DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

The prime spectrum and vanishing sets

Definition

Let $R$ be a commutative ring.

The prime spectrum of $R$ is the set

Spec (R) = {p ⊴ R : p is prime} .

If $T \subseteq R$ , define its vanishing set by

V (T) = {p \in Spec (R) : T \subseteq p} .

For an ideal $I ⊴ R$ this is simply

V (I) = {p \in Spec (R) : I \subseteq p} .

Since a prime ideal contains $T$ exactly when it contains the ideal $(T)$ generated by $T$ , one has $V (T) = V ((T))$ .

Depends on

Used by

Principal distinguished subsets of the prime spectrum Definition
The zero ring has empty prime spectrum Example
Primes of a localization avoid the denominator set Lemma
Primes of a quotient lie over the kernel Lemma
The spectrum map pulls back vanishing sets Lemma
The spectrum map respects composition and identities Lemma
Vanishing sets of arbitrary sums Lemma
Vanishing sets of finite products Lemma
Vanishing sets reverse inclusions Lemma
Vanishing-set identities 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)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 The Spectrum of a Ring (standard reference, not scraped)
The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)