Topics in Noncommutative Ring Theory

环论是代数学中的一个重要课题，与数学、计算机科学和物理学的其他分支越来越相关。这是一门研究集合的学科，称为环，在环中人们可以像算术中那样“加”和“乘”。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促进环理论的发展，通过研究其元素可以表示为和x + y的环，其中x和y分别从环的三个重要子集之一均匀选择：（1）幂等元的子集，（2）单位的子集，和（3）幂零元素的子集。有丰富的文献对这些环与各种未解决的问题。特别感兴趣的是干净的环，环中的每个元素是一个幂等元和一个单位的总和。干净环自然出现在拓扑学和泛函分析中，作为零维吉洪诺夫空间和真实的秩为零的交换C *-代数上的连续函数的环。在环理论本身中，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。干净环的研究与环论中的几个悬而未决的问题有关，包括Crawley和Jonsson在1964年提出的关于交换性质的长期悬而未决的问题以及1930年提出的著名悬而未决的问题Köthe猜想。其他重点领域包括：nil-clean环（环中的每个元素是幂等元和幂零元之和），细环（其中每个非零元是单位元与幂零元之和的环），2-好环（环中的每个元素是两个单位的和），具有2-零和性质的环（环中的每一个非中心单位是两个幂零元素的和），平野富永环（环中的每一个元素是两个幂等元的和），以及相关的主题。所有上述环是相互关联的，我们将追求新的想法，帮助推进嵌入在这些环的研究中的添加剂理论。 我们将研究目标类中的环的结构，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，并增加环理论及其应用的理解。这项研究，有价值的高级研究生和研究数学家，将有助于知识的进步，代数的基本领域，培养学生在数学科学的独特和专业技能。

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.