類似于VNL環(huán)的若干環(huán)
發(fā)布時間：2018-07-14 15:50
【摘要】：近年來.局部環(huán)和VNL環(huán)這兩類環(huán)已經成為代數學環(huán)論方向上重要的研究對象,主要研究方向有以下三個:一、直接研究局部環(huán)與VNL環(huán)的性質,對他們的根性質進行刻畫;二、研究局部環(huán)與VNL環(huán)的擴張問題.三、類似于局部環(huán)與VNL環(huán)的定義方式,提出新環(huán)的定義.研究其性質.本文主要就第三種方向進行研究,新定義了幾乎強正則環(huán),幾乎弱正則環(huán).幾乎π-正則環(huán).幾乎強π-正則環(huán).討論他們之間的關系,研究其性質及擴張問題.本文主要有以下幾個部分組成:第一章:介紹局部環(huán)和VNL環(huán)及相關環(huán)的歷史背景、發(fā)展過程和研究現狀,簡述本文問題的提出過程.簡要總結了本文的主要工作和重要結果.第二章:主要介紹強正則環(huán),弱正則環(huán).π-正則環(huán),強π-正則環(huán),幾乎單式正則環(huán),SVNL環(huán)等環(huán)的概念以及與這些環(huán)相關的一些常用結論.第三章:引入幾乎強正則環(huán)的概念,稱一個環(huán)為幾乎強正則環(huán),如果對任意a ∈R,a或1 - a是強正則的.幾乎強正則環(huán)是介于局部環(huán)和VNL環(huán)之間的一類環(huán).主要結論如下:(1)約化的VNL環(huán)是幾乎強正則環(huán);(2)Abel的VNL環(huán)是幾乎強正則環(huán);(3)一族環(huán)的直積Π {Rα :α∈I}幾乎強正則環(huán)當且僅當存在α在∈I得Rα0是幾乎強正則環(huán)環(huán)其余α ∈I- α0 ,Rα,均為強正則環(huán);(4)幾乎強正則環(huán)的直極限仍是幾乎強正則環(huán).第四章:本章主要通過引入幾乎弱正則環(huán)的概念,完善局部環(huán),VNL環(huán),幾乎強正則環(huán),幾乎弱正則環(huán).幾乎π-正則環(huán),幾乎強π-正則環(huán)環(huán)間關系,得到如下圖所示環(huán)間關系,研究他們的擴張問題.(?)第五章:綜述本文所研究的幾種類似于VNL環(huán)的相關環(huán).并對這幾類環(huán)的未來研究方向做了進一步的展望.
[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，
本文編號：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，
本文編號：2122147
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