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Z is a Euclidean domain (#86)

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Patrick Stevens
2019-12-07 13:00:18 +00:00
committed by GitHub
parent cfd9787bb8
commit e192f0e1f1
38 changed files with 1018 additions and 486 deletions
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12
Rings/Ideals/Maximal/Lemmas.agda
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@@ -16,9 +16,13 @@ open import Rings.Lemmas
open import Sets.EquivalenceRelations
open import Rings.Ideals.Definition
open import Fields.Fields
open import Fields.Lemmas
open import Rings.Cosets
open import Rings.Ideals.Maximal.Definition
open import Rings.Ideals.Lemmas
open import Rings.Ideals.Prime.Definition
open import Rings.IntegralDomains.Definition
open import Rings.Ideals.Prime.Lemmas
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
@@ -77,3 +81,11 @@ MaximalIdeal.isMaximal (quotientFieldImpliesIdealMaximal f) {bigger} biggerIdeal
u = Ideal.closedUnderInverse biggerIdeal (Ideal.isSubset biggerIdeal *Commutative (Ideal.accumulatesTimes biggerIdeal biggerA))
v : bigger 1R
v = Ideal.isSubset biggerIdeal (invTwice additiveGroup 1R) (Ideal.closedUnderInverse biggerIdeal (Ideal.isSubset biggerIdeal (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (Ideal.closedUnderPlus biggerIdeal t u)))
idealMaximalImpliesIdealPrime : ({d : _} → MaximalIdeal i {d}) → PrimeIdeal i
idealMaximalImpliesIdealPrime max = quotientIntDomImpliesIdealPrime i f'
where
f : Field (cosetRing R i)
f = idealMaximalImpliesQuotientField max
f' : IntegralDomain (cosetRing R i)
f' = fieldIsIntDom f (λ p → Field.nontrivial f (Equivalence.symmetric (Setoid.eq (cosetSetoid additiveGroup (Ideal.isSubgroup i))) p))