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.
Multiplicative subsets and the localisation as equivalence classes of fractions
Definition
Let be a commutative ring. A subset is multiplicative if and implies .
On , define The localisation of at , denoted , is the set of equivalence classes for this relation. The class of is written , and its arithmetic is The localisation map is the ring homomorphism Every maps to a unit, with . The construction permits ; in that case the localisation is the zero ring.
Depends on
Used by
- Outside a domain, the nonzero elements need not be multiplicative: 2·3=0 in ℤ/6 Counterexample
- Localisation at a prime ideal: R_mathfrak p=(R∖mathfrak p)⁻¹R Definition
- Principal localisation R_f={1,f,f²,…}⁻¹R Definition
- The field of fractions Frac(D)=(D∖{0})⁻¹D of an integral domain Definition
- The localisation relation is an equivalence relation and fraction arithmetic is well defined Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 18 results over 13 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 10.9: Localization (standard reference, not scraped)