Polynomial ring (#76)

Rings/Homomorphisms/Definition.agda

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+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+
+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.Homomorphisms.Definition where
+
+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