Polynomial ring (#76)

Rings/Definition.agda

@@ -14,49 +14,26 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 
 -- Following Part IB's course Groups, Rings, and Modules, we take rings to be commutative with one.
 module Rings.Definition where
-  record Ring {n m} {A : Set n} (S : Setoid {n} {m} A) (_+_ : A → A → A) (_*_ : A → A → A) : Set (lsuc n ⊔ m) where
-    field
-      additiveGroup : Group S _+_
-    open Group additiveGroup
-    open Setoid S
-    open Equivalence eq
-    0R : A
-    0R = 0G
-    _-R_ : A → A → A
-    a -R b = a + (inverse b)
-    field
-      *WellDefined : {r s t u : A} → (r ∼ t) → (s ∼ u) → r * s ∼ t * u
-      1R : A
-      groupIsAbelian : {a b : A} → a + b ∼ b + a
-      *Associative : {a b c : A} → (a * (b * c)) ∼ (a * b) * c
-      *Commutative : {a b : A} → a * b ∼ b * a
-      *DistributesOver+ : {a b c : A} → a * (b + c) ∼ (a * b) + (a * c)
-      identIsIdent : {a : A} → 1R * a ∼ a
-    timesZero : {a : A} → a * 0R ∼ 0R
-    timesZero {a} = symmetric (transitive (transitive (symmetric invLeft) (+WellDefined reflexive (transitive (*WellDefined {a} {a} reflexive (symmetric identRight)) *DistributesOver+))) (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft)))
 
-
-  --directSumRing : {m n : _} → {A : Set m} {B : Set n} (r : Ring A) (s : Ring B) → Ring (A && B)
-  --Ring.additiveGroup (directSumRing r s) = directSumGroup (Ring.additiveGroup r) (Ring.additiveGroup s)
-  --Ring._*_ (directSumRing r s) (a ,, b) (c ,, d) = (Ring._*_ r a c) ,, Ring._*_ s b d
-  --Ring.multWellDefined (directSumRing r s) (a ,, b) (c ,, d) = Ring.multWellDefined r a c ,, Ring.multWellDefined s b d
-  --Ring.1R (directSumRing r s) = Ring.1R r ,, Ring.1R s
-  --Ring.groupIsAbelian (directSumRing r s) = Ring.groupIsAbelian r ,, Ring.groupIsAbelian s
-  --Ring.assoc (directSumRing r s) = Ring.assoc r ,, Ring.assoc s
-  --Ring.multCommutative (directSumRing r s) = Ring.multCommutative r ,, Ring.multCommutative s
-  --Ring.multDistributes (directSumRing r s) = Ring.multDistributes r ,, Ring.multDistributes s
-  --Ring.identIsIdent (directSumRing r s) = Ring.identIsIdent r ,, Ring.identIsIdent s
-
-  record RingHom {m n o p : _} {A : Set m} {B : Set n} {SA : Setoid {m} {o} A} {SB : Setoid {n} {p} B} {_+A_ : A → A → A} {_*A_ : A → A → A} (R : Ring SA _+A_ _*A_) {_+B_ : B → B → B} {_*B_ : B → B → B} (S : Ring SB _+B_ _*B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
-    open Ring S
-    open Group additiveGroup
-    open Setoid SB
-    field
-      preserves1 : f (Ring.1R R) ∼ 1R
-      ringHom : {r s : A} → f (r *A s) ∼ (f r) *B (f s)
-      groupHom : GroupHom (Ring.additiveGroup R) additiveGroup f
-
-  --record RingIso {m n : _} {A : Set m} {B : Set n} (R : Ring A) (S : Ring B) (f : A → B) : Set (m ⊔ n) where
-  --  field
-  --    ringHom : RingHom R S f
-  --    bijective : Bijection f
+record Ring {n m} {A : Set n} (S : Setoid {n} {m} A) (_+_ : A → A → A) (_*_ : A → A → A) : Set (lsuc n ⊔ m) where
+  field
+    additiveGroup : Group S _+_
+  open Group additiveGroup
+  open Setoid S
+  open Equivalence eq
+  0R : A
+  0R = 0G
+  _-R_ : A → A → A
+  a -R b = a + (inverse b)
+  field
+    *WellDefined : {r s t u : A} → (r ∼ t) → (s ∼ u) → r * s ∼ t * u
+    1R : A
+    groupIsAbelian : {a b : A} → a + b ∼ b + a
+    *Associative : {a b c : A} → (a * (b * c)) ∼ (a * b) * c
+    *Commutative : {a b : A} → a * b ∼ b * a
+    *DistributesOver+ : {a b c : A} → a * (b + c) ∼ (a * b) + (a * c)
+    identIsIdent : {a : A} → 1R * a ∼ a
+  timesZero : {a : A} → a * 0R ∼ 0R
+  timesZero {a} = symmetric (transitive (transitive (symmetric invLeft) (+WellDefined reflexive (transitive (*WellDefined {a} {a} reflexive (symmetric identRight)) *DistributesOver+))) (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft)))
+  *DistributesOver+' : {a b c : A} → (a + b) * c ∼ (a * c) + (b * c)
+  *DistributesOver+' = transitive *Commutative (transitive *DistributesOver+ (+WellDefined *Commutative *Commutative))