Theorems · Theorem · commutative algebra

IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_mem

∀ {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) (r : R),
  (IsDedekindDomain.HeightOneSpectrum.valuation K v) ↑r < 1 ↔ r ∈ v.asIdeal

The v-adic valuation of r : R is less than 1 if and only if r ∈ v.

Defined in

Mathlib.RingTheory.DedekindDomain.AdicValuation

Cited by

3 results in Mathlib

Foundations

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

Assumes

CommRingIsDedekindDomainFieldAlgebraIsFractionRing