LemmaStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck 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.

Vanishing-set identities

Statement

Let $R$ be a commutative ring.

$V ((0)) = Spec (R)$ .
$V (R) = \emptyset$ .
For every family $(I_{λ})_{λ \in Λ}$ of ideals, $V ⁣ (\sum_{λ \in Λ} I_{λ}) = ⋂_{λ \in Λ} V (I_{λ}) .$
For every finite family $I_{1}, \dots, I_{n}$ of ideals with $n \geq 1$ , $V (I_{1} \dots I_{n}) = V (I_{1}) \cup \dots \cup V (I_{n}) .$

Facts & Assumptions

Given: A commutative ring $R$ .

[L1]

$V (K)$ is the set of prime ideals containing $K$ (The prime spectrum and vanishing sets).

[L2]

Vanishing sets turn arbitrary sums into intersections (Vanishing sets of arbitrary sums).

[L3]

Vanishing sets turn finite products into unions (Vanishing sets of finite products).

Proof

technique · direct

1.1

Every prime ideal contains $0$ , so $V ((0)) = Spec (R)$ . No prime ideal equals the whole ring, so no prime ideal contains $R$ ; hence $V (R) = \emptyset$ .

L1given

1.2

The arbitrary-sum identity is exactly [L2], and the finite-product identity is exactly [L3].

L2L3

2.1

Steps 1.1 and 1.2 supply the four listed vanishing-set identities.

step 1.1step 1.2∎

Depends on

Used by

The zero ring has empty prime spectrum Example
Vanishing sets detect radicals 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)