兰州大学机构库 >数学与统计学院
Clean 环和具有稳定秩条件的环的若干研究
Alternative TitleSome studies on clean rings and rings with stable range conditions
孙晓青
Thesis Advisor罗彦锋
2011-06-03
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keywordclean 环 半 clean 环 拟可逆 伪正交 QB-环 稳定秩条件
Abstract可逆元和幂等元是环的两类重要的元素, 它们对环的结构起着非常重要的作用.如果环R 的任意元素可以写成一个幂等元和一个可逆元的乘积, 那么称R 是单位正则环. 如果环R 的任意元素可以写成一个幂等元和一个可逆元的和, 那么称R 是clean 环.本文以clean 环为主线, 主要研究了clean 环和具有稳定秩条件的环的结构及其扩张, 得到了一些新的有意义的结果. 全文分为六章. 第一章介绍了课题的研究背景和研究进展, 概括了本论文必备的预备知识和主要结论. 第二章推广了唯一强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. 第三章定义了强半clean 环, 研究了半clean 环和强半clean 环的基本性质, 还讨论了斜Hurwitz 级数环的强半clean 性, 证明了环R 上的斜Hurwitz 级数环是半clean 的当且仅当环R 是半clean 的. 第四章用拟可逆替换clean 环里的可逆后定义了一类新的环: Q-clean 环. 本章研究了Q-clean 环的一些基本性质, 讨论了Q-clean 一般环, 同时, 证明了Q-clean 环(未必含单位元)的理想还是Q-clean 的, 最后给出了Q-clean 环和它的斜corner 之间的若干关系. 第五章研究了拟单位正则和QB-环, 伪单位正则和QB1-环的联系, 推广了Are在文献[6] 和Chen 在文献[35] 的结论, 给出了exchange 环是QB-环, exchange 环是QB1-环的许多充分必要条件. 第六章介绍了具有单位弱稳定秩1 的环的两种情况, 讨论了它们的一般线性群的基本形式, 类似的, 研究了具有I-弱稳定秩1 的exchang环和具有单位弱(I)-稳定秩的环.
Other AbstractIt is well known that the units and idempotents of a ring are key elements determining the structure of the ring. It is proved that a ring R is a unit-regular (in the sense of J.von Neu- mann) ring if and only if each element of R can be written as the product of a unit and an ide- mpotent (in any order). In parallel, if we change product to sum in this condition, that is, each element of R can be written as the sum of a unit and an idempotent, then we call R is a clean ring. This thesis does some researches on the structures and extensions of clean rings and rin- gs with stable range conditions, obtains some new meaningful results, and consists of six cha- pters. In Chapter 1, the background and major results on clean rings and rings with stable range conditions are introduced. The prepare knowledge and the main results of this thesis are pres- ented. Chapter 2 generalizes uniquely strongly clean rings to general rings. Some basic proper- ties about uniquely strongly clean general rings are obtained. Some conditions for the upper triangular matrix ring and skew power series rings over general rings being uniquely strongly clean are given. Chapter 3 defines strongly semiclean rings. Some general properties of semiclean rings and strongly semiclean rings are given. Lastly, strongly semiclean rings of skew Hurwitz seri- es are explored. The skew Hurwitz series of R is a semiclean ring if and only if R is. In chapter 4, replacing invertibility with quasi-invertibility in clean ring we discover a new class of rings, Q-clean rings. Some basic properties of Q-clean rings are obtained. The definition of Q-clean rings with or without identity is given. According this, it is shown that the ideal of a Q-clean ring(unitial or not) is also Q-clean. Some relations between Q-clean ring and its corners are obtained. In chapter 5, some relations of quasi-unit regular and QB-rings, pseudo-unit regular and -rings are obtained. It is extended the results of Are in [6] and Chen in [35]. The suffic- ient and necessary conditions of an exchange ring is a QB-ring and an exchange ring is a - ring are explored. In chapter 6, rings satisfy unit weakly stable range are introduced. The basic forms of their general linear group are investigated. Similarly, Exchange rings having ideal-weakly stable range one and unit weakly (I)-stable range rings are discussed.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225727
Collection数学与统计学院
Recommended Citation
GB/T 7714
孙晓青. Clean 环和具有稳定秩条件的环的若干研究[D]. 兰州. 兰州大学,2011.
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