Topics in noncommutative ring theory

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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.