ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 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.

A polynomial algebra is free and therefore faithfully flat over its coefficient ring

Example

Assume the Axiom of Choice for the faithfully-flat characterization used below.

For any commutative ring $R$ , the polynomial algebra $R [x]$ is a free $R$ -module with basis $1, x, x^{2}, \dots$ . Hence $R [x]$ is flat over $R$ , and the map $R \to R [x]$ is faithfully flat because the extension of any proper ideal $I ⊊ R$ is the proper ideal $I R [x]$ .

Facts & Assumptions

Given: The Axiom of Choice and a commutative ring $R$ .

[L1]

Free modules are flat (Under the stated choice boundary, free modules are projective and hence flat).

[L2]

A flat ring map is faithfully flat exactly when proper ideals remain proper (A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra).

Verification

technique · direct

1.1

As an $R$ -module, $R [x] = ⨁_{n \geq 0} R x^{n},$ so it is free on the monomial basis. By [L1], it is flat over $R$ .

L1given

1.2

If $I ⊊ R$ , then every polynomial in $I R [x]$ has all coefficients in $I$ , so $1 \notin I R [x]$ . Thus $I R [x]$ is proper. By [L2], the map $R \to R [x]$ is faithfully flat.

L2algebra

2.1

Therefore polynomial algebras give basic faithfully flat examples.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Exercise (9.8) (standard reference, not scraped)