关于γ-semiclean环

本文定义了一种新的环γ-semiclean环.环R称为γ-semiclean的是指R中的每个元素都可以写成一个正则元和一个周期元的和.本文主要利用环论的方法研究了γ-semiclean环的相关性质,推广了clean环和半-clean环的已知结果.     

拓扑空间和近Clean环

元α∈R称为近clean的,如果它是幂等元和满元察的和.若环R中的每一个元都是近clean元,则环R称为近clean环.在此定义下,证明了对Abel环R,下列结论是等价的,(1)R是近clean的;(2)(ν)α∈R,Эe=e2∈R,使得V(α)(∈)V(e)且V(1-α)(∈)V(1-e);(3)适中空间Ξ(R)是强零维的;(4)R是pm环且Max(R)是强零维的.某些近clean元的判别也可由此得到.     

唯一强clean一般环

推广了唯一强clean环的概念,定义了唯一强clean一般环,得到了唯一强clean一般环的若干性质,并且给出了一般环的三角矩阵环和斜幂级数环是唯一强clean的条件.     

关于强Clean一般环 (英)

本文介绍了强clean—般环的概念并将一些基本的结果推广到这个更广的环类．证明了强clean一般环的角落环和强π-正则一般环都是强clean的,还讨论了强clean一般环的扩张并且证明了满足条件J(I)=Q(I)的交换clean一般环的上三角矩阵环是强clean的．     

子半理想环

本文提出了环论中的一个新概念：子半理想，得到了群环为子半理想环的一个充要条件。     

拟polar环的注记（英文）

称一个环R中的元素a是拟polar元,若存在p2=P∈R满足p∈comm_R~2（a）,a＋P∈U（R）并且ap∈R~（qnil）;且称环R是拟polar的如果R中每一个元素都是拟polar元.本文证明了,任一环R中强π-正则元是拟polar的,而拟polar元是强clean的.拟polar环的一些扩张性质也作了探讨.     

Dubrovin赋值斜群环

本文中对一个斜群环为Dubrovin赋值环给出了一系列等价刻画,并且刻画了一个Dubrovin赋值斜群环的所有素理想.     

弱$r$-Clean环

As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a∈R there exist an idempotent e and a regular element r such that a = r + e or a = r-e. Some properties and examples of weakly r-clean rings are given. Furthermore, we prove the weakly clean rings and weakly r-clean rings are equivalent for abelian rings.     

J-clean环的推广

如果R中每个元素(对应地,可逆元)均可表示为一个幂等元与环R的Jacobson根中一个元素之和,则称环R是J-clean环(对应地,UJ环).所有的J-clean环都是UJ环.作为UJ环的真推广,本文引入GUJ环的概念,研究GUJ环的基本性质和应用.进一步地,研究每个元素均可表示为一个幂等元与一个方幂属于环的Jacobson根的元素之和的环.     

关于半群环与群环的链条件的等价性

给出了将半群环的链条件转化为群环的链条件的一个定理,并由此将［１］中的结果推广到半群环的情形．     

Semiclean Rings

Abstract

The notion of semiclean elements in a ring is defined. Every clean element is semiclean. A ring R is said to be semiclean if every element in R is semiclean. The group ring Z _p G with G a cyclic group of order 3 is proved to be semiclean. The n × n matrix ring M _n (R) over a semiclean ring is semiclean. If R is a torsion free semiclean ring in which every element of R can be written as a sum of periodic and ±1, then R is clean. Every element in a semiclean ring R with 2 invertible is a sum of no more than 3 units.     

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.     

具有一对零态射的Morita Context环(II)

设$(A,B,V,W,\psi,\phi)$是一个Morita Context,具有一对零态射$\psi=0$, $\phi=0$, $C =\left ( \begin{array} {cc}A & V \\W & B \end{array}\right)$是对应的Morita Context环.本文给出了$C$与$A,B,V,W$之间关于环的$\pi$-正则性、semiclean性、Mophic性和环的Exchgange性、Potent性、GM性的关系.     

Armendariz group rings

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.     

关于半局部群环

Let R be an associative ring with identity.R is said to be semilocal if R/J(R)is(semisimple)Artinian,where J(R)denotes the Jacobson radical of R.In this paper,we give necessary and sufficient conditions for the group ring RG to be semilocal,where G is a locally finite nilpotent group.     

On Hochschild Extensions of Reduced and Clean Rings

We show that an arbitrary Hochschild extension of a reduced ring by a two-sided ideal is symmetric and reversible, and that any Hochschild extension of a clean ring by an arbitrary bimodule is clean. This generalizes a result of Kim and Lee, and provides many examples of clean rings.     

Group rings which are G-Dedekind prime rings

Let R be a commutative Noetherian domain and A be a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and A when the group ring R[A] is a G-Dedekind prime ring.     

Commutative neat group rings

A ring R is called clean if every element of it is a sum of an idempotent and a unit. A ring R is neat if every proper homomorphic image of R is clean. When R is a field, then a complete characterization has been obtained for a commutative group ring RG to be neat, but not clean. And if R is not a field, then necessary conditions are obtained for a commutative group ring RG to be neat, but not clean. A counterexample is given to show that these necessary conditions are not sufficient.     

An Example of Bergman's and the Extension Problem for Clean Rings

An example of Bergman is used to show that the extension of a clean ring by another clean ring need not be clean. That is, there exists a ring R and an ideal I of R such that both R/I and I are clean and idempotents lift modulo I, but R is not clean.