Theorems · Theorem · commutative algebra

IsDedekindDomain.HeightOneSpectrum.exists_primeCompl_mul_eq_of_integer

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDedekindDomain R] {K : Type u_2} [inst_2 : Field K]
  [inst_3 : Algebra R K] [inst_4 : IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (x : K),
  (IsDedekindDomain.HeightOneSpectrum.valuation K v) x ≤ 1 → ∃ n d, x * (algebraMap R K) ↑d = (algebraMap R K) n

All x ∈ 𝓞[K] can be written as n / d with n : R and d ∈ v.asIdealᶜ.

Defined in

Mathlib.RingTheory.DedekindDomain.AdicValuation

Cited by

1 results in Mathlib

Foundations

Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDedekindDomainFieldAlgebraIsFractionRing