Prime and Maximal Ideals

Introduction

Prime and maximal ideals are special types of ideals that play a crucial role in ring theory. They correspond to integral domains and fields in quotient rings, respectively, and provide important structural information about rings.

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: Integers

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: Polynomial Rings

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.

Example 3: Zero Ideal

The zero ideal $(0)$ is prime if and only if the ring $R$ is an integral domain.

Example 4: Principal Ideals

In a principal ideal domain, the prime ideals are $(0)$ and $(p)$ where $p$ is a prime element.

Properties

Prime Ideals

The intersection of prime ideals is a radical ideal
The preimage of a prime ideal under a ring homomorphism is prime
In a Noetherian ring, every ideal is contained in a maximal ideal

Maximal Ideals

Every proper ideal is contained in a maximal ideal (Zorn's Lemma)
The preimage of a maximal ideal under a surjective ring homomorphism is maximal
In a commutative ring with identity, maximal ideals are prime

Applications

Application 1: Quotient Ring Structure

Prime and maximal ideals determine the structure of quotient rings, which is fundamental in algebra.

Application 2: Algebraic Geometry

Prime ideals correspond to irreducible varieties in algebraic geometry.

Application 3: Number Theory

In algebraic number theory, prime ideals generalize the concept of prime numbers.

Application 4: Field Theory

Maximal ideals are crucial for constructing field extensions through quotient rings.