Maximal spectrum

Let $R$ be a commutative ring . A maximal ideal of $R$ is a proper ideal $\mathfrak m\subsetneq R$ such that there is no ideal strictly between $\mathfrak m$ and $R$; equivalently, $R/\mathfrak m$ is a field .

The maximal spectrum of $R$ is the set

There is always an inclusion $\operatorname{MaxSpec}(R)\subseteq \operatorname{Spec}(R)$, since every maximal ideal is prime. One typically topologizes $\operatorname{MaxSpec}(R)$ by the subspace topology induced from the Zariski topology on \operatorname{Spec}(R) . Concretely, for an ideal $I\subseteq R$ the corresponding closed subset of $\operatorname{MaxSpec}(R)$ is

A point $\mathfrak m\in \operatorname{MaxSpec}(R)$ has residue field $R/\mathfrak m$, which agrees with the residue field at $\mathfrak m$.

Examples

1. Local rings.
   If $R$ is a local ring , it has a unique maximal ideal (see the characterization of local rings by a unique maximal ideal ), hence $\operatorname{MaxSpec}(R)$ is a single point.

2. The integers.
   For $R=\mathbb{Z}$, the maximal ideals are exactly $(p)$ for primes $p$. Thus

   $$\operatorname{MaxSpec}(\mathbb{Z})=\{(p)\mid p\ \text{prime}\},$$

   which is $\operatorname{Spec}(\mathbb{Z})$ with the generic point $(0)$ removed.

3. Polynomial rings over an algebraically closed field.
   If $k$ is algebraically closed and $R=k[x_1,\dots,x_n]$, then the Nullstellensatz identifies maximal ideals with points $a=(a_1,\dots,a_n)\in k^n$ via

   $$a \longleftrightarrow (x_1-a_1,\dots,x_n-a_n).$$

   In this sense, $\operatorname{MaxSpec}(k[x_1,\dots,x_n])$ recovers affine $n$-space over $k$ as a set, and its induced topology is the classical Zariski topology on $k^n$.