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Restructure towards ideals

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Smaug123
2019-11-20 21:20:03 +00:00
parent f2f4e867fc
commit b03a5279bc
22 changed files with 417 additions and 296 deletions
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21
Rings/Ideals/Definition.agda
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@@ -17,11 +17,13 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Ideals.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
open import Groups.Subgroups.Definition (Ring.additiveGroup R)
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)
ringKernel {T = T} R2 {f} fHom = Sg A (λ a → Setoid._∼_ T (f a) (Ring.0R R2))
ideal : {c : _} {pred : A → Set c} → (wd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) → Set (a ⊔ c)
ideal {pred = pred} wd = (pred (Ring.0R R)) & ({x y : A} → pred x → pred y → pred (x + y)) & ({x : A} → {y : A} → pred x → pred (x * y))
ideal : {c : _} (pred : A → Set c) → Set (a ⊔ b ⊔ c)
ideal pred = subgroup pred && ({x : A} → {y : A} → pred x → pred (x * y))
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
idealPredForKernel {T = T} R2 {f} fHom a = Setoid._∼_ T (f a) (Ring.0R R2)
@@ -29,17 +31,24 @@ idealPredForKernel {T = T} R2 {f} fHom a = Setoid._∼_ T (f a) (Ring.0R R2)
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)
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
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 {pred = idealPredForKernel R2 fHom} (idealPredForKernelWellDefined R2 fHom)
_&_&_.one (kernelIdealIsIdeal fHom) = imageOfIdentityIsIdentity (RingHom.groupHom fHom)
_&_&_.two (kernelIdealIsIdeal {T = T} {R2 = R2} fHom) fx=0 fy=0 = transitive (transitive (GroupHom.groupHom (RingHom.groupHom fHom)) (+WellDefined fx=0 fy=0)) identLeft
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)
_&&_.fst (_&&_.fst (kernelIdealIsIdeal {R2 = R2} fHom)) = idealPredForKernelWellDefined R2 fHom
_&_&_.one (_&&_.snd (_&&_.fst (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
where
open Ring R2
open Group (Ring.additiveGroup R2)
open Setoid T
open Equivalence eq
_&_&_.three (kernelIdealIsIdeal {T = T} {R2 = R2} fHom) fx=0 = transitive (RingHom.ringHom fHom) (transitive (Ring.*WellDefined R2 fx=0 reflexive) (transitive (Ring.*Commutative R2) (Ring.timesZero R2)))
_&_&_.two (_&&_.snd (_&&_.fst (kernelIdealIsIdeal fHom))) = imageOfIdentityIsIdentity (RingHom.groupHom fHom)
_&_&_.three (_&&_.snd (_&&_.fst (kernelIdealIsIdeal {T = T} {R2 = R2} fHom))) {x} fx=0 = zeroImpliesInverseZero (RingHom.groupHom fHom) fx=0
where
open Ring R2
open Group (Ring.additiveGroup R2)
open Setoid T
open Equivalence eq
_&&_.snd (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})))
where
open Setoid T
open Equivalence eq
-- TODO : define the quotient by an ideal; note that the result is a ring