ExampleConstruction: AI-generatedVerification: Literature-sourcedprecheck passaudited 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 zero ring has empty prime spectrum

Example

Let $R$ be the zero ring. Then $Spec (R) = \emptyset$ , and consequently $V ((0)) = V (R) = D (0) = D (1) = \emptyset$ .

Facts & Assumptions

Given: The zero ring $R$ .

[L1]

$Spec (R)$ is the set of prime ideals of $R$ (The prime spectrum and vanishing sets).

[L2]

The vanishing-set and distinguished-subset identities identify $V ((0))$ , $V (R)$ , $D (0)$ , and $D (1)$ from the prime spectrum (Vanishing-set identities, Distinguished-subset identities).

Verification

technique · direct

1.1L1givenalgebra

In the zero ring one has $0 = 1$ , so the only ideal is the whole ring. A prime ideal must be proper, so no prime ideals exist. Therefore $Spec (R) = \emptyset$ .

2.1L2step 1.1

Since every subset named in [L2] is defined as a subset of $Spec (R)$ , step 1.1 forces all of them to be empty.

3.1step 1.1step 2.1∎

Hence the zero ring has empty prime spectrum, and all the displayed $V$ and $D$ boundary sets are empty as well.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)