Topics in noncommutative ring theory

非交换环理论主题

• 批准号: RGPIN-2016-04706
• 负责人: Zhou, Yiqiang
• 金额: $ 1.31万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=669855
• 关键词: Topics; noncommutative; ring; theory

Ring theory is a subject of central importance in algebra, and is of increasing significance to other branches of mathematics, computer science and physics. It is a study of rings, which are sets in which one can "add" and "multiply" as in arithmetic. Rings arise naturally in studies of various mathematical objects. Some familiar examples are the ring of integers, the ring of polynomials, and the ring of square matrices of the same size.***The aim here is to continue the applicant's investigations of rings with a focus on a class of rings whose elements can be expressed as sums of two kinds of key elements in a ring, namely "invertible elements" and "idempotent elements". These rings naturally arise in topology and functional analysis as rings of continuous functions over zero-dimensional Tychonoff spaces and commutative C*-algebras of real rank zero. Within ring theory itself, they are tightly connected to von Neumann regular rings, Boolean rings, the exchange property of modules, the 2-sum property of rings, idempotent lifting, and direct decompositions of modules. There are many outstanding questions in ring theory which are relevant to the rings in the targeted class. For instance, the study of these rings and their variants is related to a famous 50 year old open question on the exchange property raised by Crawley and Jonsson in 1964 and the famous Kothe conjecture, which is still open, posted in 1930.***This proposal will concentrate on the study of structures and constructions of the rings in the targeted class, their connections to other important concepts in ring theory, and their links with topology and C*-algebras through utilizing new algebraic, topological and analysis methods and techniques. This research will provide new approaches for solving some fundamental problems in the literature on related rings, and contribute significantly to a deeper understanding of ring theory and its applications. This research, valuable to both advanced graduate students and research mathematicians, will contribute to the advancement of knowledge in fundamental areas of algebra and train students with unique and specialized skills in mathematical sciences, which is certainly beneficial to Canada. *** **

环理论是代数学中的一个重要课题，对数学、计算机科学和物理学的其他分支也有着越来越重要的意义。这是一个研究环，这是一个可以“加”和“乘”的集合，如算术。环在各种数学对象的研究中自然出现。一些熟悉的例子是整数环、多项式环和相同大小的方阵环。这里的目的是继续申请人对环的研究，重点是一类环，其元素可以表示为环中两种关键元素的和，即“可逆元素”和“幂等元素”。这些环自然出现在拓扑学和泛函分析中，作为零维吉洪诺夫空间和真实的秩为零的交换C*-代数上的连续函数的环。在环理论本身中，它们与冯·诺依曼正则环、布尔环、模的交换性质、环的2-和性质、幂等提升和模的直接分解紧密相连。在环理论中有许多突出的问题与目标类中的环有关。例如，对这些环及其变体的研究与克劳利和琼森在1964年提出的关于交换性质的著名的50年前的公开问题以及1930年发表的著名的Kothe猜想有关。该建议将集中于研究目标类中的环的结构和构造，它们与环理论中其他重要概念的联系，以及它们与拓扑和C*-代数的联系，通过利用新的代数，拓扑和分析方法和技术。这一研究将为解决相关环文献中的一些基本问题提供新的途径，并有助于加深对环理论及其应用的理解。这项研究对高级研究生和研究数学家都很有价值，将有助于代数基础领域知识的进步，并培养学生在数学科学方面的独特和专业技能，这对加拿大肯定是有益的。 *** **