Theorems · Theorem · commutative algebra
isField_of_isIntegral_of_isField
∀ {R : Type u_1} {S : Type u_4} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.IsIntegral R S], Function.Injective ⇑(algebraMap R S) → IsField S → IsField RIf the integral extension R → S is injective, and S is a field, then R is also a field.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- FaithfulSMulproof · cited by 340
- Algebra.IsIntegralstatement and proof · cited by 224
- IsFieldstatement and proof · cited by 103
- faithfulSMul_iff_algebraMap_injectiveproof · cited by 17
- IsLocalHom.isFieldproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- Ideal.isMaximal_comap_of_isIntegral_of_isMaximalproof · cited by 3
- Ideal.IsMaximal.ne_bot_of_isIntegral_intproof · cited by 2
- Algebra.IsIntegral.isField_iff_isFieldproof · cited by 2
- Algebra.ker_algebraMap_isMaximal_of_isIntegralproof · cited by 0