Theorems · Theorem · commutative algebra
IsFractionRing.surjective_iff_isField
∀ {R : Type u_1} [inst : CommRing R] {K : Type u_5} [inst_1 : CommRing K] [inst_2 : Algebra R K] [IsFractionRing R K]
[IsDomain R], Function.Surjective ⇑(algebraMap R K) ↔ IsField R- Cited by
- 2 results in Mathlib
- Foundations
- Depth 48 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- nonZeroDivisorsproof · cited by 895
- IsFractionRingstatement and proof · cited by 738
- IsFieldstatement and proof · cited by 103
- Ne.isUnitproof · cited by 99
- IsFractionRing.injectiveproof · cited by 70
- AlgEquiv.surjectiveproof · cited by 48
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.bijective_algebraMap_quotient_residueFieldproof · cited by 1
- Submodule.traceDual_topproof · cited by 0