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Sayantan Santra 2023-06-15 17:18:46 -05:00 committed by GitHub
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18
CommAlg/sameer(artinian-rings).lean
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@ -6,10 +6,20 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.DedekindDomain.DVR
lemma FieldisArtinian (R : Type _) [CommRing R] (h: IsField R) :
IsArtinianRing R := by sorry
lemma FieldisArtinian (R : Type _) [CommRing R] (h : IsField R) :
IsArtinianRing R := by
let inst := h.toField
rw [isArtinianRing_iff, isArtinian_iff_wellFounded]
apply WellFounded.intro
intro I
constructor
intro J hJI
constructor
intro K hKJ
exfalso
rcases Ideal.eq_bot_or_top J with rfl | rfl
. exact not_lt_bot hKJ
. exact not_top_lt hJI
lemma ArtinianDomainIsField (R : Type _) [CommRing R] [IsDomain R]
(IsArt : IsArtinianRing R) : IsField (R) := by