CorollaryStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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Primes of a localization at a prime

Statement

Let $R$ be a commutative ring and let $p \in Spec (R)$ . Contraction along $R \to R_{p}$ induces an inclusion-preserving bijection from $Spec (R_{p})$ to the set of prime ideals $q \subseteq p$ of $R$ .

Facts & Assumptions

Given: A commutative ring $R$ and a prime ideal $p \subset R$ .

[L1]

$R_{p}$ is the localization at the multiplicative set $R ∖ p$ (Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ ).

[L2]

Primes of a localization correspond to primes of the original ring disjoint from the denominator set (Primes of a localization avoid the denominator set).

Proof

technique · direct

1.1

By [L1], the denominator set is $S = R ∖ p$ . For a prime ideal $q$ of $R$ , the condition $q \cap S = \emptyset$ is equivalent to $q \subseteq p$ .

L1given

2.1

Applying [L2] to the localization at $S$ yields the claimed bijection between $Spec (R_{p})$ and the primes $q \subseteq p$ .

L2step 1.1

3.1

Hence prime ideals of the local ring $R_{p}$ are exactly the primes of $R$ lying below $p$ .

step 2.1∎

Depends on

Used by

Primes inside a localization at a prime Example
Height equals local dimension Lemma

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring (standard reference, not scraped)
The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)