The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain

Depends on

• Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions
• Zero divisor, and integral domain: a commutative ring with $1 \ne 0$ and no zero divisors

Used by

• Compute the twisted Burau determinant for a colored braid closure Algorithm
• For a field F, F(t)=Frac(F[t]) is its rational function field; in particular ℝ(t)=Frac(ℝ[t]) Corollary
• Puiseux discs normalise a reduced plane curve germ Corollary
• Rings of integers are Dedekind domains Corollary
• The normalization of an irreducible affine variety is finite Corollary
• A cusp family defeats the missing source hypothesis Counterexample
• Outside a domain, the nonzero elements need not be multiplicative: 2·3=0 in ℤ/6 Counterexample
• Quasi-finite does not imply finite Counterexample
• Fractional ideals Definition
• Integral closure in an extension ring and integrally closed domains Definition
• Intersection with an invertible sheaf and the first Chern class Definition
• Order codimension one rational function Definition
• Principal weil divisor and class group Definition
• Rational equivalence and the Chow group of cycles Definition
• regular function projective variety Definition
• Sheaf total quotient rings Definition
• Strong transcendence over a subring Definition
• The admissible Wada quotient of a deficiency-one exterior spine Definition
• The degenerate affine Hecke algebra Definition
• The function field of an irreducible classical affine variety Definition
• The function field of an irreducible classical affine variety Definition
• Twisted chain complex and based Reidemeister torsion for link exteriors Definition
• Valuative uniqueness diagram Definition
• Divisor of a rational function on the projective line Example
• Frac(ℤ) is canonically isomorphic to ℚ Example
• Normalization of a nodal affine plane curve Example
• Normalization of the cusp semigroup ring Example
• Normalization of the t³,t⁴,t⁵ monomial curve Example
• Normalizing the cuspidal plane curve Example
• Normalizing the nodal plane curve Example
• Principal divisors on the projective line have degree zero 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
• A dominant map has a surjective differential on a dense source open Lemma
• A finite rank-one projective module embeds as a fractional ideal Lemma
• A local domain has a dominating valuation overring Lemma
• Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian Lemma
• Birational morphisms restrict to isomorphisms between principal affine opens Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Dixmier's lemma: endomorphisms of a simple module over a countable-dimensional algebra Lemma

…and 39 more results.

Dependency tree · two levels

7 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)