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More rings stuff (#83)

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Patrick Stevens
2019-11-23 13:53:54 +00:00
committed by GitHub
parent 660d7aa27c
commit 2ed7bd8044
12 changed files with 260 additions and 17 deletions
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7
Rings/Ideals/Prime/Definition.agda
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@@ -18,5 +18,8 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Ideals.Prime.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {pred : A → Set c} (i : Ideal R pred) where
PrimeIdeal : Set (a ⊔ c)
PrimeIdeal = {a b : A} → pred (a * b) → ((pred a) → False) → pred b
record PrimeIdeal : Set (a ⊔ c) where
field
isPrime : {a b : A} → pred (a * b) → ((pred a) → False) → pred b
notContained : A
notContainedIsNotContained : (pred notContained) → False

50
Rings/Ideals/Prime/Lemmas.agda Normal file
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@@ -0,0 +1,50 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Lemmas
open import Groups.Definition
open import Setoids.Setoids
open import Rings.Definition
open import Sets.EquivalenceRelations
open import Rings.Ideals.Definition
open import Rings.IntegralDomains.Definition
open import Rings.Ideals.Prime.Definition
open import Rings.Cosets
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Ideals.Prime.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {pred : A → Set c} (i : Ideal R pred) where
open Ring R
open Group additiveGroup
open Setoid S
open Equivalence eq
open import Rings.Ideals.Lemmas R
idealPrimeImpliesQuotientIntDom : PrimeIdeal i → IntegralDomain (cosetRing R i)
IntegralDomain.intDom (idealPrimeImpliesQuotientIntDom isPrime) {a} {b} ab=0 a!=0 = ans
where
ab=0' : pred (a * b)
ab=0' = translate' i ab=0
a!=0' : (pred a) → False
a!=0' prA = a!=0 (translate i prA)
ans' : pred b
ans' = PrimeIdeal.isPrime isPrime ab=0' a!=0'
ans : pred (inverse (Ring.0R (cosetRing R i)) + b)
ans = translate i ans'
IntegralDomain.nontrivial (idealPrimeImpliesQuotientIntDom isPrime) 1=0 = PrimeIdeal.notContainedIsNotContained isPrime u
where
t : pred (Ring.1R (cosetRing R i))
t = translate' i 1=0
u : pred (PrimeIdeal.notContained isPrime)
u = Ideal.isSubset i identIsIdent (Ideal.accumulatesTimes i {y = PrimeIdeal.notContained isPrime} t)
quotientIntDomImpliesIdealPrime : IntegralDomain (cosetRing R i) → PrimeIdeal i
quotientIntDomImpliesIdealPrime intDom = record { isPrime = isPrime ; notContained = Ring.1R R ; notContainedIsNotContained = notCon }
where
abstract
notCon : pred 1R → False
notCon 1=0 = IntegralDomain.nontrivial intDom (translate i 1=0)
isPrime : {a b : A} → pred (a * b) → (pred a → False) → pred b
isPrime {a} {b} predAB !predA = translate' i (IntegralDomain.intDom intDom (translate i predAB) λ t → !predA (translate' i t))