Mathematical Sciences: Topics in Commutative Ring Theory

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This research concerns work on the following topics in the area of commutative associative rings and algebras. The first is birational normal spots and exceptional prime divisors over a 2- dimensional normal local domain. A specific question here is whether a 2-dimensional normal local domain that satisfies Muhly's condition (N) necessarily has torsion divisor class group. The second topic concerns finite generation of ideals contracted from a polynomial ring extension. An interesting question in this area is whether a 1-dimensional coherent domain has the property that its integral closure is a Prufer domain. The third topic concerns ring and field extensions that are not finitely generated but have the property that each proper subextension is finitely generated. Other topics in commutative ring theory concerned with the behavior of the Picard group under ring extension, the structure of rings of interest in control theory, and the multiplicity of adjacent ideals in a 2- dimensional regular local domain will be considered. Commutative rings are algebraic structures possessing a commutative addition and a commutative multiplication. These structures occur throughout mathematics and algebra and common examples include polynomial rings, and rings of algebraic integers in extension fields of the rationals. This project is concerned with a variety of questions on the structure of commutative rings.