{-# OPTIONS --safe --warning=error --without-K #-}

open import LogicalFormulae
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Rings.Homomorphisms.Definition


module Rings.Homomorphisms.Image {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

open import Groups.Homomorphisms.Image (RingHom.groupHom hom)
open import Rings.Subrings.Definition

imageGroupSubring : Subring R2 imageGroupPred
Subring.isSubgroup imageGroupSubring = imageGroupSubgroup
Subring.containsOne imageGroupSubring = Ring.1R R1 , RingHom.preserves1 hom
Subring.closedUnderProduct imageGroupSubring {x} {y} (a , fa=x) (b , fb=y) = (a *A b) , transitive ringHom (Ring.*WellDefined R2 fa=x fb=y)
  where
    open Setoid T
    open Equivalence eq
    open RingHom hom