. The minimum distance of matrix product codes constructed with several types of matrices is bounded in different ways. The duals of matrix product codes are also explicitly described in terms of matrix product codes.     

Prime Radicals of Chevalley Groups over Commutative Rings

The main purpose of this paper is the computation of prime and soluble radicals of Chevalley groups over arbitrary commutative rings with unity (except the case A₁, where restrictions 2, 3 ∈Gl(R) are essential). Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

Polynomial functions on upper triangular matrix algebras

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.     

On G-Regular Local Rings

In this article, we define a G-regular local ring as a commutative, noetherian, local ring, over which all totally reflexive modules are free. We study G-regular local rings and observe that they behave similarly to regular local rings. We extend Eisenbud's matrix factorization theorem and Knörrer's periodicity theorem to G-regular local rings.     

Invariants of finite Hopf algebras

This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings. Topics considered include integrality over the invariant rings, properties of the canonical map between the prime spectra, orbital and stabilizer algebras, projectivity over the invariant rings, and descent of Cohen-Macaulayness.     

Matrices over a commutative ring as sums of three idempotents or three involutions

Motivated by Hirano-Tominaga’s work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis’s results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices.     

Sur le produit tensoriel des groupes affines

Let A and B be two commutative affine group schemes over a field. There exists an affine group A?B such that Hom(A?B,C)?Bil(A×B,C) for any affine group C. We use technics of the commutative algebraic groups theory, in order to compute these tensor products and to characterize “flat” groups in the unipotent case. The tensor product of commutative affine groups has most properties of the usual tensor product but it is not always associative. As an application we prove a structure theorem of the category of modules over some affine connected prosmooth rings.     

Higher Level Orderings on Modules

The aim of this paper is to investigate higher level orderings on modules over commutative rings. On the basis of the theory of higher level orderings on fields and commutative rings, some results involving existence of higher level orderings are generalized to the category of modules over commutative rings. Moreover, a strict intersection theorem for higher level orderings on modules is established.     

Roth's theorems over commutative rings

In 1952, W.E. Roth showed that matrix equations of the forms AX?YB = C and AX?XB = C over fields can be solved if and only if certain block matrices built from A, B, and C are equivalent or similar. We show here that these criteria remain valid over arbitrary commutative rings. To do this, we use standard commutative algebra methods to reduce to the case of Artinian rings, where a simple argument with     

关于二次模的正交群

本文给出了交换环上二次模直交和的正交群的计算公式，对多项式环上的二次模，给出了一类子群的局部整体定理。     

Reversible Group Rings Over Commutative Rings

Let G be a torsion group and R be a commutative ring with identity. We investigate reversible group rings RG over commutative rings, extending results of Gutan and Kisielewicz which characterize all reversible group rings over fields.     

Applications of perturbation theory to iterated fibrations

For a class of spaces including simply connected spaces and classifying spaces of nilpotent groups, relatively small differential graded algebras are constructed over commutative rings with 1 which are chain homotopy equivalent to the singular cochain algebra. An application to finitely generated torsion-free nilpotent groups over the integers is given.     

The isomorphism relation between tree-automatic Structures

An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ₂¹-set nor a Π₂¹-set.