Grant: Discovery Grants Program - Individual

Modern algebra is generally considered as the universal language of natural sciences. Ring theory has become an important part of modern algebra because of its increasing significance to other branches of mathematics, computer science and physics. The aim here is to continue the applicant's investigations of rings, which are sets in which one can ``add'' and ``multiply'' with the usual rules of distributivity connecting these two operations. The classical examples are the ring of integers with the usual operations, the ring of all polynomials with integer or rational coefficients, and the ring of n by n matrices whose entries are real numbers. Many rings occur naturally in the studies of mathematical system. The applicant will particularly investigate a class of rings whose elements can be expressed through ``addition'' by two kinds of key elements in a ring, namely ``invertible elements'' and ``idempotent elements''. The ring of linear transformations of a vector space is an example. The targeted class of rings is quite large, has tight connections with several extremely important concepts in ring theory and intimate links with topology and C*-algebras in functional analysis. This proposal will concentrate on the study of structures and constructions of the rings in the targeted class through various algebraic, topological and analysis methods and techniques. As applications, this research will provide new approaches for solving some existing problems in the literature on related rings and new algebraic methods for investigating a class of C*-algebras.