Topics in noncommutative ring theory

环理论是代数中非常重要的一门学科，并且对于数学、计算机科学和物理学的其他分支也具有越来越重要的意义。它是对环的研究，环是可以像算术中一样“加”和“乘”的集合。在对各种数学对象的研究中，环自然地出现。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，重点关注一类环，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”。这些环自然地出现在拓扑和泛函分析中，作为零维吉洪诺夫空间上的连续函数环和实秩零的交换 C* 代数。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-sum 性质、幂等提升和模的直接分解紧密相连。环理论中有许多与目标课程中的环相关的悬而未决的问题。例如，对这些环及其变体的研究与 Crawley 和 Jonsson 于 1964 年提出的一个著名的 50 年前的交换性质开放问题以及 1930 年发布的著名的 Kothe 猜想有关，该猜想仍然开放。***该提案将集中于研究目标类中环的结构和构造，它们与环理论中其他重要概念的联系，以及通过利用新的代数，它们与拓扑和 C* 代数的联系，拓扑和分析方法和技术。这项研究将为解决相关环文献中的一些基本问题提供新的方法，并为更深入地理解环理论及其应用做出重大贡献。这项研究对高级研究生和研究数学家都很有价值，将有助于提高代数基础领域的知识，并培养学生在数学科学方面拥有独特和专业的技能，这对加拿大来说无疑是有利的。 *** **

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. *** **