Theorems · Theorem · commutative algebra

Ideal.mem_primesOver_iff_mem_normalizedFactors

∀ {R : Type u_1} (A : Type u_2) [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : IsDedekindDomain A] {p : Ideal R}
  [h : p.IsMaximal] [inst_3 : Algebra R A] [IsDomain R] [Module.IsTorsionFree R A],
  p ≠ ⊥ →
    ∀ {P : Ideal A}, P ∈ p.primesOver A ↔ P ∈ UniqueFactorizationMonoid.normalizedFactors (Ideal.map (algebraMap R A) p)

Defined in

Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Cited by

2 results in Mathlib

Foundations

Depth 150 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingIsDedekindDomainIdeal.IsMaximalAlgebraIsDomainModule.IsTorsionFree