CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-09-12

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 ideals (x) and (x^2) have the same zero locus but different quotient rings

Statement refuted

The underlying zero locus of an ideal determines the same quotient ring as the ideal itself.

Example

In $k [x]$ , the ideals $(x)$ and $(x^{2})$ cut out the same point set because A zero locus depends only on the generated ideal and its radical gives $V (x) = V (x^{2}) .$

The quotient rings are nevertheless different: $k [x] / (x) ≅ k$ is reduced, while in $k [x] / (x^{2})$ the class $\overline{x}$ is nonzero and satisfies ${\overline{x}}^{2} = 0$ . So the second quotient remembers a nilpotent thickening that the common zero locus does not see. This refutes the statement.

Depends on

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

J. S. Milne, Algebraic Geometry, Aside 2.24 (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, radical-ideal discussion (standard reference, not scraped)