Topics in Noncommutative Ring Theory

环理论是代数中的一个重要学科，与数学、计算机科学和物理学的其他分支越来越相关。它是对集合的研究，称为环，其中可以“加”和“乘”，就像在算术中一样。环在各种数学对象的研究中自然出现。一些经典的例子是整数环、多项式环和相同大小的方阵环。我的研究计划的目标是通过研究元素可以表示为和x+y的环来促进环理论的发展，其中x和y分别均匀地从环的三个重要子集之一中选择：(1)幂等元子集，(2)单位子集，(3)幂零元子集。关于这些环有丰富的文献和各种未解决的问题。特别有趣的是干净环，环中的每个元素都是一个幂等元素和一个单位元素的和。在拓扑学和泛函分析中，干净环作为零维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，它们与冯·诺依曼正则环、幂等提升、交换性质和模的分解紧密相连。洁净环的研究与环理论中的几个突出问题有关，包括克劳利和琼森在1964年提出的关于交换性质的长期开放问题，以及1930年提出的著名开放问题Köthe猜想。其他重点领域包括：nil-clean环(环中每一个元素是一个幂等和幂零元素的总和),细环(环每个非零元素是一个单位的总和和幂零元素),2好环(环中每一个元素是两个单位的总和),环2-nil-sum产权(戒指,每个非中央单元是两个幂零元素的总和),Hirano-Tominaga环(环中每一个元素是两个幂等)的总和,以及相关的主题。上述所有环都是相互关联的，我们将寻求新的想法，帮助推进在这些环的研究中嵌入的可加性理论。我们将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，以及它们与拓扑和分析的联系，通过各种代数、拓扑和分析方法和技术。我们将开发新的方法来解决环理论和相关领域的基本问题，增加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的知识进步，并培养学生在数学科学方面独特的专业技能。

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.