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https://github.com/Smaug123/agdaproofs
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Lots of rings (#82)
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Patrick Stevens
GitHub
@@ -19,9 +19,6 @@ module Rings.Ideals.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_
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open import Groups.Subgroups.Definition (Ring.additiveGroup R)
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ringKernel : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → Set (a ⊔ d)
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ringKernel {T = T} R2 {f} fHom = Sg A (λ a → Setoid._∼_ T (f a) (Ring.0R R2))
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record Ideal {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
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field
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isSubgroup : Subgroup pred
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@@ -30,31 +27,3 @@ record Ideal {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
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closedUnderInverse = Subgroup.closedUnderInverse isSubgroup
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containsIdentity = Subgroup.containsIdentity isSubgroup
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isSubset = Subgroup.isSubset isSubgroup
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idealPredForKernel : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → A → Set d
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idealPredForKernel {T = T} R2 {f} fHom a = Setoid._∼_ T (f a) (Ring.0R R2)
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idealPredForKernelWellDefined : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → {x y : A} → (Setoid._∼_ S x y) → (idealPredForKernel R2 fHom x → idealPredForKernel R2 fHom y)
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idealPredForKernelWellDefined {T = T} R2 {f} fHom a x=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined (RingHom.groupHom fHom) (Equivalence.symmetric (Setoid.eq S) a)) x=0
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kernelIdealIsIdeal : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} {R2 : Ring T _+2_ _*2_} {f : A → C} (fHom : RingHom R R2 f) → Ideal (idealPredForKernel R2 fHom)
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Subgroup.isSubset (Ideal.isSubgroup (kernelIdealIsIdeal {R2 = R2} fHom)) = idealPredForKernelWellDefined R2 fHom
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Subgroup.closedUnderPlus (Ideal.isSubgroup (kernelIdealIsIdeal {T = T} {R2 = R2} fHom)) {x} {y} fx=0 fy=0 = transitive (transitive (GroupHom.groupHom (RingHom.groupHom fHom)) (+WellDefined fx=0 fy=0)) identLeft
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where
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open Ring R2
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open Group (Ring.additiveGroup R2)
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open Setoid T
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open Equivalence eq
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Subgroup.containsIdentity (Ideal.isSubgroup (kernelIdealIsIdeal fHom)) = imageOfIdentityIsIdentity (RingHom.groupHom fHom)
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Subgroup.closedUnderInverse (Ideal.isSubgroup (kernelIdealIsIdeal {T = T} {R2 = R2} fHom)) {x} fx=0 = zeroImpliesInverseZero (RingHom.groupHom fHom) fx=0
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where
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open Ring R2
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open Group (Ring.additiveGroup R2)
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open Setoid T
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open Equivalence eq
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Ideal.accumulatesTimes (kernelIdealIsIdeal {T = T} {R2 = R2} {f = f} fHom) {x} {y} fx=0 = transitive (RingHom.ringHom fHom) (transitive (Ring.*WellDefined R2 fx=0 reflexive) (transitive (Ring.*Commutative R2) (Ring.timesZero R2 {f y})))
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where
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open Setoid T
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open Equivalence eq
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-- TODO : define the quotient by an ideal; note that the result is a ring
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34
Rings/Ideals/FirstIsomorphismTheorem.agda
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34
Rings/Ideals/FirstIsomorphismTheorem.agda
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Numbers.Naturals.Naturals
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open import Setoids.Orders
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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open import Rings.Homomorphisms.Definition
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open import Groups.Homomorphisms.Lemmas
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open import Rings.Subrings.Definition
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open import Rings.Isomorphisms.Definition
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open import Groups.Isomorphisms.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Rings.Ideals.FirstIsomorphismTheorem {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_+A_ _*A_ : A → A → A} {_+B_ _*B_ : B → B → B} {R1 : Ring S _+A_ _*A_} {R2 : Ring T _+B_ _*B_} {f : A → B} (hom : RingHom R1 R2 f) where
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open import Rings.Quotients.Definition R1 R2 hom
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open import Rings.Homomorphisms.Image hom
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open Setoid T
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open Equivalence eq
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open import Groups.FirstIsomorphismTheorem (RingHom.groupHom hom)
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ringFirstIsomorphismTheorem' : RingsIsomorphic quotientByRingHom (subringIsRing R2 imageGroupSubring)
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RingsIsomorphic.f ringFirstIsomorphismTheorem' a = f a , (a , reflexive)
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RingHom.preserves1 (RingIso.ringHom (RingsIsomorphic.iso ringFirstIsomorphismTheorem')) = RingHom.preserves1 hom
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RingHom.ringHom (RingIso.ringHom (RingsIsomorphic.iso ringFirstIsomorphismTheorem')) {r} {s} = RingHom.ringHom hom
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RingHom.groupHom (RingIso.ringHom (RingsIsomorphic.iso ringFirstIsomorphismTheorem')) = GroupIso.groupHom (GroupsIsomorphic.proof (groupFirstIsomorphismTheorem'))
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RingIso.bijective (RingsIsomorphic.iso ringFirstIsomorphismTheorem') = GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem')
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58
Rings/Ideals/Lemmas.agda
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58
Rings/Ideals/Lemmas.agda
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Numbers.Naturals.Naturals
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open import Setoids.Orders
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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open import Rings.Homomorphisms.Definition
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open import Groups.Homomorphisms.Lemmas
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open import Groups.Subgroups.Definition
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open import Rings.Homomorphisms.Kernel
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open import Rings.Cosets
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Rings.Ideals.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
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open import Rings.Ideals.Definition R
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idealPredForKernel : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → A → Set d
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idealPredForKernel {T = T} R2 {f} fHom a = Setoid._∼_ T (f a) (Ring.0R R2)
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idealPredForKernelWellDefined : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → {x y : A} → (Setoid._∼_ S x y) → (idealPredForKernel R2 fHom x → idealPredForKernel R2 fHom y)
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idealPredForKernelWellDefined {T = T} R2 {f} fHom a x=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined (RingHom.groupHom fHom) (Equivalence.symmetric (Setoid.eq S) a)) x=0
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kernelIdealIsIdeal : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} {R2 : Ring T _+2_ _*2_} {f : A → C} (fHom : RingHom R R2 f) → Ideal (idealPredForKernel R2 fHom)
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Subgroup.isSubset (Ideal.isSubgroup (kernelIdealIsIdeal {R2 = R2} fHom)) = idealPredForKernelWellDefined R2 fHom
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Subgroup.closedUnderPlus (Ideal.isSubgroup (kernelIdealIsIdeal {T = T} {R2 = R2} fHom)) {x} {y} fx=0 fy=0 = transitive (transitive (GroupHom.groupHom (RingHom.groupHom fHom)) (+WellDefined fx=0 fy=0)) identLeft
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where
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open Ring R2
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open Group (Ring.additiveGroup R2)
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open Setoid T
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open Equivalence eq
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Subgroup.containsIdentity (Ideal.isSubgroup (kernelIdealIsIdeal fHom)) = imageOfIdentityIsIdentity (RingHom.groupHom fHom)
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Subgroup.closedUnderInverse (Ideal.isSubgroup (kernelIdealIsIdeal {T = T} {R2 = R2} fHom)) {x} fx=0 = zeroImpliesInverseZero (RingHom.groupHom fHom) fx=0
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where
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open Ring R2
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open Group (Ring.additiveGroup R2)
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open Setoid T
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open Equivalence eq
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Ideal.accumulatesTimes (kernelIdealIsIdeal {T = T} {R2 = R2} {f = f} fHom) {x} {y} fx=0 = transitive (RingHom.ringHom fHom) (transitive (Ring.*WellDefined R2 fx=0 reflexive) (transitive (Ring.*Commutative R2) (Ring.timesZero R2 {f y})))
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where
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open Setoid T
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open Equivalence eq
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open Setoid S
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open Ring R
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open Group additiveGroup
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open Equivalence eq
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open import Groups.Lemmas additiveGroup
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idealIsKernelMap : {c : _} {pred : A → Set c} → (i : Ideal pred) → {x : A} → pred x → ringKernel {R1 = R} {R2 = cosetRing R i} (cosetRingHom R i)
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idealIsKernelMap {c} {pred} i {x} predX = x , (Ideal.isSubset i (transitive (symmetric identLeft) (+WellDefined (symmetric invIdent) reflexive)) predX)
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22
Rings/Ideals/Prime/Definition.agda
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22
Rings/Ideals/Prime/Definition.agda
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Lemmas
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open import Groups.Definition
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open import Numbers.Naturals.Naturals
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open import Orders
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open import Setoids.Setoids
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open import Functions
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open import Rings.Definition
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open import Rings.Lemmas
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open import Sets.EquivalenceRelations
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open import Fields.Fields
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open import Rings.Ideals.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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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
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PrimeIdeal : Set (a ⊔ c)
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PrimeIdeal = {a b : A} → pred (a * b) → ((pred a) → False) → pred b
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24
Rings/Ideals/Principal/Definition.agda
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24
Rings/Ideals/Principal/Definition.agda
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Numbers.Naturals.Naturals
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open import Setoids.Orders
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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open import Rings.Homomorphisms.Definition
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open import Groups.Homomorphisms.Lemmas
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Rings.Ideals.Principal.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
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open import Rings.Ideals.Definition R
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open Setoid S
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PrincipalIdeal : {c : _} {pred : A → Set c} (ideal : Ideal pred) → Set (a ⊔ b ⊔ c)
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PrincipalIdeal {pred = pred} ideal = Sg A (λ a → {x : A} → (pred x) → Sg A (λ c → (a * c) ∼ x))
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