Theorems · Theorem · commutative algebra
isFractionRing_of_exists_eq_algebraMap_or_inv_eq_algebraMap_of_injective
∀ {𝒪 : Type u} {K : Type v} [inst : CommRing 𝒪] [inst_1 : Field K] [inst_2 : Algebra 𝒪 K],
(∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a) →
Function.Injective ⇑(algebraMap 𝒪 K) → IsFractionRing 𝒪 K- Cited by
- 2 results in Mathlib
- Foundations
- Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- IsDomainproof · cited by 2,196
- map_zeroproof · cited by 1,614
- nonZeroDivisorsproof · cited by 895
- map_oneproof · cited by 861
Cited by2
Results whose statement or proof uses this declaration.
- Valuation.Integers.isFractionRingproof · cited by 1
- ValuationRing.isFractionRing_iffproof · cited by 0