Topics in Noncommutative Ring Theory

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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.