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https://github.com/Smaug123/agdaproofs
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20 lines
912 B
Agda
20 lines
912 B
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import Setoids.Setoids
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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open import Rings.Homomorphisms.Definition
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open import Rings.Ideals.Definition
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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
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open import Groups.Homomorphisms.Kernel (RingHom.groupHom fHom)
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ringKernelIsIdeal : Ideal R1 groupKernelPred
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Ideal.isSubgroup ringKernelIsIdeal = groupKernelIsSubgroup
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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)))
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where
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open Setoid T
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open Equivalence eq
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