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Merge pull request #54 from ssavkar1/main

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wrote proofs of two lemmas
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ssavkar1 2023-06-14 11:56:00 -07:00 committed by GitHub
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1 changed files with 10 additions and 9 deletions
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CommAlg/sameer(artinian-rings).lean
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@ -6,7 +6,9 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.DedekindDomain.DVR import Mathlib.RingTheory.DedekindDomain.DVR
lemma FieldisArtinian (R : Type _) [CommRing R] (IsField : ):= by sorry lemma FieldisArtinian (R : Type _) [CommRing R] (h: IsField R) :
IsArtinianRing R := by sorry
lemma ArtinianDomainIsField (R : Type _) [CommRing R] [IsDomain R] lemma ArtinianDomainIsField (R : Type _) [CommRing R] [IsDomain R]
@ -47,8 +49,7 @@ lemma isArtinianRing_of_quotient_of_artinian (R : Type _) [CommRing R]
lemma IsPrimeMaximal (R : Type _) [CommRing R] (P : Ideal R) lemma IsPrimeMaximal (R : Type _) [CommRing R] (P : Ideal R)
(IsArt : IsArtinianRing R) (isPrime : Ideal.IsPrime P) : Ideal.IsMaximal P := (IsArt : IsArtinianRing R) (isPrime : Ideal.IsPrime P) : Ideal.IsMaximal P :=
by by
-- if R is Artinian and P is prime then R/P is Integral Domain -- if R is Artinian and P is prime then R/P is Artinian Domain
-- which is Artinian Domain
-- R⧸P is a field by the above lemma -- R⧸P is a field by the above lemma
-- P is maximal -- P is maximal
@ -56,13 +57,13 @@ by
have artRP : IsArtinianRing (R⧸P) := by have artRP : IsArtinianRing (R⧸P) := by
exact isArtinianRing_of_quotient_of_artinian R P IsArt exact isArtinianRing_of_quotient_of_artinian R P IsArt
have artRPField : IsField (R⧸P) := by
exact ArtinianDomainIsField (R⧸P) artRP
have h := Ideal.Quotient.maximal_of_isField P artRPField
exact h
-- Then R/I is Artinian -- Then R/I is Artinian
-- have' : IsArtinianRing R ∧ Ideal.IsPrime I → IsDomain (R⧸I) := by -- have' : IsArtinianRing R ∧ Ideal.IsPrime I → IsDomain (R⧸I) := by
-- R⧸I.IsArtinian → monotone_stabilizes_iff_artinian.R⧸I -- R⧸I.IsArtinian → monotone_stabilizes_iff_artinian.R⧸I
-- Use Stacks project proof since it's broken into lemmas