Prime correspondence under localization

Let $R$ be a commutative ring and let $S\subseteq R$ be a multiplicative set . Write $S^{-1}R$ for the localization .

Theorem (prime correspondence)

The map

induces an inclusion-preserving bijection between:

• prime ideals $\mathfrak p\subseteq R$ with $\mathfrak p\cap S=\varnothing$, and
• prime ideals $\mathfrak q\subseteq S^{-1}R$.

The inverse bijection is contraction:

In other words, “primes survive localization exactly when they do not meet the set of denominators.” This refines the fact that localization preserves primality .

Geometric form

On the prime spectrum , this correspondence identifies $\operatorname{Spec}(S^{-1}R)$ with the open subset

for the Zariski topology . For $S=\{1,f,f^2,\dots\}$, this is the basic open set $D(f)$.

Special case: localization at a prime

If $S=R\setminus \mathfrak p$ for a prime ideal $\mathfrak p$, then $S^{-1}R$ is the ring $R_\mathfrak p$ . The primes of $R_\mathfrak p$ correspond exactly to primes $\mathfrak q\subseteq R$ with $\mathfrak q\subseteq \mathfrak p$.

Examples

1. Inverting a prime in $\mathbb{Z}$.
   Take $R=\mathbb{Z}$ and $S=\{1,p,p^2,\dots\}$. Then $S^{-1}R=\mathbb{Z}[1/p]$. A prime ideal of $\mathbb{Z}$ meets $S$ iff it contains $p$, so the primes of $\mathbb{Z}[1/p]$ correspond to $(0)$ and $(\ell)$ for primes $\ell\neq p$.

2. Localizing away from a hypersurface.
   Let $R=k[x,y]$ and $S=\{1,x,x^2,\dots\}$, so $S^{-1}R=R_x$. Primes of $R_x$ correspond to primes of $k[x,y]$ that do not contain $x$. For instance, $(y)$ survives (since $x\notin(y)$), while $(x,y)$ does not survive (since it contains $x$).

3. Localization at a maximal ideal.
   In $R=k[x,y]$, localize at $\mathfrak m=(x,y)$ to get $R_\mathfrak m$ . The primes of $R_\mathfrak m$ correspond to primes contained in $(x,y)$, namely $(0)$, $(x)$, $(y)$, and $(x,y)$.