Maximal spectrum

The set of maximal ideals of a commutative ring, denoted MaxSpec(R).

Maximal spectrum

Definition (maximal spectrum)

Let $R$ be a commutative ring (with $1$ ). The maximal spectrum of $R$ is

\operatorname{MaxSpec}(R)\;:=\;\{\ \mathfrak m \subset R \mid \mathfrak m \text{ is a } maximal ideal \ \}.

Since every maximal ideal is prime, there is a natural inclusion

\operatorname{MaxSpec}(R)\ \subseteq\ \operatorname{Spec}(R)=Spec(R) .

The Zariski topology on $\operatorname{MaxSpec}(R)$ is the subspace topology induced from the Zariski topology on Spec(R) .

Equivalently, closed sets in $\operatorname{MaxSpec}(R)$ are of the form

V(I)\cap \operatorname{MaxSpec}(R)

for ideals $I\subset R$ .

A field.
If $k$ is a field, then $\operatorname{MaxSpec}(k)=\{(0)\}$ .
The integers.
$\operatorname{MaxSpec}(\mathbb Z)=\{(p)\mid p \text{ prime}\},$
since the maximal ideals of $\mathbb Z$ are exactly the principal ideals generated by primes.
Affine line over an algebraically closed field.
If $k$ is algebraically closed, then maximal ideals of $k[x]$ are exactly $(x-a)$ for $a\in k$ . Hence
$\operatorname{MaxSpec}(k[x]) \cong k$
as sets (and the Zariski closed sets correspond to finite subsets and the whole set).