Principal Ideal Domains

Introduction

Principal Ideal Domains (PIDs) are integral domains where every ideal is principal, meaning it can be generated by a single element. This property makes PIDs particularly well-behaved and important in algebra.

Definition

Definition 10.3: An integral domain $R$ is a Principal Ideal Domain (PID) if every ideal in $R$ is a principal ideal, meaning it can be generated by a single element. That is, for every ideal $I$ , there exists an $a \in R$ such that $I = (a) = {r a ∣ r \in R}$ .

Properties

Ideal Structure

In a PID, every ideal is principal, which means the ideal structure is particularly simple and well-understood.

Divisibility

In a PID, the divisibility relation $a ∣ b$ corresponds to the containment relation $(b) \subseteq (a)$ .

Greatest Common Divisors

In a PID, any two elements have a greatest common divisor, and it can be written as a linear combination of the two elements.

Unique Factorization

Every PID is a Unique Factorization Domain (UFD), meaning every non-zero, non-unit element can be written uniquely as a product of irreducible elements.

Examples

Example 1: The Ring of Integers

The ring $Z$ is a PID. Every ideal in $Z$ is of the form $(n)$ for some integer $n$ .

Example 2: Polynomial Rings over Fields

For any field $F$ , the polynomial ring $F [x]$ is a PID. Every ideal in $F [x]$ is of the form $(f (x))$ for some polynomial $f (x)$ .

Example 3: Gaussian Integers

The ring of Gaussian integers $Z [i] = {a + b i ∣ a, b \in Z}$ is a PID.

Example 4: A PID that is not Euclidean

The ring $Z [\frac{1 + \sqrt{- 19}}{2}]$ is a PID but not a Euclidean domain. This shows that the converse of the implication "Euclidean Domain $⟹$ PID" is false.

Relationship to Other Domains

The relationship between the classes of domains is a beautiful hierarchy:

Euclidean Domains ⟹ Principal Ideal Domains ⟹ Unique Factorization Domains

Each implication is strict, meaning there are examples of PIDs that are not EDs, and UFDs that are not PIDs. For example, $Z [x]$ is a UFD but not a PID, as the ideal $(2, x)$ is not principal.

Applications

Application 1: Number Theory

PIDs are important in algebraic number theory for understanding the structure of rings of integers in number fields.

Application 2: Linear Algebra

The structure theorem for finitely generated modules over PIDs is fundamental in linear algebra.

Application 3: Polynomial Theory

PIDs provide a natural setting for studying polynomial factorization and divisibility.

Application 4: Algebraic Geometry

PIDs are important in algebraic geometry for understanding coordinate rings of curves.

Advanced Properties

Noetherian Property

Every PID is Noetherian, meaning every ascending chain of ideals stabilizes.

Dedekind Domains

While PIDs are important, many rings of integers in number fields are not PIDs but are Dedekind domains, which have a more general unique factorization property for ideals.