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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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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