diff --git a/CommAlg/sameer(artinian-rings).lean b/CommAlg/sameer(artinian-rings).lean index 417cc96..e619c07 100644 --- a/CommAlg/sameer(artinian-rings).lean +++ b/CommAlg/sameer(artinian-rings).lean @@ -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