Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13
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.
Localisation at a prime ideal:
Definition
Let be a commutative ring and let be a prime ideal. Since and the product of two elements outside remains outside , the complement is multiplicative. The localisation of at is Its elements are fractions with .
Depends on
Used by
- Assuming the Axiom of Choice, a local ring R is canonically isomorphic to R_mathfrak m at its maximal ideal Corollary
- F[x]₍ₓ₎ is the ring of rational functions defined at 0, with maximal ideal generated by x and residue field F Example
- ℤ₍ₚ₎ consists of rationals with denominator not divisible by p, has maximal ideal pℤ₍ₚ₎, and residue field Fₚ Example
- R_mathfrak p is local with unique maximal ideal mathfrak pR_mathfrak p Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 19 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- The Stacks Project, Section 10.18: Local rings (standard reference, not scraped)