Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.iff_forall_isPrincipal_of_height_eq_one
∀ {R : Type u_1} [inst : CommRing R] [IsDomain R] [IsNoetherianRing R],
UniqueFactorizationMonoid R ↔ ∀ (p : Ideal R) [p.IsPrime], p.height = 1 → Submodule.IsPrincipal pLet R be a Noetherian domain. Then R is a UFD if and only if every height 1 prime ideal is
principal.
- Defined in
- Mathlib.RingTheory.Ideal.UFD
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- ENatstatement · cited by 4,985
- Idealstatement and proof · cited by 4,748
- IsDomainstatement and proof · cited by 2,196
- Ideal.IsPrimestatement and proof · cited by 827
- UniqueFactorizationMonoidstatement and proof · cited by 279
- IsNoetherianRingstatement and proof · cited by 268
- Submodule.IsPrincipalstatement · cited by 129
- Ideal.heightstatement · cited by 83
- UniqueFactorizationMonoid.isPrincipal_of_height_eq_oneproof · cited by 2
- UniqueFactorizationMonoid.of_forall_isPrincipal_of_height_eq_oneproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.iff_of_isLocalizationAway_of_primeproof · cited by 1