Topics in noncommutative ring theory

环理论在代数中是一个非常重要的学科，对数学、计算机科学和物理学的其他分支也越来越重要。它是对环的研究，环是一组可以像算术一样“加”和“乘”的集合。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”。在拓扑学和泛函分析中，这些环作为零维Tychonoff空间上的连续函数环和实秩为零的可交换C*-代数自然出现。在环理论本身，它们与von Neumann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。环理论中有许多悬而未决的问题都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一个著名的50年前的开放问题以及1930年发表的著名的Kothe猜想有关。***本课题将利用新的代数、拓扑和分析方法和技术，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，以及它们与拓扑和C*-代数的联系。本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，并对环理论及其应用的深入理解作出重要贡献。这项研究对高级研究生和研究数学家都很有价值，将有助于代数基础领域的知识进步，并培养具有独特和专业数学科学技能的学生，这对加拿大无疑是有益的。* * * * *

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. *** **