CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck 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

1.1L1given

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 $q \subseteq p$ .

2.1L2step 1.1

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

3.1step 2.1∎

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

Primes of a localization at a prime

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources