类似于VNL环的若干环
发布时间：2018-07-14 15:50
【摘要】：近年来.局部环和VNL环这两类环已经成为代数学环论方向上重要的研究对象,主要研究方向有以下三个:一、直接研究局部环与VNL环的性质,对他们的根性质进行刻画;二、研究局部环与VNL环的扩张问题.三、类似于局部环与VNL环的定义方式,提出新环的定义.研究其性质.本文主要就第三种方向进行研究,新定义了几乎强正则环,几乎弱正则环.几乎π-正则环.几乎强π-正则环.讨论他们之间的关系,研究其性质及扩张问题.本文主要有以下几个部分组成:第一章:介绍局部环和VNL环及相关环的历史背景、发展过程和研究现状,简述本文问题的提出过程.简要总结了本文的主要工作和重要结果.第二章:主要介绍强正则环,弱正则环.π-正则环,强π-正则环,几乎单式正则环,SVNL环等环的概念以及与这些环相关的一些常用结论.第三章:引入几乎强正则环的概念,称一个环为几乎强正则环,如果对任意a ∈R,a或1 - a是强正则的.几乎强正则环是介于局部环和VNL环之间的一类环.主要结论如下:(1)约化的VNL环是几乎强正则环;(2)Abel的VNL环是几乎强正则环;(3)一族环的直积Π {Rα :α∈I}几乎强正则环当且仅当存在α在∈I得Rα0是几乎强正则环环其余α ∈I- α0 ,Rα,均为强正则环;(4)几乎强正则环的直极限仍是几乎强正则环.第四章:本章主要通过引入几乎弱正则环的概念,完善局部环,VNL环,几乎强正则环,几乎弱正则环.几乎π-正则环,几乎强π-正则环环间关系,得到如下图所示环间关系,研究他们的扩张问题.(?)第五章:综述本文所研究的几种类似于VNL环的相关环.并对这几类环的未来研究方向做了进一步的展望.
[Abstract]:In recent years Local rings and VNL rings have become important research objects in the direction of algebraic ring theory. The main research directions are as follows: first, the properties of local ring and VNL ring are studied directly, and their root properties are described. The extension of local rings and VNL rings is studied. Thirdly, the definition of new ring is proposed, which is similar to that of local ring and VNL ring. To study its properties. In this paper, we study the third direction and define almost strong regular rings and almost weak regular rings. Almost 蟺 -regular rings. Almost strongly 蟺 -regular rings. The relations between them are discussed, and their properties and expansion problems are studied. The main contents of this paper are as follows: chapter 1: introduce the historical background, development process and research status of local ring, VNL ring and related ring, and briefly introduce the process of putting forward the problem in this paper. The main work and important results of this paper are summarized briefly. In chapter 2, we introduce the concepts of strongly regular rings, weakly regular rings, 蟺 -regular rings, strongly 蟺 -regular rings, almost simple regular rings and SVNL rings, and some common conclusions related to these rings. Chapter 3: the concept of almost strongly regular ring is introduced, and a ring is called an almost strongly regular ring if it is strongly regular for any a 鈭，
本文编号：2122147
[Abstract]:In recent years Local rings and VNL rings have become important research objects in the direction of algebraic ring theory. The main research directions are as follows: first, the properties of local ring and VNL ring are studied directly, and their root properties are described. The extension of local rings and VNL rings is studied. Thirdly, the definition of new ring is proposed, which is similar to that of local ring and VNL ring. To study its properties. In this paper, we study the third direction and define almost strong regular rings and almost weak regular rings. Almost 蟺 -regular rings. Almost strongly 蟺 -regular rings. The relations between them are discussed, and their properties and expansion problems are studied. The main contents of this paper are as follows: chapter 1: introduce the historical background, development process and research status of local ring, VNL ring and related ring, and briefly introduce the process of putting forward the problem in this paper. The main work and important results of this paper are summarized briefly. In chapter 2, we introduce the concepts of strongly regular rings, weakly regular rings, 蟺 -regular rings, strongly 蟺 -regular rings, almost simple regular rings and SVNL rings, and some common conclusions related to these rings. Chapter 3: the concept of almost strongly regular ring is introduced, and a ring is called an almost strongly regular ring if it is strongly regular for any a 鈭，
本文编号：2122147
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