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Quotient ring (#81)

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Patrick Stevens
2019-11-21 07:28:25 +00:00
committed by GitHub
parent d4f51a3efe
commit b33baa5fb7
12 changed files with 125 additions and 64 deletions
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20
Rings/Quotients/Definition.agda
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@@ -20,20 +20,20 @@ module Rings.Quotients.Definition {a b c d : _} {A : Set a} {B : Set b} {S : Set
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))))
quotientByRingHom : Ring (quotientGroupSetoid (Ring.additiveGroup R) (RingHom.groupHom f)) _+A_ _*A_
Ring.additiveGroup quotientByRingHom = quotientGroupByHom (Ring.additiveGroup R) (RingHom.groupHom f)
Ring.*WellDefined quotientByRingHom 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))
Ring.1R quotientByRingHom = Ring.1R R
Ring.groupIsAbelian quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.groupIsAbelian R))
Ring.*Associative quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*Associative R))
Ring.*Commutative quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*Commutative R))
Ring.*DistributesOver+ quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*DistributesOver+ R))
Ring.identIsIdent quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.identIsIdent R))
projectionMapIsHom : RingHom R quotientRing id
projectionMapIsHom : RingHom R quotientByRingHom 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