ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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 separating prime for an element outside a radical

Example

Assume the Axiom of Choice.

In the polynomial ring $k [x, y]$ , take the ideal $I = (x^{2})$ and the element $f = y$ . Then no positive power of $y$ lies in $I$ , so the corollary produces a prime ideal containing $I$ but avoiding $y$ ; one such prime is $(x)$ .

Facts & Assumptions

Given: A field $k$ , the ideal $I = (x^{2}) \subseteq k [x, y]$ , the element $f = y$ , and the Axiom of Choice.

[L1]

Assuming the Axiom of Choice, if no positive power of an element lies in an ideal, then some prime ideal contains the ideal while avoiding the element (Separating an element from an ideal by a prime).

Verification

technique · direct

1.1

Every element of $I = (x^{2})$ is divisible by $x^{2}$ , so no power $y^{n}$ lies in $I$ .

givenalgebra

2.1

Applying [L1] to step 1.1 yields a prime ideal containing $I$ but avoiding $y$ . Concretely, $(x)$ works: it contains $x^{2}$ and does not contain $y$ .

L1step 1.1algebra

3.1

Thus this example exhibits the separating-prime conclusion directly.

step 2.1∎

Depends on

Separating an element from an ideal by a prime

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

M. Hochster, Introduction to Commutative Algebra, Math 614 notes (2020) (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 and §17 (standard reference, not scraped)