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.

The quotient by a nonidempotent ideal is not flat

Example

In $R = k [ε] / (ε^{2})$ , the quotient by the ideal $I = (ε)$ is not flat.

Facts & Assumptions

Given: A field $k$ , the ring $R = k [ε] / (ε^{2})$ , and the ideal $I = (ε)$ .

[L1]

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

[L2]

Flatness is detected by the ideal-injection criterion (Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests).

Verification

technique · direct

1.1

Here $I^{2} = (ε^{2}) = 0$ , while $I \neq 0$ . So $I \neq I^{2}$ .

givenalgebra

1.2

By [L1], the quotient $R / I$ cannot be flat. Equivalently, the ideal criterion [L2] fails for the inclusion $I ↪ R$ after tensoring with $R / I$ .

L1L2

2.1

Thus quotients by nonidempotent ideals need not be flat.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Mihnea Mustata, Graduate Commutative Algebra, §10 (standard reference, not scraped)