Alphabeta MathHow 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.
Integral affine schemes
Definition
An affine scheme is integral when is a nonzero integral domain. Equivalently, is nonempty, reduced, and irreducible: reduced says the nilradical is zero, and irreducibility says that nilradical is prime, hence is prime; nonemptiness excludes .
Depends on
- Irreducible topological spaces and irreducible subsets in the subspace topology
- Reduced affine schemes
- Zero divisor, and integral domain: a commutative ring with $1 \ne 0$ and no zero divisors
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
Used by
Nothing in the library uses this result yet.
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
- James S. Milne, Algebraic Geometry, 10.28 (standard reference, not scraped)