Prime and Maximal Ideals

Prime and maximal ideals are the central structural features of commutative rings, providing the link between ring theory and geometry via the spectrum construction.

Definitions

Definition4.2Prime ideal

An ideal $\mathfrak{p}$ of a commutative ring $R$ is prime if $\mathfrak{p} \neq R$ and whenever $ab \in \mathfrak{p}$ , then $a \in \mathfrak{p}$ or $b \in \mathfrak{p}$ . Equivalently, $R/\mathfrak{p}$ is an integral domain.

Definition4.3Maximal ideal

An ideal $\mathfrak{m}$ of $R$ is maximal if $\mathfrak{m} \neq R$ and there is no ideal $I$ with $\mathfrak{m} \subsetneq I \subsetneq R$ . Equivalently, $R/\mathfrak{m}$ is a field.

Theorem4.2Maximal implies prime

Every maximal ideal is prime. The converse is false in general but holds in PIDs.

Examples

ExamplePrime and maximal ideals

In $\mathbb{Z}$ : $(p)$ is both prime and maximal for prime $p$ . $(0)$ is prime but not maximal.
In $\mathbb{Z}[x]$ : $(x)$ is prime (quotient $\cong \mathbb{Z}$ , an integral domain) but not maximal. $(x, 2) = \{f : f(0) \text{ even}\}$ is maximal (quotient $\cong \mathbb{F}_2$ ).
In $k[x,y]$ ( $k$ field): $(x-a, y-b)$ is maximal (quotient $\cong k$ ). $(x)$ is prime but not maximal.
In $\mathbb{Z}/6\mathbb{Z}$ : the prime ideals are $(\bar{2})$ and $(\bar{3})$ , both maximal.

The Spectrum

Definition4.4Prime spectrum

The spectrum of a commutative ring $R$ is the set $\mathrm{Spec}(R) = \{\mathfrak{p} \subseteq R : \mathfrak{p} \text{ is prime}\}$ , equipped with the Zariski topology: closed sets are $V(I) = \{\mathfrak{p} \in \mathrm{Spec}(R) : I \subseteq \mathfrak{p}\}$ for ideals $I$ .

ExampleSpectra of rings

$\mathrm{Spec}(\mathbb{Z}) = \{(0), (2), (3), (5), (7), \ldots\}$ . The point $(0)$ is the generic point (its closure is all of $\mathrm{Spec}(\mathbb{Z})$ ).
$\mathrm{Spec}(k[x]) = \{(0)\} \cup \{(x-a) : a \in k\}$ when $k$ is algebraically closed. This is the "affine line."
$\mathrm{Spec}(k[x,y]) = \{(0)\} \cup \{(f) : f \text{ irreducible}\} \cup \{(x-a,y-b) : a,b \in k\}$ : points, curves, and the generic point.

RemarkConnection to algebraic geometry

$\mathrm{Spec}(R)$ is the foundation of modern algebraic geometry (Grothendieck). The prime ideals correspond to "points" of a geometric space, maximal ideals to "closed points," and the Zariski topology encodes the incidence structure. Ring homomorphisms $R \to S$ induce continuous maps $\mathrm{Spec}(S) \to \mathrm{Spec}(R)$ , providing a contravariant functor from rings to topological spaces.