Alphabeta Math

The Field of Fractions and Localisation

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✓ 19 results · all verified · 12 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Multiplicative subsets and the localisation S⁻¹R as equivalence classes of fractions

Theorem

The localisation relation is an equivalence relation and fraction arithmetic is well defined

Proposition

Equality, vanishing, and the kernel of the localisation map

Theorem

Universal property of localisation: maps that invert S factor uniquely through S⁻¹R

Corollary

A localisation is unique up to a unique isomorphism compatible with the map from R

Proposition

A fraction r/s is a unit in S⁻¹R exactly when ar∈ S for some a∈ R

Definition

Principal localisation R_f={1,f,f²,…}⁻¹R

Proposition

Localising twice is localising once at the multiplicative set generated by both denominator sets

Definition

The field of fractions Frac(D)=(D∖{0})⁻¹D of an integral domain

Lemma

Gauss lemma over a UFD

Theorem

Frac(D) is a field and d↦ d/1 embeds the integral domain D

Theorem

Every injective ring map from a domain into a field factors uniquely through its field of fractions

Corollary

Fields of fractions are uniquely isomorphic over their embedded domain

Corollary

For a field F, F(t)=Frac(F[t]) is its rational function field; in particular ℝ(t)=Frac(ℝ[t])

Definition

A local ring is a nonzero commutative ring with a unique maximal ideal

Theorem

Assuming the Axiom of Choice, a nonzero commutative ring is local exactly when its nonunits form an ideal, exactly when one of x and 1-x is a unit for every x

Definition

Localisation at a prime ideal: Rₚ=(R setminusp)⁻¹R

Theorem

Rₚ is local with unique maximal ideal pRₚ

Corollary

Assuming the Axiom of Choice, a local ring R is canonically isomorphic to Rₘ at its maximal ideal

Corollary

Assuming the Axiom of Choice, a local ring has no idempotents other than 0 and 1

Theorem

Ideals of S⁻¹R correspond to S-saturated ideals of R, and prime ideals correspond to primes disjoint from S

Theorem

Localisation commutes with quotient rings: S⁻¹R/S⁻¹I≅ S̄⁻¹(R/I)

Corollary

Rₚ/pRₚcongFrac(R/p) is the residue field at p

Example

F[x]₍ₓ₎ is the ring of rational functions defined at 0, with maximal ideal generated by x and residue field F

Sources