TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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 classical affine variety has a domain coordinate ring, and conversely

Statement

Assume the Axiom of Choice, inherited from the Nullstellensatz route. For an affine algebraic set $X$ , $X$ is a classical affine variety if and only if $k [X]$ is a nonzero integral domain.

Facts & Assumptions

Given: AC, an algebraically closed field $k$ , and an affine algebraic set $X \subseteq k^{n}$ .

[F1]

Nonempty irreducible algebraic sets correspond to prime vanishing ideals (Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals).

[F2]

The coordinate ring is the quotient by the vanishing ideal (The coordinate ring of a classical affine algebraic set).

[F3]

A quotient is a domain exactly when the ideal is prime ( $R / P$ is an integral domain if and only if $P$ is a prime ideal).

Proof

technique · direct

1.1

If $X$ is a variety, F1 makes $I (X)$ prime. The polynomial ring is commutative, so F3 applied to $I (X)$ and the quotient in F2 says $k [X]$ is a nonzero domain.

F1F2F3given

2.1

If $k [X]$ is a nonzero domain, F3 makes $I (X)$ prime, and F1 makes $X$ nonempty irreducible. In particular the zero ring $k [\emptyset]$ is excluded on both sides.

F1F2F3given∎

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, Proposition 2.27 and §2i, pp. 45–48. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Used by

Dependency tree · two levels

18 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 v6.10, Proposition 2.27 and §2i, pp. 45–48 (standard reference, not scraped)