Topics in Noncommutative Ring Theory

非交换环理论专题

• 批准号: RGPIN-2022-03783
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759195
• 关键词: Topics; Noncommutative; Ring; Theory

Ring theory, a subject of central importance in algebra, is becoming increasingly relevant to other branches of mathematics, computer science and physics. It is the study of sets, called rings, in which one can ``add" and ``multiply" as in arithmetic. Rings arise naturally in the studies of various mathematical objects. Some classical examples are the ring of integers, the ring of polynomials, and the ring of square matrices of the same size. The goal of my research program is to contribute to the development of ring theory, by investigating the rings whose elements can be expressed as sums x+y, where x and y, respectively, are uniformly chosen from one of the three significant subsets of the ring: (1) the subset of idempotents, (2) the subset of units, and (3) the subset of nilpotent elements. There is a rich literature on these rings with various unsolved questions. Of particular interest are clean rings, rings in which every element is the sum of an idempotent and a unit. Clean 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, idempotent lifting, the exchange property, and decomposition of modules. The investigation of clean rings is related to several outstanding questions in ring theory including a long-standing open question on the exchange property raised by Crawley and Jonsson in 1964 and the Köthe conjecture, a famous open problem posed in 1930. Other areas of focus include: nil-clean rings (rings in which every element is the sum of an idempotent and a nilpotent element), fine rings (rings in which every nonzero element is the sum of a unit and a nilpotent element), 2-good rings (rings in which every element is the sum of two units), rings with the 2-nil-sum property (rings in which every non central-unit is a sum of two nilpotent elements), Hirano-Tominaga rings (rings in which every element is the sum of two idempotents), and related topics. All the aforementioned rings are interrelated and we will pursue new ideas that help advance the additive theory embedded in the study of these rings. We will study the structure, classification, and construction of the rings in the targeted classes, their connections with other important concepts in ring theory, and their links with topology and analysis through various algebraic, topological and analytic methods and techniques. We will develop new approaches for solving fundamental problems in ring theory and related areas, and augment the  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.

环论是代数中的核心课题，与数学、计算机科学和物理学的其他分支正变得越来越相关。它是对集合的研究，称为环，其中一个人可以像算术一样‘加’和‘乘’。环是在研究各种数学对象时自然产生的。一些经典的例子有整数环、多项式环和相同大小的方阵环。我的研究项目的目的是通过研究环的元素可以表示为和x+y的环，其中x和y分别从环的三个重要子集之一中均匀地选择：(1)幂等元的子集，(2)单位的子集，和(3)幂零元的子集，从而为环理论的发展做出贡献。关于这些环的文献很丰富，有各种悬而未决的问题。特别令人感兴趣的是干净环，其中每个元素都是幂等元和单位的和。清洁环在拓扑学和泛函分析中自然地作为零维Tychonoff空间上的连续函数环和实秩零的交换C*-代数出现。在环论本身中，它们与von Neumann正则环、幂等提升、交换性和模的分解密切相关。清洁环的研究涉及到环论中的几个悬而未决的问题，包括1964年Crawley和Jonsson提出的关于交换性的一个长期悬而未决的问题和1930年提出的著名的公开问题Köthe猜想。其他关注的领域包括：nil-lean环(其中每个元素都是一个幂零元素和一个幂零元素的和的环)，精细环(其中每个非零元素都是一个单位和一个幂零元素的环)，2-好环(其中每个元素都是两个单位的和的环)，具有2-零和性质的环(其中每个非中心单位都是两个幂零元素的和的环)，Hirano-Tominaga环(其中每个元素都是两个幂零元素的和的环)，以及相关的主题。所有上述环都是相互关联的，我们将寻求新的想法，以帮助推进嵌入到这些环的研究中的加法理论。我们将通过各种代数、拓扑和分析方法和技巧，研究目标类中环的结构、分类和构造，它们与环理论中其他重要概念的联系，以及它们与拓扑学和分析的联系。开拓解决环论及相关领域基本问题的新途径，加深对环论及其应用的深入理解。这项研究对高级研究生和研究型数学家都有价值，将有助于提高代数基础领域的知识，并培养学生在数学科学方面具有独特的专业技能。