决定有限结合环构造的一个递归方法

本文中,“环”总是指结合环,对于自然数 n,用 R(n)表示两两互不同构的所有 n阶环之集合,N(n)表示集合 R(n)中元素的个数,由于一个有限环可以唯一地分解成为素数幂阶的环的直和,所以研究有限环的构造就归结为研究集合 R(p~m)和找出 N(p~m)的解析表达式,这里 p 是素数,m 是自然数.     

循环矩阵的逆

称为n阶循环矩阵,易知,一个n阶循环矩阵有n个循环矩阵,用Aa_0,Aa_1,…Aa_(n-1)分别表示主对角线上元素为a_0,a_1,…a_(n-1),的n阶循环矩阵,设     

矩阵环的单边理想

设R是有单位元1的环,(R)_n表示R上全体n阶方阵关于矩阵加法与乘法作成的环,则(R)_n的所有理想作成的格(见[1]P217)与R的所有理想作成的格同构(见[2]P178)。本文讨论(R)_n的左理想,得出如下结论: 1) (R)_n的左理想格同构于左R-模的子模格。     

有限級有序加羣及有序環的表現

在另一文中,我们讨论了由全体2维實向量所成的有序环,在该文最後並说當维数n>2时(n为有限)也可类似地作初步讨论.为了显示这种向量环的用途,我们考虑用向量环来表现一般有序环的问题.在本文中我们证明:任一“n级的”(见以下定义)有序环都能与一个由若干n维實向量所组成的有序环同构.(主要在於证出关於n级有序加羣的类似结果.)我们希望有较好的结果,即:任一n级有序环都能与由全体n维實向量所成的一个有序环的一个子环同构,但未能证明或否定.     

XST-环和Morita-Like等价

XST-环的概念于1999年由García和Marín[1]引进.本文主要研究XST-环的Morita-Like等价.证明了在环R中,由两个右q-稠密的XST-环确定的两个完全可加的范畴是同构的.通过描述生成元AM的自同态环End(AM)的q-稠密右理想和稠密右理想的关系,得到了,对于有单位元的环A而言,位于FMг(A)和FCг(A)间的中间矩阵环上的所有完全可加范畴是同构的.如此,扩张了中间矩阵环的Morita-Like等价链[2].再则,改进了文[1]中的主要结果,刻画了一个右XST-环Morita-Like等价于一个有单位元的环的条件.     

循环准整环的构造

研究了一类特殊循环环即循环准整环的构造,得到的主要结论有:1)所有的无限循环准整环就是M~0和 1)的标准分解式为n=p₁^α₁p₂^α₂…p_s^α_s,ⅰ)若s=1,则n阶循环准整环共有α₁+1个,它们是dZ/ndZ,其中d=p₁^β₁,0≤β₁≤α₁;ⅱ)若s>1,则n阶循环准整环共有α₁α₂…α_s个,它们是dZ/ndZ,其中d=p₁^β₁p₂^β₂…p_s^β_s,...     

强Θ类算子生成的C*-代数及其K-理论

本文研究由强类算子T生成的C^*-代数,证明了C^*(T)的换子理想I(T)是一列*同构于C₀(Z⁰)上n阶矩阵代数的n齐次代数和的闭包,特别当T完全非正常时,I(T)与C₀(Z₀)K(l²)*同构。本文第二部分得到完全非正常强类算子的谱与其近似点谱相等的一些等价条件,并计算了由该类算子生成的C^*-代数的一些K群。     

关于整表代数的同构

对任意整表基为T 的整表代数, 研究了代数整数环的一个子环R上的整R-代数RT, 证明了两个整表代数之间的一个R-代数同构必是一个整表代数同构, 只要它与一种标准化变换相容.     

关于三角形Toeplitz系统的复杂性

目前,已有结果表明,作两个n阶上(或下)三角形T矩阵的乘积以及做n阶三角形T矩阵乘n维列向量的算术运算次数,均不超过O(nlog_2n);而求n阶三角形T矩阵的逆,其工作量则不超过O(nlog_2~2n). 本文给出三角形T矩阵求逆与求解三角形Toeplitz线性方程组的快速算法.该算     

关于Sz''''asz的一个问题

1981年Sz′asz提出了如下的问题: “In which ring are the distinct subrings always non-isomorphic?” 为讨论此问题,先引入如下的 定义 一个(结合)环R,叫做内同构环(inner isomorphic ring),若R的所有真子环都是同构的。 一个(结合)环R,叫做内异环(inner non-isomorphic ring),若R的不同子环也不同构。 本文共分三节,在§§1—2中,分别给出了一个环R是内同构环和内异环的充要条件,并且也容易看它们的结构;在§3中还给出了一个有限多单环R是内异环的一个充重条件。 下面的环,都是给合环。     

分次单环的结构

A characterization of gr-simple rings is given by using the notion of componentwise-dense subrings of a full matrix ring over a division ring. As a consequence, any G-graded full matrix ring over a division ring is isomorphic to a dense subring of a full matrix ring with a good G-grading. Some conditions for a grading of a full matrix ring to be isomorphic to a good one are given, which generalize some results in: Dascascu, S., Lon, B., Nastasescu, C. and Montes, J. R., Group gradings on full matrix rings, J. Algebra, 220(1999), 709-728.     

Homomorphic images of subdirectly irreducible rings

We prove that every ring is a proper homomorphic image of some subdirectly irreducible ring. We also show that a finite ring R does not need to be isomorphic to the factor of a subdirectly irreducible ring by its monolith as well as R does not need to be a homomorphic image of a finite subdirectly irreducible ring. We provide an analogous characterization also for varieties of rings with unity, for the quasiregular rings, for the rings with involution and for their subvarieties of commutative rings.     

A generalization of right pci rings

By a well-known result of Osofsky [6, Theorem] a ring R is semisimple (i.e. R is right artinian and the Jacobson radical of R is zero) if and only if every cyclic right R-module is injective. Starting from this, a larger class of rings has been introduced and investigated, namely the class of right PCI rings. A ring R is called right PCI if every proper cyclic right R- module is injective (proper here means not being isomorphic to RR). By [l] and [Z], a right PCI ring is either semisimple or it is a right noetherian, right hereditary simple ring. The latter ring is usually called a right PCI domain. In this paper we consider the similar question in studying rings whose cyclic right modules satisfy some decomposition property. The starting point is a theorem recently proved in 13, Theorem 1.1): A ring R is right artinian if and only if every cyclic right R- module is a direct sum of an injective module and a finitely cogenerated module.     

一类特殊的有限循环环

证明了一类n阶(n=P_1P_2…p_m,p_i(i=1,2,…,m)互异为素数)环是有限循环环,并讨论了他们的结构及相关性质,最后给出了这类n阶环有零因子或有子域的充要条件.主要结果:P_1P_2…P_m阶环共有2^m个,它们是(p_(1m个,它们是(p_(1^k_1) p_(2k_1) p_(2^k_2)…p_(mk_2)…p_(m^k_m)Z)/(p_(1k_m)Z)/(p_(1^k_1+1)p_(2k_1+1)p_(2^k_2+1)…p_(mk_2+1)…p_(m^k_m+1)Z),其中k_i=0或1,1≤i≤m;阶是n=P_1P_2…p_m的环R可唯一分解为m个素数阶理想的直和,即R=〈α〉=(?);含pi(1≤i≤m)阶子域的P_1P_2…P_m阶环共有2k_m+1)Z),其中k_i=0或1,1≤i≤m;阶是n=P_1P_2…p_m的环R可唯一分解为m个素数阶理想的直和,即R=〈α〉=(?);含pi(1≤i≤m)阶子域的P_1P_2…P_m阶环共有2^(m-1)个,它们是p_(1(m-1)个,它们是p_(1^k_1) p_(2k_1) p_(2^k_2)…p_(mk_2)…p_(m^k_m)Z)/(p_(1k_m)Z)/(p_(1^k_1+1)p_(2k_1+1)p_(2^k_2+1)…p_(mk_2+1)…p_(m^k_m+1)Z),其.中k_i=0,k_j=0或1,1≤j≤m,j≠i.     

M-McCoy环和M-Armendariz环的多项式扩张

研究非交换环上的相对于幺半群的McCoy环和Armendariz环的多项式扩张.对于包含无限循环子幺半群的交换可消幺半群M,证明了若R是M-McCoy（或M-Armendariz）环,则R上的洛朗多项式环R[x,x-1]是M-McCoy（或M-Armendariz）环.     

关于C-环

主要讨论了 C-环的结构以及它与其它环类之间的重要关系 ,从而较深刻地刻划这种环     

Nil-Clean Rings with Involution

A *-ring R is called a nil *-clean ring if every element of R is a sum of a projection and a nilpotent.Nil *-clean rings are the *-version of nil-clean rings introduced by Diesl.This paper is about the nil *-clean property of rings with emphasis on matrix rings.We show that a *-ring R is nil *-clean if and only if J(R) is nil and R/J(R) is nil*-clean.For a 2-primal *-ring R,with the induced involution given by (aij)* =(a*ij)T,the nil *-clean property of Mn(R) is completely reduced to that of Mn(Z2).Consequently,Mn(R) is not a nil *-clean ring for n =3,4,and M2(R) is a nil *-clean ring if and only if J(R) is nil,R/J(R) is a Boolean ring and a*-a ∈ J(R) for all a ∈ R.     

Spectral partially ordered sets

We examine some topics related to (gold)spectral partially ordered sets, i.e., those that are order isomorphic to (Goldman) prime spectra of commutative rings. Among other results, we characterize the partially ordered sets that are isomorphic to prime spectra of rings satisfying the descending chain condition on radical ideals, and we show that a dual of a tree is isomorphic to the Goldman prime spectrum of a ring if and only if every chain has an upper bound. We also give some new methods for constructing (gold)spectral partially ordered sets.     

Isomorphisms of formal matrix incidence rings

Abstract—In a paper published in 2008 P. A. Krylov showed that formal matrix rings Ks(R) and Kt(R) are isomorphic if and only if the elements s and t differ up to an automorphism by an invertible element. Similar dependence takes place in many cases. In this paper we consider formal matrix rings (and algebras) which have the same structure as incidence rings. We show that the isomorphism problem for formal matrix incidence rings can be reduced to the isomorphism problem for generalized incidence algebras. For these algebras, the direct assertion of Krylov’s theorem holds, but the converse is not true. In particular, we obtain a complete classification of isomorphisms of generalized incidence algebras of order 4 over a field. We also consider the isomorphism problem for special classes of formal matrix rings, namely, formal matrix rings with zero trace ideals.     

α-对称环

引入α-对称环的概念,讨论了它与其它相关环的关系,证明环R是α-对称环当且仅当R上的n×n上三角矩阵环T_n(R)是α-对称环;若R是α-对称环,则R[x]/(x~n)是α-对称环,其中(x~n)是由x~n生成的理想,n为任意正整数.