Theorems · Theorem · commutative algebra
Ideal.isPrincipal_of_isPrincipal_localizationAway_of_prime
∀ {R : Type u_1} [inst : CommRing R] [IsDomain R] [WfDvdMonoid R] {x : R},
Prime x →
∀ {p : Ideal R} [p.IsPrime],
x ∉ p → Submodule.IsPrincipal (Ideal.map (algebraMap R (Localization.Away x)) p) → Submodule.IsPrincipal p- Defined in
- Mathlib.RingTheory.Ideal.UFD
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites13
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
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.mapstatement and proof · cited by 692
- Submonoid.powersstatement · cited by 408
- Primestatement and proof · cited by 277
- Localization.Awaystatement and proof · cited by 162
- Submodule.IsPrincipalstatement and proof · cited by 129
- WfDvdMonoidstatement and proof · cited by 37
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