{-# 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