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

• For a field F, F(t)=Frac(F[t]) is its rational function field; in particular ℝ(t)=Frac(ℝ[t]) Corollary
• The normalization of an irreducible affine variety is finite Corollary
• 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
• regular function projective variety Definition
• Strong transcendence over a subring Definition
• The function field of an irreducible classical affine variety Definition
• The function field of an irreducible classical affine variety Definition
• Valuative uniqueness diagram Definition
• 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
• 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 finite rank-one projective module embeds as a fractional ideal Lemma
• A local domain has a dominating valuation overring Lemma
• Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes Lemma
• Finite normalization commutes with principal localization Lemma
• Finite purely inseparable rational extensions admit a finite Frobenius envelope Lemma
• Finite torsion-free modules over Dedekind domains are projective Lemma
• Finite-variable polynomial algebras over fields are integrally closed Lemma
• Gauss lemma over a UFD Lemma
• Integral closure in a purely inseparable rational envelope is finite Lemma
• Local DVRs at the nonzero primes force global normality Lemma
• Only one saturated step can lie over a fixed contracted prime in R[x] Lemma
• Polynomial rings over normal domains are normal Lemma
• Quasi-finite local fibres transfer through quotients and intermediate rings Lemma
• Extension to the fraction field recovers the free rank of a finitely generated PID module Proposition
• A finite-type domain over a field has finite normalization Theorem
• Finite separable integral closures over normal Noetherian domains are module-finite Theorem
• Frac(D) is a field and d↦ d/1 embeds the integral domain D Theorem
• Polynomial algebras over fields have finite integral closures Theorem
• The ring of holomorphic germs is a UFD Theorem

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)