Date of Award
8-2022
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
School of Mathematical and Statistical Sciences
Committee Chair/Advisor
Dr. Jim Coykendall
Committee Member
Dr. Kevin James
Committee Member
Dr. Felice Manganiello
Committee Member
Dr. Keri Sather-Wagstaff
Abstract
Given an integral domain D with quotient field K, an element x in K is called integral over D if x is a root of a monic polynomial with coefficients in D. The notion of integrality has roots in Dedekind's work with algebraic integers, and was later developed more rigorously by Emmy Noether. Different variations or generalizations of integrality have since been studied, including almost integrality and pseudo-integrality. In this work we give a brief history of integrality and almost integrality before developing the basic theory of these two notions. We will continue the theory of almost integrality further by examining anchor ideals of almost integral elements and by presenting a domain which sheds light on iterations of complete integral closure. Some time is also spent on developing pseudo-integrality and other generalizations.
Recommended Citation
Fenstermacher, Todd, "On Complete Integral Closure of Integral Domains" (2022). All Dissertations. 3143.
https://tigerprints.clemson.edu/all_dissertations/3143