ExampleConstruction: AI-generatedVerification: Literature-sourcedprecheck passaudited 2026-09-01

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 polynomial-dimension formula at fields, Artinian rings, and the zero-ring boundary

Example

The theorem $\dim R [x] = \dim R + 1$ behaves as expected for fields and nonreduced Artinian rings, and it deliberately excludes the zero ring.

Facts & Assumptions

Given: A field $k$ , the Artinian ring $A = k [t] / (t^{2})$ , and the zero ring $0$ .

[L1]

For a finite-dimensional Noetherian ring, adjoining one polynomial variable raises dimension by one (A Noetherian polynomial ring has dimension one larger).

[L2]

Krull dimension is defined by prime chains, with the zero ring convention recorded separately (Krull dimension of a nonzero ring).

Verification

technique · cases

1.1

For the field $k$ , one has $\dim k = 0$ , so [L1] gives $\dim k [x] = 1$ .

L1givencases

1.2

The ring $A = k [t] / (t^{2})$ is Artinian local with the single prime $(\overset{ˉ}{t})$ , so $\dim A = 0$ . Applying [L1] again gives $\dim A [x] = 1$ . The nilpotent element $\overset{ˉ}{t}$ does not change the one-step dimension jump.

L1L2givencases

2.1

The zero ring is excluded from the statement because it has no prime ideals at all. Its polynomial ring is again the zero ring, so the expression $\dim 0 + 1$ does not describe its behavior. Thus the theorem is intentionally stated only for ordinary Noetherian rings with the established zero-ring convention left separate.

L2step 1.1step 1.2givencases-exhaustive∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., §21 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, §§18, 21 (standard reference, not scraped)