Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.of_forall_isPrincipal_of_height_eq_one
∀ {R : Type u_1} [inst : CommRing R] [IsDomain R] [IsNoetherianRing R],
(∀ (p : Ideal R) [p.IsPrime], p.height = 1 → Submodule.IsPrincipal p) → UniqueFactorizationMonoid R- Defined in
- Mathlib.RingTheory.Ideal.UFD
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 138 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
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
- Bot.botproof · cited by 4,720
- IsDomainstatement and proof · cited by 2,196
- le_antisymmproof · cited by 2,068
- Ideal.spanproof · cited by 948
- Ideal.IsPrimestatement and proof · cited by 827
- LE.le.trans_eqproof · cited by 328
- UniqueFactorizationMonoidstatement · cited by 279
- IsNoetherianRingstatement and proof · cited by 268
- Submodule.IsPrincipalstatement and proof · cited by 129
Cited by1
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.iff_forall_isPrincipal_of_height_eq_oneproof · cited by 1