倍幂等元与倍对合元的换位子的可逆性

本文研究在一个有单位元的环中两个倍幂等元的换位子与两个倍对合元的换位子的可逆性问题.利用倍幂等元和倍对合元的性质,得到了两个倍幂等元的换位子与两个倍对合元的换位子可逆的几个充要条件,揭示了倍幂等元和倍对合元与它们的换位子的内在联系.     

正反幂等元和差的可逆性

研究在一个有单位元的环中,将幂等元的和与差的可逆性的一些已知结果,推广到正反幂等元的和与差可逆性上,并在矩阵环中得到一些运用.     

一类幂等元模J(R)可强提升的环

将Nicholson提出的幂等元强提升概念进行了推广,定义了L-环,弱L-环,使用通常环论方法研究了L-环中本原幂等元的Local性和L-环与potent环之间的关系,证明了一个环是L-环的充分必要条件是R/J(R)是Boole环,且幂等元模J(R)可强提升,同时对具有一对零同态的Morita Context环C=A VW B,关于L-性讨论了C与A,B之间的关系.     

正则幂环的同态与同构

随着模糊数学的发展,各种代数结构的提升为得越来越重要,李洪兴教授在「１,２」中首次提出并研究了幂群及ＨＸ环,本文在文「３～６」的基础上深入讨论了幂环的一些性质,并在正则幂环中建立了几个同态与同构定理。     

半群环的幂等元

讨论了半群环R[S]的幂等元问题．对[1]提出的公开问题9作了一个肯定回答，同时就一般半群环的幂等元的具体形式作了深入的研究，给出了若干情形下的幂等元刻划．     

具有弱中间幂等元的正则半群

本文在正则半群上引入弱中间幂等元和拟中间幂等元,着重探讨了这两类幂等元的性质特征.构造了若干具有弱(拟)中间幂等元的正则半群,确定了弱中间幂等元和拟中间幂等元之间的关系,给出了弱中间幂等元和拟中间幂等元各自的等价判定,利用拟中间幂等元刻画了纯正半群.最后,得到了构造具有拟中间幂等元的正则半群的一般途径,并在此基础上进一步给出了判定正则半群是否具有乘逆断面的方法.     

本原环的秩等于1的幂等元的分类及其应用

本文提出了含非零基座本原环的全体秩等于１的幂等元的两种分类，据此进一步提示了此类本原环的结构。     

拟fine环和2-fine环（英文）

称环R是fine环,如果R中每个非零元素均可以表示为一个可逆元与一个幂零元之和.Fine环的概念由Cǎlugǎreanu和Lam在[J.Algebr Appl.,2016,15(9):1650173,18 pp.]中给出,fine环和单环密切相关.本文研究如下两类环:每个非幂零元素均可表示为一个可逆元与一个幂等元之和的环以及每个元素均可表示为一个可逆元与两个幂零元之和的环.     

由环的幂零性所确定的根

结合环中的环的幂零性不是根性质。为此，本文将结合环中的幂零理想概念扩展为次拟幂零理想和拟幂零理想，定义次拟幂零根ＳＮ和拟幂零根ＱＮ，证明它们均为Ａｍｉｔｓｕｒ－Ｋｕｒｏｓｈ根，且二者相等，进一步，我们给出了ＱＮ－半单环的构造命题和ＱＮ－根的模刻划。     

有幺单Г——环类所确定的上根

结合环根论中由具有单位元的单环类所确定的上根就是Ｂｒｏｗｎ－ＭｃＣｏｙ根。本文指出在Г－环中，鉴于Г－环单位元的复杂性质使得有幺单Г－环类确定的上根演变成８个，进而研究了这些根之间及这些根与Г－环其它根之间的大小关系。     

ON THE INVERTIBILITY OF HILBERT SPACE IDEMPOTENTS

This note is to present some results on the group invertibility of linear combina- tions of idempotents when the difference of two idempotents is group invertible.     

On the invertibility of Hilbert space idempotents

This note is to present some results on the group invertibility of linear combinations of idempotents when the difference of two idempotents is group invertible.     

Characterizations and representations of the group inverse involving idempotents

This paper is to present some results on the group invertibility of products and differences of idempotents. In addition, some formulae for the group inverse of sums, differences and products of idempotents are established by using some given idempotents.     

一个特征标环素谱的连通性

本文确定了一个有限群特征标环通过代数整数环扩张后素谱的结构，在此基础之上，利用与[1]中类似的方法证明了这个扩张后的特征标环素谱的连通性。同时还计算了有限群的复类函数空间的幂等元．     

Characterizations and representations of the Drazin inverse involving idempotents

This paper is to present some results on the Drazin invertibility of products and differences of idempotents. In addition, some formulae for the Drazin inverse of sums, differences and products of idempotents are also established.     

Invertibility of the Sum of Idempotents

We study necessary and sufficient conditions for the invertibility of the sum f+g when f and g are idempotents in a unital ring or bounded linear operators in Hilbert or Banach spaces. We describe the relation between the invertibility of f+g and f -g.     

Invertibility of the Sum of Idempotents

We study necessary and sufficient conditions for the invertibility of the sum f+g when f and g are idempotents in a unital ring or bounded linear operators in Hilbert or Banach spaces. We describe the relation between the invertibility of f+g and f m g.     

Invertibility of the Difference of Idempotents

We study conditions equivalent to the invertibility of f -g when f and g are idempotents in a unital ring, and give applications to bounded linear operators in Banach and Hilbert spaces. In the setting of rings we are able to show that many conditions previously linked to finite dimensionality, rank equalities, norm topology of bounded linear operators or to properties of C *-algebras can be in fact proved by simple algebraic arguments.