模消去的Morita Contexts

证明了置换环上n-稳定秩条件关于Morita Context是遗传的;进一步把该结果推广到了强可分环,从而提供了新的具有模消去的置换环.     

幂理想消去的置换环

引进了一类新的部分幂消去的置换环, 得到了置换环幂理想消去的等价刻画, 推广了幂消去的相关结果.     

置换环上的稳定矩阵

本文证明了置换环上的正则稳定矩阵是幂等矩阵和可逆矩阵的积,进一步证明了置换环上的正则稳定矩阵可以对角化。     

置换模,ρ置换模与Conlon比析

本文对特征P的基域F引入适当的Galois群T,讨论p置换模的Green环的Conlon比析在Galois群T作用下的动态,证明了在T作用下p置换模的Green环的不动点集重合于置换模的Green环。     

广义幂比较置换理想

在置换理想上,我们引进了一类广义幂比较;得到了置换理想为广义cu-理想的等价刻画,并推广了置换广义cu-环的相关结果．     

关于适中稳定环

引入了一类新的稳定环--适中稳定环,证明了如果R是Artin本原因子的置换环,则R是适中稳定环当且仅当1R=α+β,其中α,β∈U(R).如果R是适中稳定环,则K1(R)(≌)U(R)/θ(R),这里θ(R)是V(R)与U(R)'之间的适中子群.作为应用,计算了Artin本原因子的置换环的K1群.     

置换QB-环上的有限生成投射模

研究了置换QB-环上的有限生成投射模,证明了QB-环可以通过有限投射生成子的外替换和内替换来刻画.同时,还研究了QB-环上投射模的消去性.     

有限置换模自同态环的可分性质

设Q是有限置换右R模,则EndR(Q)是可分环当且仅当对所有A,B∈FP(Q),A A≌A B≌B B A≤ B或B≤ A.作为应用得到了EndR(P Q)是可分环当且仅当EndRP和EndRQ为可分环,其中P,Q为有限置换右R模.     

有限置换模自同态环的可分性质

设Q是有限置换右R模,则End_R(Q)是可分环当且仅当对所有A,B∈FP(Q),A AA B B B A≤ B或 B≤A,作为应用得到了 End_R(P Q)是可分环当且仅当End_R P和End_R Q为可分环,其中P,Q为有限置换右R模。     

几乎半正则环（英文）

设R是环,J(R)是R的Jacobson根.R的元素a称为半正则元,如果存在正则元b∈R使得a-b∈J(R).环R称为几乎半正则环,如果对R的任意元a,有a或者1-a是半正则的.本文引入了几乎半正则环作为VNL-环和半正则环的推广.构造了一些例子,证明了几乎半正则环是置换环;将半正则环的许多性质推广到了几乎半正则环上.     

Separative cancellation for projective modules over exchange rings

A separative ring is one whose finitely generated projective modules satisfy the propertyA⊕A⋟A⊕B⋟B⊕B⇒A⋟B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ringR has an idealI withI andR/I both separative, thenR is separative. The research of the first and fourth authors was partially supported by a grant from the DGICYT (Spain) and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. That of the second author was partially supported by a grant from the NSF (USA). The final version of this paper was prepared while he was visiting the Centre de Recerca Matemàtica, Institut d'Estudis Catalans in Barcelona, and he thanks the CRM for its hospitality.     

模的r-比较与r-幂比较

本文研究模的r-比较性和r-幂比较性,给出了r-比较性的新刻划,得到了模的r-幂比较结构与其自同态环的r-幂比较性间的关系,证明了如果模M为r-幂比较的则它的任意直和项也是r-幂比较的．对于可分置换环,给出了r-幂比较性与r-比较性间的联系．     

SELF-CANCELLATION OF MODULES HAVING THE FINITE EXCHANGE PROPERTY

Self-cancellation of modules having the finite exchange property is introduced. If a right R-module M has the finite exchange property, it is shown that M has self-cancellation if and only if EndR(M) is a strongly separative ring. Using this result, some new characterizations of strong separativity are obtained.     

Comparability,separativity, and exchange rings

There are several long-standing open problems which ask whether regular rings, and C^?-algebras of real rank zero, satisfy certain module cancellation properties. Ara, Goodearl, O'Meara and Pardo recently observed that both types of rings are exchange rings, and showed that separative exchange rings have these good cancellation properties, thus answering the questions affirmatively in the separative case. In this article, we prove that, for any positive integer s, exchange rings satisfying s-comparability are separative, thus answering the questions affirmatively in the s-comparable case. We also introduce the weaker, more technical, notion of generalized s-comparability, and show that this condition still implies separativity for exchange rings. On restricting to directly finite regular rings, we recover results of Ara, O'Meara and Tyukavkin.     

Related comparability over exchange rings

It is shown that every exchange ring satisfying related comparability is separative. This yields that related comparability over exchange rings is Morita invariant. Also we investigate pseudosimilarity over exchange rings satisfying related comparability.     

形式三角矩阵环的可分性和稳定度

In this paper we study the formal triangular matrix ring T =and give some necessary and sufficient conditions for T to be (strongly) separative, m-fold stable and unit 1-stable. Moreover, a condition for finitely generated projec-tive T-modules to have n in the stable range is given under the assumption that A and B are exchange rings.     

Separative Cancellation over Regular Rings

This article develops element-wise characterizations of separative ideals in regular rings. These give connections among separativity, annihilators, cokernels and unit-regularity. For strongly separativity, the analogs are also obtained.     

Countable Exchange and Full Exchange Rings

We show that a Dedekind-finite, semi-π-regular ring with a “nice” topology is an ?₀-exchange ring, and the same holds true for a strongly clean ring with a “nice” topology. We generalize the argument to show that a Dedekind-finite, semi-regular ring with a “nice” topology is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite exchange has countable exchange, and all modules with Dedekind-finite, semi-regular endomorphism rings are full exchange modules. These results are generalized further.     

Exchange环的比较结构

本文得到了相关比较Exchange环上模的替代性,并研究了幂比较Exchange 环上模的结构,进而提供了一类新的相关比较:Exchange环.