Unique Factorization Domains and Generalizations

Abstract

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime or the product of primes. This product is unique up to rearranging the order of the primes. Unique factorization is a property of the integers that is often taken for granted. What kind of algebraic systems have this property? What other properties of the integers do we also take for granted? And, most importantly, do there exist algebraic systems with different properties than the integers? These questions are the heart of this project. This paper presents the conditions for unique factorization and loosens these conditions to create a hierarchy of integral domains.