Theorems · Theorem · commutative algebra
Ideal.isPrincipal_of_isPrincipal_isLocalizationAway_of_prime
∀ {R : Type u_1} [inst : CommRing R] [IsDomain R] [WfDvdMonoid R] {x : R},
Prime x →
∀ {p : Ideal R} [p.IsPrime],
x ∉ p →
∀ (S : Type u_2) [inst_4 : CommRing S] [inst_5 : Algebra R S] [IsLocalization.Away x S],
Submodule.IsPrincipal (Ideal.map (algebraMap R S) p) → Submodule.IsPrincipal p- Defined in
- Mathlib.RingTheory.Ideal.UFD
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites39
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- SetLike.coeproof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- Algebra.algebraMapstatement and proof · cited by 4,706
- Disjointproof · cited by 2,201
- IsDomainstatement and proof · cited by 2,196
- IsUnitproof · cited by 1,602
- Submodule.spanproof · cited by 1,504
Cited by2
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.iff_of_isLocalizationAway_of_primeproof · cited by 1
- Ideal.isPrincipal_of_isPrincipal_localizationAway_of_primeproof · cited by 0