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ideals-generated-by-subsets-commutative-rings.lagda.md
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ideals-generated-by-subsets-commutative-rings.lagda.md
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# Ideals generated by subsets of commutative rings
```agda
module commutative-algebra.ideals-generated-by-subsets-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.subsets-commutative-rings
open import foundation.identity-types
open import foundation.powersets
open import foundation.universe-levels
open import lists.concatenation-lists
open import ring-theory.ideals-generated-by-subsets-rings
```
</details>
## Idea
The **ideal generated by a subset** `S` of a commutative ring `R` is the least
ideal that contains `S`.
## Definitions
### The universal property of the ideal generated by a subset
```agda
module _
{l1 l2 l3 : Level} (R : Commutative-Ring l1)
(S : subset-Commutative-Ring l2 R)
(I : ideal-Commutative-Ring l3 R) (H : S ⊆ subset-ideal-Commutative-Ring R I)
where
is-ideal-generated-by-subset-Commutative-Ring :
(l : Level) → UU (l1 ⊔ l2 ⊔ l3 ⊔ lsuc l)
is-ideal-generated-by-subset-Commutative-Ring l =
(J : ideal-Commutative-Ring l R) →
S ⊆ subset-ideal-Commutative-Ring R J →
subset-ideal-Commutative-Ring R I ⊆ subset-ideal-Commutative-Ring R J
```
```agda
module _
{l1 l2 : Level} (R : Commutative-Ring l1) (S : subset-Commutative-Ring l2 R)
where
formal-combination-subset-Commutative-Ring : UU (l1 ⊔ l2)
formal-combination-subset-Commutative-Ring =
formal-combination-subset-Ring (ring-Commutative-Ring R) S
ev-formal-combination-subset-Commutative-Ring :
formal-combination-subset-Commutative-Ring → type-Commutative-Ring R
ev-formal-combination-subset-Commutative-Ring =
ev-formal-combination-subset-Ring (ring-Commutative-Ring R) S
preserves-concat-ev-formal-combination-subset-Commutative-Ring :
(u v : formal-combination-subset-Commutative-Ring) →
Id
( ev-formal-combination-subset-Commutative-Ring (concat-list u v))
( add-Commutative-Ring R
( ev-formal-combination-subset-Commutative-Ring u)
( ev-formal-combination-subset-Commutative-Ring v))
preserves-concat-ev-formal-combination-subset-Commutative-Ring =
preserves-concat-ev-formal-combination-subset-Ring
( ring-Commutative-Ring R)
( S)
mul-formal-combination-subset-Commutative-Ring :
type-Commutative-Ring R →
formal-combination-subset-Commutative-Ring →
formal-combination-subset-Commutative-Ring
mul-formal-combination-subset-Commutative-Ring =
mul-formal-combination-subset-Ring (ring-Commutative-Ring R) S
preserves-mul-ev-formal-combination-subset-Commutative-Ring :
(r : type-Commutative-Ring R)
(u : formal-combination-subset-Commutative-Ring) →
Id
( ev-formal-combination-subset-Commutative-Ring
( mul-formal-combination-subset-Commutative-Ring r u))
( mul-Commutative-Ring R r
( ev-formal-combination-subset-Commutative-Ring u))
preserves-mul-ev-formal-combination-subset-Commutative-Ring =
preserves-mul-ev-formal-combination-subset-Ring
( ring-Commutative-Ring R)
( S)
subset-ideal-subset-Commutative-Ring' : type-Commutative-Ring R → UU (l1 ⊔ l2)
subset-ideal-subset-Commutative-Ring' =
subset-left-ideal-subset-Ring'
( ring-Commutative-Ring R)
( S)
subset-ideal-subset-Commutative-Ring : subset-Commutative-Ring (l1 ⊔ l2) R
subset-ideal-subset-Commutative-Ring =
subset-left-ideal-subset-Ring (ring-Commutative-Ring R) S
contains-zero-ideal-subset-Commutative-Ring :
contains-zero-subset-Commutative-Ring R subset-ideal-subset-Commutative-Ring
contains-zero-ideal-subset-Commutative-Ring =
contains-zero-left-ideal-subset-Ring (ring-Commutative-Ring R) S
is-closed-under-addition-ideal-subset-Commutative-Ring :
is-closed-under-addition-subset-Commutative-Ring R
subset-ideal-subset-Commutative-Ring
is-closed-under-addition-ideal-subset-Commutative-Ring =
is-closed-under-addition-left-ideal-subset-Ring
( ring-Commutative-Ring R)
( S)
is-closed-under-left-multiplication-ideal-subset-Commutative-Ring :
is-closed-under-left-multiplication-subset-Commutative-Ring R
subset-ideal-subset-Commutative-Ring
is-closed-under-left-multiplication-ideal-subset-Commutative-Ring =
is-closed-under-left-multiplication-ideal-subset-Ring
( ring-Commutative-Ring R)
( S)
is-closed-under-negatives-ideal-subset-Commutative-Ring :
is-closed-under-negatives-subset-Commutative-Ring R
subset-ideal-subset-Commutative-Ring
is-closed-under-negatives-ideal-subset-Commutative-Ring =
is-closed-under-negatives-left-ideal-subset-Ring
( ring-Commutative-Ring R)
( S)
ideal-subset-Commutative-Ring : ideal-Commutative-Ring (l1 ⊔ l2) R
ideal-subset-Commutative-Ring =
ideal-left-ideal-Commutative-Ring R
subset-ideal-subset-Commutative-Ring
contains-zero-ideal-subset-Commutative-Ring
is-closed-under-addition-ideal-subset-Commutative-Ring
is-closed-under-negatives-ideal-subset-Commutative-Ring
is-closed-under-left-multiplication-ideal-subset-Commutative-Ring
contains-subset-ideal-subset-Commutative-Ring :
S ⊆ subset-ideal-subset-Commutative-Ring
contains-subset-ideal-subset-Commutative-Ring =
contains-subset-left-ideal-subset-Ring
( ring-Commutative-Ring R)
( S)
contains-formal-combinations-ideal-subset-Commutative-Ring :
{l3 : Level} (I : ideal-Commutative-Ring l3 R) →
S ⊆ subset-ideal-Commutative-Ring R I →
(x : formal-combination-subset-Commutative-Ring) →
is-in-ideal-Commutative-Ring R I
( ev-formal-combination-subset-Commutative-Ring x)
contains-formal-combinations-ideal-subset-Commutative-Ring I =
contains-formal-combinations-left-ideal-subset-Ring
( ring-Commutative-Ring R)
( S)
( left-ideal-ideal-Commutative-Ring R I)
is-ideal-generated-by-subset-ideal-subset-Commutative-Ring :
(l : Level) →
is-ideal-generated-by-subset-Commutative-Ring R S
( ideal-subset-Commutative-Ring)
( contains-subset-ideal-subset-Commutative-Ring)
( l)
is-ideal-generated-by-subset-ideal-subset-Commutative-Ring l I =
is-left-ideal-generated-by-subset-left-ideal-subset-Ring
( ring-Commutative-Ring R)
( S)
( l)
( left-ideal-ideal-Commutative-Ring R I)
```