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Irreducible and maximal (#87)

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Patrick Stevens
2019-12-07 18:53:08 +00:00
committed by GitHub
parent e192f0e1f1
commit 99c38495ce
9 changed files with 123 additions and 19 deletions
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12
Rings/PrincipalIdealDomains/Definition.agda
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Numbers.Naturals.Naturals
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Rings.Homomorphisms.Definition
open import Groups.Homomorphisms.Lemmas
open import Rings.IntegralDomains.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.PrincipalIdealDomains.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
module Rings.PrincipalIdealDomains.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (intDom : IntegralDomain R) where
open import Rings.Ideals.Principal.Definition R
open import Rings.Ideals.Definition R

53
Rings/PrincipalIdealDomains/Lemmas.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Sets.EquivalenceRelations
open import Setoids.Setoids
open import Rings.Definition
open import Rings.PrincipalIdealDomains.Definition
open import Rings.IntegralDomains.Definition
open import Rings.Ideals.Maximal.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.PrincipalIdealDomains.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (intDom : IntegralDomain R) (pid : {c : _} → PrincipalIdealDomain intDom {c}) where
open import Rings.Ideals.Definition R
open import Rings.Irreducibles.Definition intDom
open import Rings.Ideals.Principal.Definition R
open import Rings.Divisible.Definition R
open import Rings.Associates.Lemmas intDom
open import Rings.Ideals.Lemmas R
open import Rings.Units.Definition R
open import Rings.Irreducibles.Lemmas intDom
open import Rings.Units.Lemmas R
open Ring R
open Setoid S
open Equivalence eq
irreducibleImpliesMaximalIdeal : (r : A) → Irreducible r → {d : _} → MaximalIdeal (generatedIdeal r) {d}
MaximalIdeal.notContained (irreducibleImpliesMaximalIdeal r irred {d}) = 1R
MaximalIdeal.notContainedIsNotContained (irreducibleImpliesMaximalIdeal r irred {d}) = Irreducible.nonunit irred
MaximalIdeal.isMaximal (irreducibleImpliesMaximalIdeal r irred {d}) {biggerPred} bigger biggerContains (outsideR , (biggerContainsOutside ,, notInR)) {x} = biggerPrincipal' (unitImpliesGeneratedIdealEverything w {x})
where
biggerGen : A
biggerGen = PrincipalIdeal.generator (pid bigger)
biggerPrincipal : {x : A} → biggerPred x → biggerGen ∣ x
biggerPrincipal = PrincipalIdeal.genGenerates (pid bigger)
bp : biggerPred biggerGen
bp = PrincipalIdeal.genIsInIdeal (pid bigger)
biggerPrincipal' : {x : A} → biggerGen ∣ x → biggerPred x
biggerPrincipal' {y} bg|y = memberDividesImpliesMember bigger bp bg|y
u : biggerGen ∣ r
u = biggerPrincipal (biggerContains (1R , transitive *Commutative identIsIdent))
biggerGenNonzero : biggerGen ∼ 0R → False
biggerGenNonzero bg=0 = notInR (Ideal.isSubset (generatedIdeal r) (symmetric t) (Ideal.containsIdentity (generatedIdeal r)))
where
t : outsideR ∼ 0R
t with biggerPrincipal {outsideR} biggerContainsOutside
... | mult , pr = transitive (symmetric pr) (transitive (*WellDefined bg=0 reflexive) (transitive *Commutative timesZero))
v : (r ∣ biggerGen) → False
v r|bg with mutualDivisionImpliesAssociate r|bg u (Irreducible.nonzero irred)
v r|bg | assoc = notInR (associateImpliesGeneratedIdealsEqual' assoc (PrincipalIdeal.genGenerates (pid bigger) biggerContainsOutside))
w : Unit biggerGen
w = dividesIrreducibleImpliesUnit irred u v