TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1L1

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

1.2L2

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.

2.1step 1.1step 1.2∎

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

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources