Special Ideals and Domains

Introduction

The properties of a ring are often reflected in the types of ideals it contains. We now explore a "zoo" of special ideals and the well-behaved rings they define.

Prime and Maximal Ideals

Definitions

Definition 10.1: Let $R$ be a commutative ring.

A proper ideal $P \subset R$ is a prime ideal if for any $a, b \in R$ , whenever $a b \in P$ , then either $a \in P$ or $b \in P$ .
A proper ideal $M \subset R$ is a maximal ideal if there is no other ideal $I$ such that $M \subset I \subset R$ .

Characterizations

These definitions have elegant characterizations in terms of quotient rings:

An ideal $P$ is prime if and only if the quotient ring $R / P$ is an integral domain (a commutative ring with no zero divisors).
An ideal $M$ is maximal if and only if the quotient ring $R / M$ is a field.

Since every field is an integral domain, it follows that every maximal ideal is also a prime ideal. The converse is not always true; for example, in the ring of integers $Z$ , the ideal $(0)$ is prime but not maximal.

Examples

Example 1: In $Z$ , the prime ideals are $(0)$ and $(p)$ for prime numbers $p$ . The maximal ideals are exactly $(p)$ for prime numbers $p$ .

Example 2: In $R [x]$ , the prime ideals are $(0)$ and $(f (x))$ where $f (x)$ is irreducible. The maximal ideals are exactly $(f (x))$ where $f (x)$ is irreducible.

The Chinese Remainder Theorem

The Chinese Remainder Theorem is a classical result in number theory that can be elegantly generalized to the language of ring theory. It provides a powerful tool for solving systems of congruences.

Statement

Theorem 10.2 (Chinese Remainder Theorem for Rings): Let $R$ be a commutative ring and let $I_{1}, I_{2}, \dots, I_{k}$ be ideals of $R$ . If the ideals are pairwise coprime (or comaximal), meaning $I_{i} + I_{j} = R$ for all $i \neq j$ , then the natural ring homomorphism

ϕ : R \to (R / I_{1}) \times (R / I_{2}) \times \dots \times (R / I_{k})

defined by $ϕ (x) = (x + I_{1}, x + I_{2}, \dots, x + I_{k})$ is surjective. Furthermore, its kernel is the intersection of the ideals, $\ker (ϕ) = ⋂_{j = 1}^{k} I_{j}$ . By the First Isomorphism Theorem for rings, this induces an isomorphism:

R / (⋂_{j = 1}^{k} I_{j}) ≅ (R / I_{1}) \times (R / I_{2}) \times \dots \times (R / I_{k})

Since for pairwise coprime ideals, their intersection is equal to their product, we have $⋂ I_{j} = \prod I_{j}$ .

Applications

The Chinese Remainder Theorem is fundamental in number theory and has applications in:

Solving systems of congruences
Understanding the structure of rings
Cryptography (RSA algorithm)
Error-correcting codes

Special Types of Domains

We now focus on integral domains with increasingly strong properties related to their ideal structure.

Principal Ideal Domains (PIDs)

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 $I = (a) = {r a ∣ r \in R}$ .

Euclidean Domains (EDs)

Definition 10.4: An integral domain $R$ is a Euclidean Domain (ED) if there exists a function $N : R ∖ {0} \to Z_{\geq 0}$ (called a Euclidean norm) such that for any $a, b \in R$ with $b \neq 0$ , there exist $q, r \in R$ where $a = q b + r$ and either $r = 0$ or $N (r) < N (b)$ .

The existence of a division algorithm in Euclidean domains makes their structure particularly transparent. For example:

The integers $Z$ (with norm $N (a) = | a |$ )
Polynomial rings over a field $F [x]$ (with norm $N (f) = \deg (f)$ )

are both Euclidean domains. It can be shown that every Euclidean domain is a PID, but the converse is not true.

Relationship Between Domains

The relationship between the classes of domains introduced so far 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.

Examples

Example 1: The Ring of Integers

The ring $Z$ is a Euclidean domain with the absolute value as the Euclidean norm. Therefore, it is also a PID and a UFD. The prime ideals in $Z$ are $(0)$ and $(p)$ for prime numbers $p$ .

Example 2: Polynomial Rings over Fields

For any field $F$ , the polynomial ring $F [x]$ is a Euclidean domain with the degree function as the Euclidean norm. Therefore, it is also a PID and a UFD.

Example 3: Gaussian Integers

The ring of Gaussian integers $Z [i] = {a + b i ∣ a, b \in Z}$ is a Euclidean domain with the norm $N (a + b i) = a^{2} + b^{2}$ . Therefore, it is also a PID and a UFD.

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.

Applications

Application 1: Number Theory

The Chinese Remainder Theorem is fundamental in number theory for solving systems of congruences. For example, to solve the system:

x \equiv 2 (\mod 3)

x \equiv 3 (\mod 5)

x \equiv 2 (\mod 7)

We can use the Chinese Remainder Theorem to find a solution modulo $3 \times 5 \times 7 = 105$ .

Application 2: Algebraic Number Theory

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

Application 3: Cryptography

The Chinese Remainder Theorem is used in the RSA cryptosystem and other cryptographic protocols.

Advanced Topics

Dedekind Domains

A Dedekind domain is an integral domain where every non-zero proper ideal factors uniquely into a product of prime ideals. Dedekind domains are important in algebraic number theory and include:

Rings of integers in number fields
Coordinate rings of smooth affine curves

Noetherian Rings

A ring $R$ is Noetherian if every ascending chain of ideals stabilizes. Many important rings are Noetherian:

Fields
Principal ideal domains
Polynomial rings over Noetherian rings
Rings of integers in number fields

Krull Domains

A Krull domain is an integral domain that is the intersection of a family of discrete valuation rings. Krull domains generalize the concept of unique factorization to more general settings.

Summary

Special ideals and domains provide a rich framework for understanding the structure of rings. Prime and maximal ideals correspond to integral domains and fields in quotient rings, respectively. The Chinese Remainder Theorem provides a powerful tool for understanding the structure of rings with multiple ideals.

The hierarchy of domains (Euclidean $⟹$ PID $⟹$ UFD) provides a systematic way to understand the properties of different types of rings. These concepts are fundamental to commutative algebra and have applications throughout mathematics, from number theory to algebraic geometry to cryptography.