ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

dual numbers not regular

Example

For every field $k$ , the dual-number ring $R = k [ε] / (ε^{2})$ is local with $\dim R = 0$ and $edim R = 1$ , so is not regular.

Facts & Assumptions

Given: The objects and hypotheses in the example. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

embedding dimension and regular local ring: For a nonzero commutative Noetherian local ring $(R, m, k)$ , define $edim R = \dim_{k} (m / m^{2})$ . The ring is regular local when $edim R = \dim R$ . The cotangent space is intrinsic, and is finite-dimensional because $m$ is finitely generated.

Verification

1.1

Every element has a unique form $a + b ε$ . It is a unit precisely when $a \neq 0$ , with inverse $a^{- 1} - b a^{- 2} ε$ . Hence the unique maximal ideal is $(ε)$ . Every prime contains the nilpotent $ε$ , so this is the only prime and the dimension is zero.

givenalgebra

2.1

The square of the maximal ideal is zero and the class of $ε$ is a nonzero $k$ -basis of it. Therefore the cotangent dimension is one, strictly larger than Krull dimension. The ring is finite-dimensional over $k$ and hence Noetherian, so the regularity definition applies and fails.

F1step 1.1algebra∎

Depends on

embedding dimension and regular local ring

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Definitions 12.3–12.5 and Example 12.6, p.115 (standard reference, not scraped)