ExampleConstruction: AI-generatedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

In a concrete Artinian local quotient, maximal radical forces primaryity

Example

Let $A = k [x, y] / (x^{2}, x y, y^{2}), m = (\overset{ˉ}{x}, \overset{ˉ}{y}) .$ Then every proper ideal $J ⊊ A$ with $\sqrt{J} = m$ is $m$ -primary.

Facts & Assumptions

Given: A field $k$ , the Artinian local ring $A = k [x, y] / (x^{2}, x y, y^{2})$ with maximal ideal $m = (\overset{ˉ}{x}, \overset{ˉ}{y})$ , and a proper ideal $J ⊊ A$ satisfying $\sqrt{J} = m$ .

[L1]

A proper submodule is primary exactly when every zero divisor on the quotient acts nilpotently (Primary submodules and primary ideals).

Verification

technique · direct

1.1

In $A$ , every quadratic monomial vanishes, so $m^{2} = (\overset{ˉ}{x}, \overset{ˉ}{y})^{2} = 0.$ Consequently $(m / J)^{2} = 0$ in the quotient ring $A / J$ .

givenalgebra

2.1

The quotient $A / J$ is local with maximal ideal $m / J$ . Any zero divisor in $A / J$ is a nonunit, hence lies in the maximal ideal $m / J$ . By step 1.1 every element of $m / J$ is square-zero, so every zero divisor on $A / J$ acts nilpotently.

step 1.1algebra

3.1

Fact [L1] now shows that $J$ is primary, and its radical is $m$ by assumption. Hence $J$ is $m$ -primary.

L1step 2.1∎

Depends on

Primary submodules and primary ideals

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

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §18 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, §19 (standard reference, not scraped)