Theorems · Theorem · commutative algebra
IsArtinianRing.isField_of_isReduced_of_isLocalRing
∀ (R : Type u_1) [inst : CommRing R] [IsArtinianRing R] [IsReduced R] [IsLocalRing R], IsField R
Commutative Artinian reduced local ring is a field.
- Defined in
- Mathlib.RingTheory.Artinian.Ring
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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
- HasQuotient.Quotientproof · cited by 2,301
- IsLocalRingstatement and proof · cited by 339
- AlgEquiv.toRingEquivproof · cited by 137
- IsFieldstatement · cited by 103
- IsArtinianRingstatement and proof · cited by 98
- IsReducedstatement and proof · cited by 98
- MaximalSpectrumproof · cited by 73
- MaximalSpectrum.asIdealproof · cited by 58
- RingEquiv.transproof · cited by 54
- RingEquiv.toMulEquivproof · cited by 26
- Field.toIsFieldproof · cited by 18
Cited by1
Results whose statement or proof uses this declaration.
- Algebra.IsUnramifiedAt.not_minpoly_sq_dvdproof · cited by 0