Integral Domains, An Introduction

Introduction

An integral domain is a ring with two important constraints: multiplication is commutative, and the product of two nonzero elements isn't going to give you zero. These constraints, modeled after the integers, force a great deal of structure on the ring, but they aren't really a straight jacket either.

There are many classes of integral domains, some finite and some infinite, some based on Z and some based on Z_p, some exhibiting unique factorization and some with several factorizations for each "composite number". One class of integral domains contains ideals that behave like prime numbers, complete with unique factorization. (This assumes you are comfortable with the notion of multiplying ideals together.)

I assume you are familiar with ideals, prime ideals, and prime and irreducible elements in a commutative ring. The theorems that relate to noncommutative rings are, obviously, not required here.

Some of the subtopics listed at the top of this page, e.g. valuation rings and dedekind domains, are indeed specializations of integral domains, as you would expect. However, other subtopics, such as integral extensions and localizations, require only commutative rings. Still, these concepts are easier to understand if you have reviewed integral domains; and when they are restricted to integral domains, some nice properties emerge. So I've placed them all here. I apologize for any confusion.

for now, let's begin with the rings that look like the integers. These are the euclidean domains, because Euclid's proofs, spelled out some 2,000 years ago, apply to euclidean domains with just the slightest change in wording. It's a beautiful generalization.