Topics in noncommutative ring theory

环论是代数中的核心课题，对数学、计算机科学和物理学的其他分支具有越来越重要的意义。它是对环的研究，在环中，人们可以像在算术中一样，在其中进行“加法”和“乘法”。环在研究各种数学对象时自然而然地出现。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，重点是环的一类环，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、布尔环、模的交换性、环的2-和性质、幂等提升以及模的直接分解密切相关。在环论中有许多与目标类中的环相关的突出问题。例如，对这些环及其变种的研究涉及到Crawley和Jonsson在1964年提出的关于交换性的一个著名的50年前的公开问题，以及著名的Kothe猜想，该猜想在1930年发表，至今仍未解决。*这个建议将集中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，以及它们与拓扑学和C*-代数的联系。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，并培养具有独特和专业数学科学技能的学生，这肯定对加拿大有利。*

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. *** **