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 quotient by an idempotent ideal is flat

Example

Let $R = A \times B$ and let $I = A \times 0 = R (1, 0)$ . Then $(1, 0)$ is idempotent, so the quotient

R / I ≅ 0 \times B ≅ B

is a flat $R$ -module.

Facts & Assumptions

Given: A product ring $R = A \times B$ and the ideal $I = A \times 0$ .

[L1]

Quotients by idempotent-generated ideals are flat (If $R / I$ is flat then $I = I^{2}$ , and for finitely generated $I$ this is equivalent to generation by an idempotent).

Verification

technique · direct

1.1

The element $e = (1, 0) \in A \times B$ satisfies $e^{2} = e$ , and $I = R e$ .

givenalgebra

1.2

Therefore [L1] applies and shows that $R / I$ is flat. Concretely, $R / I ≅ B$ as the second factor.

2.1

This is the standard idempotent-quotient example.

algebra∎

Depends on

If $R/I$ is flat then $I = I^2$, and for finitely generated $I$ this is equivalent to generation by an idempotent

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, Exercise (9.9) (standard reference, not scraped)