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.
is a field and embeds the integral domain
Statement
For every integral domain , the localisation is a field. Its canonical map is an injective unital ring homomorphism.
Facts & Assumptions
Given: An integral domain .
The field of fractions is the localisation at (The field of fractions of an integral domain).
A localisation map is injective exactly when every denominator has trivial annihilator (Equality, vanishing, and the kernel of the localisation map).
Localisation is a commutative ring with the stated fraction arithmetic (The localisation relation is an equivalence relation and fraction arithmetic is well defined).
A field is a nonzero commutative ring in which every nonzero element is a unit (Field).
Proof
Every nonzero element of the domain has trivial annihilator. Hence [F2], applied to , makes the canonical homomorphism injective. In particular , so the localisation is nonzero.
Let be nonzero. By the vanishing criterion in [F2], , because otherwise . Thus , and [F3] gives .
Every nonzero element is therefore a unit, and [F3] supplies the commutative-ring structure. By [F4], is a field.
Depends on
Used by
- For a field F, F(t)=Frac(F[t]) is its rational function field; in particular ℝ(t)=Frac(ℝ[t]) Corollary
- Rₚ/pRₚcongFrac(R/p) is the residue field at p Corollary
- The generic point of the affine line has no relative k-valued coordinate Counterexample
- regular function projective variety Definition
- The function field of an irreducible classical affine variety Definition
- The p-primary quotient Q/Z_(p) over Z_(p) shows finite generation is essential in Nakayama Example
- The subring k[x,y,x/y,x/y²,…] of k(x,y) has a strictly ascending chain of principal ideals Example
- False statement: every subring of a Noetherian ring is Noetherian False statement
- Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes Lemma
- The eventual Hilbert function of a zero-dimensional projective quotient equals its total length Lemma
- Extension to the fraction field recovers the free rank of a finitely generated PID module Proposition
- Every injective ring map from a domain into a field factors uniquely through its field of fractions Theorem
- Lying over for integral ring maps Theorem
Dependency tree · two levels
6 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
- The CRing Project, Chapter 13: Fields and Extensions (standard reference, not scraped)