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Smaug123
2019-11-20 21:53:53 +00:00
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Rings/Quotients/Definition.agda Normal file
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{-# OPTIONS --warning=error --safe --without-K #-}
open import Functions
open import Sets.FinSet
open import LogicalFormulae
open import Groups.Definition
open import Groups.Groups
open import Groups.FiniteGroups.Definition
open import Rings.Homomorphisms.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Abelian.Definition
open import Setoids.Setoids
open import Rings.Definition
open import Fields.FieldOfFractions.Setoid
open import Sets.EquivalenceRelations
open import Groups.Lemmas
open import Groups.QuotientGroup.Definition
module Rings.Quotients.Definition {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ _*A_ : A → A → A} {_+B_ _*B_ : B → B → B} (R : Ring S _+A_ _*A_) (R2 : Ring T _+B_ _*B_) {underf : A → B} (f : RingHom R R2 underf) where
open import Groups.QuotientGroup.Lemmas (Ring.additiveGroup R) (Ring.additiveGroup R2) (RingHom.groupHom f)
quotientRing : Ring (quotientGroupSetoid (Ring.additiveGroup R) (RingHom.groupHom f)) _+A_ _*A_
Ring.additiveGroup quotientRing = quotientGroupByHom (Ring.additiveGroup R) (RingHom.groupHom f)
Ring.*WellDefined quotientRing fr=ft fs=fu = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (transitive (RingHom.ringHom f) (transitive (Ring.*WellDefined R2 (quotientGroupLemma' (Ring.additiveGroup R) (RingHom.groupHom f) fr=ft) (quotientGroupLemma' (Ring.additiveGroup R) (RingHom.groupHom f) fs=fu)) (symmetric (RingHom.ringHom f))))
where
open Setoid T
open Equivalence eq
Ring.1R quotientRing = Ring.1R R
Ring.groupIsAbelian quotientRing = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.groupIsAbelian R))
Ring.*Associative quotientRing = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*Associative R))
Ring.*Commutative quotientRing = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*Commutative R))
Ring.*DistributesOver+ quotientRing = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*DistributesOver+ R))
Ring.identIsIdent quotientRing = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.identIsIdent R))
projectionMapIsHom : RingHom R quotientRing id
RingHom.preserves1 projectionMapIsHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (Equivalence.reflexive (Setoid.eq T))
RingHom.ringHom projectionMapIsHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (Equivalence.reflexive (Setoid.eq T))
RingHom.groupHom projectionMapIsHom = projectionMapIsGroupHom