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full-ideals-commutative-rings.lagda.md
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full-ideals-commutative-rings.lagda.md
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# Full ideals of commutative rings
```agda
module commutative-algebra.full-ideals-commutative-rings where
```
<details><summary>Imports</summary>
```agda
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.radical-ideals-commutative-rings
open import commutative-algebra.subsets-commutative-rings
open import foundation.dependent-pair-types
open import foundation.propositions
open import foundation.unit-type
open import foundation.universe-levels
open import order-theory.top-elements-large-posets
open import ring-theory.full-ideals-rings
```
</details>
## Idea
A **full ideal** in a
[commutative ring](commutative-algebra.commutative-rings.md) `A` is an
[ideal](commutative-algebra.ideals-commutative-rings.md) that contains every
element of `A`.
## Definitions
### The predicate of being a full ideal
```agda
module _
{l1 l2 : Level} (A : Commutative-Ring l1) (I : ideal-Commutative-Ring l2 A)
where
is-full-ideal-Commutative-Ring-Prop : Prop (l1 ⊔ l2)
is-full-ideal-Commutative-Ring-Prop =
is-full-ideal-Ring-Prop (ring-Commutative-Ring A) I
is-full-ideal-Commutative-Ring : UU (l1 ⊔ l2)
is-full-ideal-Commutative-Ring =
is-full-ideal-Ring (ring-Commutative-Ring A) I
is-prop-is-full-ideal-Commutative-Ring :
is-prop is-full-ideal-Commutative-Ring
is-prop-is-full-ideal-Commutative-Ring =
is-prop-is-full-ideal-Ring (ring-Commutative-Ring A) I
```
### The (standard) full ideal
```agda
module _
{l1 : Level} (A : Commutative-Ring l1)
where
subset-full-ideal-Commutative-Ring : subset-Commutative-Ring lzero A
subset-full-ideal-Commutative-Ring =
subset-full-ideal-Ring (ring-Commutative-Ring A)
is-in-full-ideal-Commutative-Ring : type-Commutative-Ring A → UU lzero
is-in-full-ideal-Commutative-Ring =
is-in-full-ideal-Ring (ring-Commutative-Ring A)
contains-zero-full-ideal-Commutative-Ring :
contains-zero-subset-Commutative-Ring A subset-full-ideal-Commutative-Ring
contains-zero-full-ideal-Commutative-Ring =
contains-zero-full-ideal-Ring (ring-Commutative-Ring A)
is-closed-under-addition-full-ideal-Commutative-Ring :
is-closed-under-addition-subset-Commutative-Ring A
subset-full-ideal-Commutative-Ring
is-closed-under-addition-full-ideal-Commutative-Ring {x} {y} =
is-closed-under-addition-full-ideal-Ring (ring-Commutative-Ring A) {x} {y}
is-closed-under-negatives-full-ideal-Commutative-Ring :
is-closed-under-negatives-subset-Commutative-Ring A
subset-full-ideal-Commutative-Ring
is-closed-under-negatives-full-ideal-Commutative-Ring {x} =
is-closed-under-negatives-full-ideal-Ring (ring-Commutative-Ring A) {x}
is-additive-subgroup-full-ideal-Commutative-Ring :
is-additive-subgroup-subset-Commutative-Ring A
subset-full-ideal-Commutative-Ring
is-additive-subgroup-full-ideal-Commutative-Ring =
is-additive-subgroup-full-ideal-Ring (ring-Commutative-Ring A)
is-closed-under-left-multiplication-full-ideal-Commutative-Ring :
is-closed-under-left-multiplication-subset-Commutative-Ring A
subset-full-ideal-Commutative-Ring
is-closed-under-left-multiplication-full-ideal-Commutative-Ring =
is-closed-under-left-multiplication-full-ideal-Ring
( ring-Commutative-Ring A)
is-closed-under-right-multiplication-full-ideal-Commutative-Ring :
is-closed-under-right-multiplication-subset-Commutative-Ring A
subset-full-ideal-Commutative-Ring
is-closed-under-right-multiplication-full-ideal-Commutative-Ring =
is-closed-under-right-multiplication-full-ideal-Ring
( ring-Commutative-Ring A)
is-left-ideal-full-ideal-Commutative-Ring :
is-left-ideal-subset-Commutative-Ring A subset-full-ideal-Commutative-Ring
is-left-ideal-full-ideal-Commutative-Ring =
is-left-ideal-full-ideal-Ring (ring-Commutative-Ring A)
full-left-ideal-Commutative-Ring : left-ideal-Commutative-Ring lzero A
full-left-ideal-Commutative-Ring =
full-left-ideal-Ring (ring-Commutative-Ring A)
is-right-ideal-full-ideal-Commutative-Ring :
is-right-ideal-subset-Commutative-Ring A subset-full-ideal-Commutative-Ring
is-right-ideal-full-ideal-Commutative-Ring =
is-right-ideal-full-ideal-Ring (ring-Commutative-Ring A)
full-right-ideal-Commutative-Ring : right-ideal-Commutative-Ring lzero A
full-right-ideal-Commutative-Ring =
full-right-ideal-Ring (ring-Commutative-Ring A)
is-ideal-full-ideal-Commutative-Ring :
is-ideal-subset-Commutative-Ring A subset-full-ideal-Commutative-Ring
is-ideal-full-ideal-Commutative-Ring =
is-ideal-full-ideal-Ring (ring-Commutative-Ring A)
full-ideal-Commutative-Ring : ideal-Commutative-Ring lzero A
full-ideal-Commutative-Ring = full-ideal-Ring (ring-Commutative-Ring A)
is-full-full-ideal-Commutative-Ring :
is-full-ideal-Commutative-Ring A full-ideal-Commutative-Ring
is-full-full-ideal-Commutative-Ring =
is-full-full-ideal-Ring (ring-Commutative-Ring A)
```
## Properties
### Any ideal is full if and only if it contains `1`
```agda
module _
{l1 l2 : Level} (A : Commutative-Ring l1) (I : ideal-Commutative-Ring l2 A)
where
is-full-contains-one-ideal-Commutative-Ring :
is-in-ideal-Commutative-Ring A I (one-Commutative-Ring A) →
is-full-ideal-Commutative-Ring A I
is-full-contains-one-ideal-Commutative-Ring =
is-full-contains-one-ideal-Ring (ring-Commutative-Ring A) I
contains-one-is-full-ideal-Commutative-Ring :
is-full-ideal-Commutative-Ring A I →
is-in-ideal-Commutative-Ring A I (one-Commutative-Ring A)
contains-one-is-full-ideal-Commutative-Ring =
contains-one-is-full-ideal-Ring (ring-Commutative-Ring A) I
```
### Any ideal is full if and only if it is a top element in the large poset of ideals
```agda
module _
{l1 l2 : Level} (A : Commutative-Ring l1) (I : ideal-Commutative-Ring l2 A)
where
is-full-is-top-element-ideal-Commutative-Ring :
is-top-element-Large-Poset (ideal-Commutative-Ring-Large-Poset A) I →
is-full-ideal-Commutative-Ring A I
is-full-is-top-element-ideal-Commutative-Ring =
is-full-is-top-element-ideal-Ring (ring-Commutative-Ring A) I
is-top-element-is-full-ideal-Commutative-Ring :
is-full-ideal-Commutative-Ring A I →
is-top-element-Large-Poset (ideal-Commutative-Ring-Large-Poset A) I
is-top-element-is-full-ideal-Commutative-Ring =
is-top-element-is-full-ideal-Ring (ring-Commutative-Ring A) I
module _
{l1 : Level} (A : Commutative-Ring l1)
where
is-top-element-full-ideal-Commutative-Ring :
is-top-element-Large-Poset
( ideal-Commutative-Ring-Large-Poset A)
( full-ideal-Commutative-Ring A)
is-top-element-full-ideal-Commutative-Ring =
is-top-element-full-ideal-Ring (ring-Commutative-Ring A)
has-top-element-ideal-Commutative-Ring :
has-top-element-Large-Poset (ideal-Commutative-Ring-Large-Poset A)
has-top-element-ideal-Commutative-Ring =
has-top-element-ideal-Ring (ring-Commutative-Ring A)
```
### The full ideal of a commutative ring is radical
```agda
module _
{l1 : Level} (A : Commutative-Ring l1)
where
is-radical-full-ideal-Commutative-Ring :
is-radical-ideal-Commutative-Ring A (full-ideal-Commutative-Ring A)
is-radical-full-ideal-Commutative-Ring x n H = raise-star
full-radical-ideal-Commutative-Ring : radical-ideal-Commutative-Ring lzero A
pr1 full-radical-ideal-Commutative-Ring =
full-ideal-Commutative-Ring A
pr2 full-radical-ideal-Commutative-Ring =
is-radical-full-ideal-Commutative-Ring
is-top-element-full-radical-ideal-Commutative-Ring :
is-top-element-Large-Poset
( radical-ideal-Commutative-Ring-Large-Poset A)
( full-radical-ideal-Commutative-Ring)
is-top-element-full-radical-ideal-Commutative-Ring I =
is-top-element-full-ideal-Commutative-Ring A
( ideal-radical-ideal-Commutative-Ring A I)
has-top-element-radical-ideal-Commutative-Ring :
has-top-element-Large-Poset
( radical-ideal-Commutative-Ring-Large-Poset A)
top-has-top-element-Large-Poset
has-top-element-radical-ideal-Commutative-Ring =
full-radical-ideal-Commutative-Ring
is-top-element-top-has-top-element-Large-Poset
has-top-element-radical-ideal-Commutative-Ring =
is-top-element-full-radical-ideal-Commutative-Ring
```