Radicals of ideals of commutative rings

Content created by Egbert Rijke, Fredrik Bakke and Maša Žaucer.

Created on 2023-03-19.
Last modified on 2023-11-24.

module commutative-algebra.radicals-of-ideals-commutative-rings where
Imports
open import commutative-algebra.binomial-theorem-commutative-rings
open import commutative-algebra.commutative-rings
open import commutative-algebra.ideals-commutative-rings
open import commutative-algebra.poset-of-ideals-commutative-rings
open import commutative-algebra.poset-of-radical-ideals-commutative-rings
open import commutative-algebra.powers-of-elements-commutative-rings
open import commutative-algebra.radical-ideals-commutative-rings
open import commutative-algebra.subsets-commutative-rings

open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.multiplication-natural-numbers
open import elementary-number-theory.natural-numbers

open import foundation.dependent-pair-types
open import foundation.existential-quantification
open import foundation.logical-equivalences
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.subtypes
open import foundation.universe-levels

open import order-theory.galois-connections-large-posets
open import order-theory.order-preserving-maps-large-posets
open import order-theory.order-preserving-maps-large-preorders
open import order-theory.reflective-galois-connections-large-posets

Idea

The radical √ I of an ideal I in a commutative ring A is the least radical ideal of A containing I. The radical √ I is constructed as the ideal consisting of all elements f for which there exists an n such that fⁿ ∈ I.

The operation I ↦ √ I is a reflective Galois connection from the large poset of ideals in A into the large poset of radical ideals in A.

Definitions

The condition of being the radical of an ideal

module _
  {l1 l2 l3 : Level} (A : Commutative-Ring l1)
  (I : ideal-Commutative-Ring l2 A)
  (J : radical-ideal-Commutative-Ring l3 A)
  (H :
    leq-ideal-Commutative-Ring A I (ideal-radical-ideal-Commutative-Ring A J))
  where

  is-radical-of-ideal-Commutative-Ring : UUω
  is-radical-of-ideal-Commutative-Ring =
    {l4 : Level} (K : radical-ideal-Commutative-Ring l4 A) 
    leq-ideal-Commutative-Ring A I (ideal-radical-ideal-Commutative-Ring A K) 
    leq-ideal-Commutative-Ring A
      ( ideal-radical-ideal-Commutative-Ring A J)
      ( ideal-radical-ideal-Commutative-Ring A K)

The radical Galois connection on ideals of a commutative ring

The radical of an ideal

module _
  {l1 l2 : Level} (A : Commutative-Ring l1) (I : ideal-Commutative-Ring l2 A)
  where

  subset-radical-of-ideal-Commutative-Ring : type-Commutative-Ring A  Prop l2
  subset-radical-of-ideal-Commutative-Ring f =
    ∃-Prop 
      ( λ n  is-in-ideal-Commutative-Ring A I (power-Commutative-Ring A n f))

  is-in-radical-of-ideal-Commutative-Ring : type-Commutative-Ring A  UU l2
  is-in-radical-of-ideal-Commutative-Ring =
    is-in-subtype subset-radical-of-ideal-Commutative-Ring

  contains-ideal-radical-of-ideal-Commutative-Ring :
    (f : type-Commutative-Ring A) 
    is-in-ideal-Commutative-Ring A I f 
    is-in-radical-of-ideal-Commutative-Ring f
  contains-ideal-radical-of-ideal-Commutative-Ring f H = intro-∃ 1 H

  contains-zero-radical-of-ideal-Commutative-Ring :
    is-in-radical-of-ideal-Commutative-Ring (zero-Commutative-Ring A)
  contains-zero-radical-of-ideal-Commutative-Ring =
    contains-ideal-radical-of-ideal-Commutative-Ring
      ( zero-Commutative-Ring A)
      ( contains-zero-ideal-Commutative-Ring A I)

  is-closed-under-addition-radical-of-ideal-Commutative-Ring :
    is-closed-under-addition-subset-Commutative-Ring A
      ( subset-radical-of-ideal-Commutative-Ring)
  is-closed-under-addition-radical-of-ideal-Commutative-Ring {x} {y} H K =
    apply-universal-property-trunc-Prop H
      ( subset-radical-of-ideal-Commutative-Ring (add-Commutative-Ring A x y))
      ( λ (n , p) 
        apply-universal-property-trunc-Prop K
          ( subset-radical-of-ideal-Commutative-Ring
            ( add-Commutative-Ring A x y))
          ( λ (m , q) 
            intro-∃
              ( n +ℕ m)
              ( is-closed-under-eq-ideal-Commutative-Ring' A I
                ( is-closed-under-addition-ideal-Commutative-Ring A I
                  ( is-closed-under-right-multiplication-ideal-Commutative-Ring
                    ( A)
                    ( I)
                    ( _)
                    ( _)
                    ( q))
                  ( is-closed-under-right-multiplication-ideal-Commutative-Ring
                    ( A)
                    ( I)
                    ( _)
                    ( _)
                    ( p)))
                ( is-linear-combination-power-add-Commutative-Ring A n m x y))))

  is-closed-under-negatives-radical-of-ideal-Commutative-Ring :
    is-closed-under-negatives-subset-Commutative-Ring A
      ( subset-radical-of-ideal-Commutative-Ring)
  is-closed-under-negatives-radical-of-ideal-Commutative-Ring {x} H =
    apply-universal-property-trunc-Prop H
      ( subset-radical-of-ideal-Commutative-Ring (neg-Commutative-Ring A x))
      ( λ (n , p) 
        intro-∃ n
          ( is-closed-under-eq-ideal-Commutative-Ring' A I
            ( is-closed-under-left-multiplication-ideal-Commutative-Ring A I _
              ( power-Commutative-Ring A n x)
              ( p))
            ( power-neg-Commutative-Ring A n x)))

  is-closed-under-right-multiplication-radical-of-ideal-Commutative-Ring :
    is-closed-under-right-multiplication-subset-Commutative-Ring A
      ( subset-radical-of-ideal-Commutative-Ring)
  is-closed-under-right-multiplication-radical-of-ideal-Commutative-Ring x y H =
    apply-universal-property-trunc-Prop H
      ( subset-radical-of-ideal-Commutative-Ring (mul-Commutative-Ring A x y))
      ( λ (n , p) 
        intro-∃ n
          ( is-closed-under-eq-ideal-Commutative-Ring' A I
            ( is-closed-under-right-multiplication-ideal-Commutative-Ring A I
              ( power-Commutative-Ring A n x)
              ( power-Commutative-Ring A n y)
              ( p))
            ( distributive-power-mul-Commutative-Ring A n x y)))

  is-closed-under-left-multiplication-radical-of-ideal-Commutative-Ring :
    is-closed-under-left-multiplication-subset-Commutative-Ring A
      ( subset-radical-of-ideal-Commutative-Ring)
  is-closed-under-left-multiplication-radical-of-ideal-Commutative-Ring x y H =
    apply-universal-property-trunc-Prop H
      ( subset-radical-of-ideal-Commutative-Ring (mul-Commutative-Ring A x y))
      ( λ (n , p) 
        intro-∃ n
          ( is-closed-under-eq-ideal-Commutative-Ring' A I
            ( is-closed-under-left-multiplication-ideal-Commutative-Ring A I
              ( power-Commutative-Ring A n x)
              ( power-Commutative-Ring A n y)
              ( p))
            ( distributive-power-mul-Commutative-Ring A n x y)))

  ideal-radical-of-ideal-Commutative-Ring : ideal-Commutative-Ring l2 A
  ideal-radical-of-ideal-Commutative-Ring =
    ideal-right-ideal-Commutative-Ring A
      subset-radical-of-ideal-Commutative-Ring
      contains-zero-radical-of-ideal-Commutative-Ring
      is-closed-under-addition-radical-of-ideal-Commutative-Ring
      is-closed-under-negatives-radical-of-ideal-Commutative-Ring
      is-closed-under-right-multiplication-radical-of-ideal-Commutative-Ring

  is-radical-radical-of-ideal-Commutative-Ring :
    is-radical-ideal-Commutative-Ring A ideal-radical-of-ideal-Commutative-Ring
  is-radical-radical-of-ideal-Commutative-Ring x n H =
    apply-universal-property-trunc-Prop H
      ( subset-radical-of-ideal-Commutative-Ring x)
      ( λ (m , K) 
        intro-∃
          ( mul-ℕ n m)
          ( is-closed-under-eq-ideal-Commutative-Ring' A I K
            ( power-mul-Commutative-Ring A n m)))

  radical-of-ideal-Commutative-Ring :
    radical-ideal-Commutative-Ring l2 A
  pr1 radical-of-ideal-Commutative-Ring =
    ideal-radical-of-ideal-Commutative-Ring
  pr2 radical-of-ideal-Commutative-Ring =
    is-radical-radical-of-ideal-Commutative-Ring

  is-radical-of-ideal-radical-of-ideal-Commutative-Ring :
    is-radical-of-ideal-Commutative-Ring A I
      ( radical-of-ideal-Commutative-Ring)
      ( contains-ideal-radical-of-ideal-Commutative-Ring)
  is-radical-of-ideal-radical-of-ideal-Commutative-Ring J H x K =
    apply-universal-property-trunc-Prop K
      ( subset-radical-ideal-Commutative-Ring A J x)
      ( λ (n , L) 
        is-radical-radical-ideal-Commutative-Ring A J x n
          ( H (power-Commutative-Ring A n x) L))

The operation I ↦ √ I as an order preserving map

module _
  {l1 : Level} (A : Commutative-Ring l1)
  where

  preserves-order-radical-of-ideal-Commutative-Ring :
    {l2 l3 : Level}
    (I : ideal-Commutative-Ring l2 A) (J : ideal-Commutative-Ring l3 A) 
    leq-ideal-Commutative-Ring A I J 
    leq-radical-ideal-Commutative-Ring A
      ( radical-of-ideal-Commutative-Ring A I)
      ( radical-of-ideal-Commutative-Ring A J)
  preserves-order-radical-of-ideal-Commutative-Ring I J H =
    is-radical-of-ideal-radical-of-ideal-Commutative-Ring A I
      ( radical-of-ideal-Commutative-Ring A J)
      ( transitive-leq-ideal-Commutative-Ring A I J
        ( ideal-radical-of-ideal-Commutative-Ring A J)
        ( contains-ideal-radical-of-ideal-Commutative-Ring A J)
        ( H))

  radical-of-ideal-hom-large-poset-Commutative-Ring :
    hom-Large-Poset
      ( λ l  l)
      ( ideal-Commutative-Ring-Large-Poset A)
      ( radical-ideal-Commutative-Ring-Large-Poset A)
  map-hom-Large-Preorder
    radical-of-ideal-hom-large-poset-Commutative-Ring =
    radical-of-ideal-Commutative-Ring A
  preserves-order-hom-Large-Preorder
    radical-of-ideal-hom-large-poset-Commutative-Ring =
    preserves-order-radical-of-ideal-Commutative-Ring

The radical Galois connection

module _
  {l1 : Level} (A : Commutative-Ring l1)
  where

  adjoint-relation-radical-of-ideal-Commutative-Ring :
    {l2 l3 : Level}
    (I : ideal-Commutative-Ring l2 A)
    (J : radical-ideal-Commutative-Ring l3 A) 
    leq-radical-ideal-Commutative-Ring A
      ( radical-of-ideal-Commutative-Ring A I)
      ( J) 
    leq-ideal-Commutative-Ring A
      ( I)
      ( ideal-radical-ideal-Commutative-Ring A J)
  pr1 (adjoint-relation-radical-of-ideal-Commutative-Ring I J) H =
    transitive-leq-ideal-Commutative-Ring A I
      ( ideal-radical-of-ideal-Commutative-Ring A I)
      ( ideal-radical-ideal-Commutative-Ring A J)
      ( H)
      ( contains-ideal-radical-of-ideal-Commutative-Ring A I)
  pr2 (adjoint-relation-radical-of-ideal-Commutative-Ring I J) =
    is-radical-of-ideal-radical-of-ideal-Commutative-Ring A I J

  radical-of-ideal-galois-connection-Commutative-Ring :
    galois-connection-Large-Poset  l  l)  l  l)
      ( ideal-Commutative-Ring-Large-Poset A)
      ( radical-ideal-Commutative-Ring-Large-Poset A)
  lower-adjoint-galois-connection-Large-Poset
    radical-of-ideal-galois-connection-Commutative-Ring =
    radical-of-ideal-hom-large-poset-Commutative-Ring A
  upper-adjoint-galois-connection-Large-Poset
    radical-of-ideal-galois-connection-Commutative-Ring =
    ideal-radical-ideal-hom-large-poset-Commutative-Ring A
  adjoint-relation-galois-connection-Large-Poset
    radical-of-ideal-galois-connection-Commutative-Ring =
    adjoint-relation-radical-of-ideal-Commutative-Ring

The radical reflective Galois connection

module _
  {l1 : Level} (A : Commutative-Ring l1)
  where

  is-reflective-radical-of-ideal-Commutative-Ring :
    is-reflective-galois-connection-Large-Poset
      ( ideal-Commutative-Ring-Large-Poset A)
      ( radical-ideal-Commutative-Ring-Large-Poset A)
      ( radical-of-ideal-galois-connection-Commutative-Ring A)
  pr1 (is-reflective-radical-of-ideal-Commutative-Ring I) =
    is-radical-of-ideal-radical-of-ideal-Commutative-Ring A
      ( ideal-radical-ideal-Commutative-Ring A I)
      ( I)
      ( refl-leq-radical-ideal-Commutative-Ring A I)
  pr2 (is-reflective-radical-of-ideal-Commutative-Ring I) =
    contains-ideal-radical-of-ideal-Commutative-Ring A
      ( ideal-radical-ideal-Commutative-Ring A I)

  radical-of-ideal-reflective-galois-connection-Commutative-Ring :
    reflective-galois-connection-Large-Poset
      ( ideal-Commutative-Ring-Large-Poset A)
      ( radical-ideal-Commutative-Ring-Large-Poset A)
  galois-connection-reflective-galois-connection-Large-Poset
    radical-of-ideal-reflective-galois-connection-Commutative-Ring =
    radical-of-ideal-galois-connection-Commutative-Ring A
  is-reflective-reflective-galois-connection-Large-Poset
    radical-of-ideal-reflective-galois-connection-Commutative-Ring =
    is-reflective-radical-of-ideal-Commutative-Ring

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