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69 lines
2.5 KiB
Text
69 lines
2.5 KiB
Text
import Mathlib.RingTheory.Ideal.Basic
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import Mathlib.RingTheory.Noetherian
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import Mathlib.RingTheory.Artinian
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import Mathlib.RingTheory.Ideal.Quotient
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import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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import Mathlib.RingTheory.DedekindDomain.DVR
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lemma FieldisArtinian (R : Type _) [CommRing R] (h: IsField R) :
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IsArtinianRing R := by sorry
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lemma ArtinianDomainIsField (R : Type _) [CommRing R] [IsDomain R]
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(IsArt : IsArtinianRing R) : IsField (R) := by
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-- Assume P is nonzero and R is Artinian
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-- Localize at P; Then R_P is Artinian;
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-- Assume R_P is not a field
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-- Then Jacobson Radical of R_P is nilpotent so it's maximal ideal is nilpotent
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-- Maximal ideal is zero since local ring is a domain
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-- a contradiction since P is nonzero
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-- Therefore, R is a field
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have maxIdeal := Ideal.exists_maximal R
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obtain ⟨m,hm⟩ := maxIdeal
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have h:= Ideal.primeCompl_le_nonZeroDivisors m
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have artRP : IsDomain _ := IsLocalization.isDomain_localization h
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have h' : IsArtinianRing (Localization (Ideal.primeCompl m)) := inferInstance
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have h' : IsNilpotent (Ideal.jacobson (⊥ : Ideal (Localization
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(Ideal.primeCompl m)))):= IsArtinianRing.isNilpotent_jacobson_bot
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have := LocalRing.jacobson_eq_maximalIdeal (⊥ : Ideal (Localization
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(Ideal.primeCompl m))) bot_ne_top
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rw [this] at h'
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have := IsNilpotent.eq_zero h'
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rw [Ideal.zero_eq_bot, ← LocalRing.isField_iff_maximalIdeal_eq] at this
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by_contra h''
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--by_cases h'' : m = ⊥
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have := Ring.ne_bot_of_isMaximal_of_not_isField hm h''
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have := IsLocalization.AtPrime.not_isField R this (Localization (Ideal.primeCompl m))
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contradiction
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#check Ideal.IsPrime
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#check IsDomain
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lemma isArtinianRing_of_quotient_of_artinian (R : Type _) [CommRing R]
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(I : Ideal R) (IsArt : IsArtinianRing R) : IsArtinianRing (R ⧸ I) :=
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isArtinian_of_tower R (isArtinian_of_quotient_of_artinian R R I IsArt)
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lemma IsPrimeMaximal (R : Type _) [CommRing R] (P : Ideal R)
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(IsArt : IsArtinianRing R) (isPrime : Ideal.IsPrime P) : Ideal.IsMaximal P :=
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by
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-- if R is Artinian and P is prime then R/P is Artinian Domain
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-- R⧸P is a field by the above lemma
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-- P is maximal
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have : IsDomain (R⧸P) := Ideal.Quotient.isDomain P
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have artRP : IsArtinianRing (R⧸P) := by
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exact isArtinianRing_of_quotient_of_artinian R P IsArt
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have artRPField : IsField (R⧸P) := by
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exact ArtinianDomainIsField (R⧸P) artRP
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have h := Ideal.Quotient.maximal_of_isField P artRPField
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exact h
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-- Then R/I is Artinian
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-- have' : IsArtinianRing R ∧ Ideal.IsPrime I → IsDomain (R⧸I) := by
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-- R⧸I.IsArtinian → monotone_stabilizes_iff_artinian.R⧸I
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