Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


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


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.

Included in

Algebra Commons



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