Two questions on semi-clean group rings

It was proved in [4] that every group ring of a torsion abelian group over a commutative local ring is a semi-clean ring. It was asked in [4] whether every group ring of a torsion abelian group over a commutative clean ring is a semi-clean ring and whether every group ring of a torsion abelian group over a commutative semi-clean ring is a semi-clean ring. In this paper, we give a positive answer to question 1 and a negative answer to question 2.     

Reduction of Matrices over Bezout Rings of Stable Rank not Higher than 2

We prove that a commutative Bezout ring is an Hermitian ring if and only if it is a Bezout ring of stable rank 2. It is shown that a noncommutative Bezout ring of stable rank 1 is an Hermitian ring. This implies that a noncommutative semilocal Bezout ring is an Hermitian ring. We prove that the Bezout domain of stable rank 1 with two-element group of units is a ring of elementary divisors if and only if it is a duo-domain.     

一类环上HX环的结构

自李洪兴１９９１年提出了ＨＸ环以来，人们一直有这么一个问题没解决，就是是否存在非平凡的ＨＸ环的例子？但至今既没找到非平凡的ＨＸ环，也没有证明任一环Ｒ仅存在平凡的ＨＸ环。针对这个问题，本文提出并证明了一类环仅有平凡ＨＸ环，还给出了一系列的结构定理。这样，既为证明任一环Ｒ仅有平凡的ＨＸ环的猜想有新的启示，也为人们指明无须在这一类环上寻找非平凡ＨＸ环。     

Trivial Ring Extensions Defined by Arithmetical-Like Properties

In this article we investigate the transfer of the notions of elementary divisor ring, Hermite ring, Bezout ring, and arithmetical ring to trivial ring extensions of commutative rings by modules. Namely, we prove that the trivial ring extension R: = A ? B defined by extension of integral domains is an elementary divisor ring if and only if A is an elementary divisor ring and B = qf(A); and R is an Hermite ring if and only if R is a Bezout ring if and only if A is a Bezout domain and qf(A) = B. We provide necessary and sufficient conditions for R = A ? E to be an arithmetical ring when E is a nontorsion or a finitely generated A ? module. As an immediate consequences, we show that A ? A is an arithmetical ring if and only if A is a von Neumann regular ring, and A ? Q(A) is an arithmetical ring if and only if A is a semihereditary ring.     

右对称环

本文在左对称环的基础上提出了右对称环的概念,分别给出了是右对称环但不是左对称环和是左对称环但不是右对称环的例子．证明了（1）如果R是Armendariz环,则R是右对称环的充要条件R[x]是右对称环;（2）如果R是约化环,则R[x]/（x^n）是右对称环,其中（xn）是由xn生成的理想．     

Rings of almost unit stable rank 1

We introduce the notion of a ring of almost unit stable rank 1 as a generalization of the notion of a ring of unit stable rank 1. We prove that a ring of almost unit stable rank 1 with nonzero Jacobson radical is a ring of unit stable rank 1 and is a 2-good ring. We introduce the notion of almost 2-good ring and show that an adequate domain is an almost 2-good ring.     

Clean一般环的几个结果

证明了一般环I是Clean一般环当且仅当I上的形式幂级数一般环I[[x]]是Clean一般环;一般环I上的多项式环I[x]是Clean一般环当且仅当I是诣零的.引入了强Clean一般环的概念,它是强Clean环的推广.并证明了强π-正则的一般环是强Clean一般环.     

投射可迁环上矩阵环的若当同态

设R′是一个环,Mn′（R′）是R′上的n′×n′矩阵环.如果环R有不变基数性质并且每个有限生成的投射左R-模是自由模,则R是一个投射自由环.如果环R≌Mr（S）,其中S是一个投射自由环,则R是一个投射可迁环.当R是一个投射可迁环时,给出了从Mn′（R′）到Mn（R）（n′≥n≥2）的若当同态的代数公式.     

Coverings and Ring-Groupoids

We prove that the set of homotopy classes of the paths in a topological ring is a ring object (called ring groupoid). Using this concept we show that the ring structure of a topological ring lifts to a simply connected covering space.     

分次FS-环(英)

本文引进群分次环上分次模的分次FS－模的概念,利用分次极大分次左理想给出分次FS－环的几个刻画,得到了环R和群环RG,分次环R和分次环的群环R[G]间的几个等价条件．     

THE MAXIMAL LEFT QUOTIENT RINGS OF ALTERNATIVE RINGS

ABSTRACT

We study the notion of a (general) left quotient ring of an alternative ring and show the existence of a maximal left quotient ring for every alternative ring that is a left quotient ring of itself.     

强symmetric环

为了统一交换环和约化环的层表示,Lambek引进了Symmetric环.继续symmetric环的研究,定义引入了强symmetric环的概念,研究它的一些扩张性质.证明环R是强symmetric环当且仅当R[x]是强symmetric环当且仅当R[x;x~(-1)]是强symmetric环.也证明对于右Ore环R的经典右商环Q,R是强symmetric环当且仅当Q是强symmetric环.     

Periodic jordan rings and order structure

The aim of this paper is to obtain information about a periodic Jordan ring by using only properties of its idempotent elements. Osborn proves that a power-associative periodic ring having only one nonzero idempotent element is a division ring. so associative. He also proves that a periodic Jordan ring is a subdirect product of simple periodic Jordan rings and that a simple periodic Jordan ring is either a periodic field or a Jordan ring of capacity 2. Using these results we obtain some necessary and suficient conditions for a periodic Jordan ring to be associative, and these conditions are only given in terms of the idempotent elements. We also characterize the periodic Jordan ring which are a direct product of periodic fields and simple periodic Jordan rings of capacity two.     

关于伪准素环的注记

一个有单位元的交换环Ｒ称为伪准素环，如果Ｒ的每个非零理想都是某个准素理想之幂．本文证明了环Ｒ是伪准素环当且仅当Ｒ是准素环或Ｒ是两个域的直和或Ｒ是至多具有三个素理想的一维局部环，并且每个非零理想或是某个极小素理想之幂或是某个属于极大理想的准不比理想之幂.     

与线性变换的完全环同构的环理论(Ⅱ)

<正> 为了进一步对本原环结构的研究,本文引进规范环的概念,我们说环R是规范的,若R是一个线性变换完全环并且及的基座对于任一对应基{E_i}皆有=∑RE_i=∑E_iR.容易知道,满足单侧理想极小条件的单纯环必是规范的.     

二面体群的Grothendieck环结构

二面体群的表示范畴为对称半单monoidal范畴,因而其Grothendieck环为有限多个元素生成的交换环.本文确定了该Grothendieck环的极小生成元,并且进一步证明了该Grothendieck环与某一多项式环的商环同构.     

拟-中心半交换环

环$R$称为拟-中心半交换的(简称QCS环)如果对$a,b\in R$, $ab=0$蕴含$aRb\subseteq Q(R)$, 其中$Q(R)$为$R$的拟中心.证明了如果$R$ 为QCS环, 那么$R$的幂零元集恰好是它的Wedderburn根, 且对$n\geq 2$, 上三角矩阵环$R=T_n(S)$ 是QCS 环当且仅当$n=2$ 且$S$ 是duo 环, 而$T_{2k+2}^k$是QCS环如果$R$是约化的duo环.     

Differential projective modules over differential rings

Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.     

Noether环理想的性质

Noether交换环是一类非常重要的环,本文主要对Noether交换环进行了研究和讨论,得到了Noether交换环、Noether整环的若干性质;并推广了文[1]中的部分结果.     

环F_p+uF_p+vF_p+uvF_p上的循环码

讨论了非有限链环R=F_p+uF_p+vF_p+uvF_p上的循环码.通过环R上的循环码与多项式环R_n=(F_p+uF_p+vF_p+uvF_p)[x]/(xⁿ-1)的理想的对应关系及对R_n的研究给出了R上循环码的刻画.最后定义了一个Gray映射,并刻画了F_p+uF_p+vF_p+uvF_p上的循环码在该映射下的像.