Theorems · Theorem · commutative algebra
ValuationRing.isFractionRing_iff
∀ {𝒪 : Type u} {K : Type v} [inst : CommRing 𝒪] [inst_1 : Field K] [inst_2 : Algebra 𝒪 K] [inst_3 : IsDomain 𝒪]
[ValuationRing 𝒪],
IsFractionRing 𝒪 K ↔
(∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a) ∧ Function.Injective ⇑(algebraMap 𝒪 K)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 50 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- IsFractionRingstatement and proof · cited by 738
- IsFractionRing.injectiveproof · cited by 70
- IsLocalization.IsIntegerproof · cited by 39
- ValuationRingstatement and proof · cited by 28
- ValuationRing.isInteger_or_isIntegerproof · cited by 3
Cited by0
Results whose statement or proof uses this declaration.
Nothing cites this yet.