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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Prime ideals of a localization are exactly the primes disjoint from the denominator set

Statement

Let $R$ be a commutative ring, let $S \subseteq R$ be a multiplicative subset, and let $λ : R \to S^{- 1} R$ be the localization map. Then contraction along $λ$ induces an inclusion-preserving bijection $Spec (S^{- 1} R) \to {p \in Spec (R) : p \cap S = \emptyset}$ . Its inverse sends $p$ to $S^{- 1} p$ .

Facts & Assumptions

Given: A commutative ring $R$ , a multiplicative subset $S \subseteq R$ , and the localization map $λ : R \to S^{- 1} R$ .

[L1]

Every localization map induces a spectrum map by contraction (A ring map induces a contraction map on prime spectra).

[L2]

Primes of the localization correspond exactly to primes of $R$ disjoint from $S$ (Primes of a localization avoid the denominator set).

Proof

technique · direct

1.1

By [L1], contraction along $λ$ gives a map from $Spec (S^{- 1} R)$ to $Spec (R)$ .

1.2

The localization-prime correspondence [L2] says that this map lands exactly in the primes disjoint from $S$ , and that extension and contraction are inverse inclusion-preserving bijections on that subset.

2.1

Therefore $Spec (S^{- 1} R)$ is identified with the primes of $R$ that avoid the denominator set.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 and §17 (standard reference, not scraped)