# The poset of ideals of a ring

Content created by Egbert Rijke, Fredrik Bakke, Julian KG, Maša Žaucer, fernabnor, Gregor Perčič and louismntnu.

Created on 2023-06-08.

module ring-theory.poset-of-ideals-rings where
Imports
open import foundation.binary-relations
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.powersets
open import foundation.propositions
open import foundation.subtypes
open import foundation.universe-levels

open import order-theory.large-posets
open import order-theory.large-preorders
open import order-theory.order-preserving-maps-large-posets
open import order-theory.order-preserving-maps-large-preorders
open import order-theory.similarity-of-elements-large-posets

open import ring-theory.ideals-rings
open import ring-theory.rings

## Idea

The ideals of a ring form a large poset ordered by inclusion.

## Definition

### The inclusion relation on ideals

module _
{l1 : Level} (R : Ring l1)
where

leq-prop-ideal-Ring :
{l2 l3 : Level}  ideal-Ring l2 R  ideal-Ring l3 R  Prop (l1  l2  l3)
leq-prop-ideal-Ring I J =
leq-prop-subtype
( subset-ideal-Ring R I)
( subset-ideal-Ring R J)

leq-ideal-Ring :
{l2 l3 : Level}  ideal-Ring l2 R  ideal-Ring l3 R  UU (l1  l2  l3)
leq-ideal-Ring I J =
subset-ideal-Ring R I  subset-ideal-Ring R J

is-prop-leq-ideal-Ring :
{l2 l3 : Level} (I : ideal-Ring l2 R) (J : ideal-Ring l3 R)
is-prop (leq-ideal-Ring I J)
is-prop-leq-ideal-Ring I J =
is-prop-leq-subtype (subset-ideal-Ring R I) (subset-ideal-Ring R J)

refl-leq-ideal-Ring :
{l2 : Level}  is-reflexive (leq-ideal-Ring {l2})
refl-leq-ideal-Ring I =
refl-leq-subtype (subset-ideal-Ring R I)

transitive-leq-ideal-Ring :
{l2 l3 l4 : Level}
(I : ideal-Ring l2 R)
(J : ideal-Ring l3 R)
(K : ideal-Ring l4 R)
leq-ideal-Ring J K
leq-ideal-Ring I J
leq-ideal-Ring I K
transitive-leq-ideal-Ring I J K =
transitive-leq-subtype
( subset-ideal-Ring R I)
( subset-ideal-Ring R J)
( subset-ideal-Ring R K)

antisymmetric-leq-ideal-Ring :
{l2 : Level}  is-antisymmetric (leq-ideal-Ring {l2})
antisymmetric-leq-ideal-Ring I J U V =
eq-has-same-elements-ideal-Ring R I J  x  U x , V x)

### The large poset of ideals

module _
{l1 : Level} (R : Ring l1)
where

ideal-Ring-Large-Preorder :
Large-Preorder  l2  l1  lsuc l2)  l2 l3  l1  l2  l3)
type-Large-Preorder ideal-Ring-Large-Preorder l = ideal-Ring l R
leq-prop-Large-Preorder ideal-Ring-Large-Preorder = leq-prop-ideal-Ring R
refl-leq-Large-Preorder ideal-Ring-Large-Preorder = refl-leq-ideal-Ring R
transitive-leq-Large-Preorder ideal-Ring-Large-Preorder =
transitive-leq-ideal-Ring R

ideal-Ring-Large-Poset :
Large-Poset  l2  l1  lsuc l2)  l2 l3  l1  l2  l3)
large-preorder-Large-Poset ideal-Ring-Large-Poset = ideal-Ring-Large-Preorder
antisymmetric-leq-Large-Poset ideal-Ring-Large-Poset =
antisymmetric-leq-ideal-Ring R

### The similarity relation on ideals in a ring

module _
{l1 : Level} (R : Ring l1)
where

sim-prop-ideal-Ring :
{l2 l3 : Level} (I : ideal-Ring l2 R) (J : ideal-Ring l3 R)
Prop (l1  l2  l3)
sim-prop-ideal-Ring =
sim-prop-Large-Poset (ideal-Ring-Large-Poset R)

sim-ideal-Ring :
{l2 l3 : Level} (I : ideal-Ring l2 R) (J : ideal-Ring l3 R)
UU (l1  l2  l3)
sim-ideal-Ring = sim-Large-Poset (ideal-Ring-Large-Poset R)

is-prop-sim-ideal-Ring :
{l2 l3 : Level} (I : ideal-Ring l2 R) (J : ideal-Ring l3 R)
is-prop (sim-ideal-Ring I J)
is-prop-sim-ideal-Ring =
is-prop-sim-Large-Poset (ideal-Ring-Large-Poset R)

eq-sim-ideal-Ring :
{l2 : Level} (I J : ideal-Ring l2 R)  sim-ideal-Ring I J  I  J
eq-sim-ideal-Ring = eq-sim-Large-Poset (ideal-Ring-Large-Poset R)

## Properties

### The forgetful function from ideals to subsets preserves inclusions

module _
{l : Level} (R : Ring l)
where

preserves-order-subset-ideal-Ring :
{l1 l2 : Level} (I : ideal-Ring l1 R) (J : ideal-Ring l2 R)
leq-ideal-Ring R I J  subset-ideal-Ring R I  subset-ideal-Ring R J
preserves-order-subset-ideal-Ring I J H = H

subset-ideal-hom-large-poset-Ring :
hom-Large-Poset
( λ l  l)
( ideal-Ring-Large-Poset R)
( powerset-Large-Poset (type-Ring R))
map-hom-Large-Preorder subset-ideal-hom-large-poset-Ring =
subset-ideal-Ring R
preserves-order-hom-Large-Preorder subset-ideal-hom-large-poset-Ring =
preserves-order-subset-ideal-Ring