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

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


module Rings.Homomorphisms.Kernel {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_+1_ _*1_ : A → A → A} {_+2_ _*2_ : B → B → B} {R1 : Ring S _+1_ _*1_} {R2 : Ring T _+2_ _*2_} {f : A → B} (fHom : RingHom R1 R2 f) where

open import Groups.Homomorphisms.Kernel (RingHom.groupHom fHom)

ringKernelIsIdeal : Ideal R1 groupKernelPred
Ideal.isSubgroup ringKernelIsIdeal = groupKernelIsSubgroup
Ideal.accumulatesTimes ringKernelIsIdeal {x} {y} fx=0 = transitive (RingHom.ringHom fHom) (transitive (Ring.*WellDefined R2 fx=0 reflexive) (transitive (Ring.*Commutative R2) (Ring.timesZero R2)))
  where
    open Setoid T
    open Equivalence eq