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.
regular noetherian ring
Definition
A commutative Noetherian ring is regular if for every prime ideal , the local ring is regular local. This includes the zero ring vacuously. The maximal-localization test is proved in the localization and polynomial-extension theorem.
Depends on
Used by
- Regular field factors can have a nonregular tensor product Counterexample
- Geometrically regular algebras and geometrically regular fibres Definition
- Smoothness over a field by geometric regularity Definition
- A nondegenerate projective quadric Example
- Field tests for geometric regularity Lemma
- Invertible Jacobian minor gives regular parameters in a polynomial fibre Lemma
- localisation and polynomial extension of regular rings Theorem
- Locally standard smooth iff flat with geometrically regular fibres Theorem
- Regular algebras over a perfect field are geometrically regular Theorem
- Regular equals smooth over a perfect field Theorem
Dependency tree · two levels
4 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
- Definition 12.14 and Corollary 12.34 (standard reference, not scraped)