Ideals of semirings

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

Created on 2023-03-18.
Last modified on 2023-11-24.

module ring-theory.ideals-semirings where
Imports
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.propositions
open import foundation.universe-levels

open import group-theory.submonoids

open import ring-theory.semirings
open import ring-theory.subsets-semirings

Idea

An ideal (resp. a left/right ideal) of a semiring R is an additive submodule of R that is closed under multiplication by elements of R (from the left/right).

Note

This is the standard definition of ideals in semirings. However, such two-sided ideals do not correspond uniquely to congruences on R. If we ask in addition that the underlying additive submodule is normal, then we get unique correspondence to congruences. We will call such ideals normal.

Definitions

Additive submonoids

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

  is-additive-submonoid-Semiring :
    {l2 : Level}  subset-Semiring l2 R  UU (l1  l2)
  is-additive-submonoid-Semiring =
    is-submonoid-subset-Monoid (additive-monoid-Semiring R)

Left ideals

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

  is-left-ideal-subset-Semiring :
    {l2 : Level}  subset-Semiring l2 R  UU (l1  l2)
  is-left-ideal-subset-Semiring P =
    is-additive-submonoid-Semiring R P ×
    is-closed-under-left-multiplication-subset-Semiring R P

left-ideal-Semiring :
  (l : Level) {l1 : Level} (R : Semiring l1)  UU ((lsuc l)  l1)
left-ideal-Semiring l R =
  Σ (subset-Semiring l R) (is-left-ideal-subset-Semiring R)

module _
  {l1 l2 : Level} (R : Semiring l1) (I : left-ideal-Semiring l2 R)
  where

  subset-left-ideal-Semiring : subset-Semiring l2 R
  subset-left-ideal-Semiring = pr1 I

  is-in-left-ideal-Semiring : type-Semiring R  UU l2
  is-in-left-ideal-Semiring x = type-Prop (subset-left-ideal-Semiring x)

  type-left-ideal-Semiring : UU (l1  l2)
  type-left-ideal-Semiring = type-subset-Semiring R subset-left-ideal-Semiring

  inclusion-left-ideal-Semiring : type-left-ideal-Semiring  type-Semiring R
  inclusion-left-ideal-Semiring =
    inclusion-subset-Semiring R subset-left-ideal-Semiring

  ap-inclusion-left-ideal-Semiring :
    (x y : type-left-ideal-Semiring)  x  y 
    inclusion-left-ideal-Semiring x  inclusion-left-ideal-Semiring y
  ap-inclusion-left-ideal-Semiring =
    ap-inclusion-subset-Semiring R subset-left-ideal-Semiring

  is-in-subset-inclusion-left-ideal-Semiring :
    (x : type-left-ideal-Semiring) 
    is-in-left-ideal-Semiring (inclusion-left-ideal-Semiring x)
  is-in-subset-inclusion-left-ideal-Semiring =
    is-in-subset-inclusion-subset-Semiring R subset-left-ideal-Semiring

  is-closed-under-eq-left-ideal-Semiring :
    {x y : type-Semiring R}  is-in-left-ideal-Semiring x 
    (x  y)  is-in-left-ideal-Semiring y
  is-closed-under-eq-left-ideal-Semiring =
    is-closed-under-eq-subset-Semiring R subset-left-ideal-Semiring

  is-closed-under-eq-left-ideal-Semiring' :
    {x y : type-Semiring R}  is-in-left-ideal-Semiring y 
    (x  y)  is-in-left-ideal-Semiring x
  is-closed-under-eq-left-ideal-Semiring' =
    is-closed-under-eq-subset-Semiring' R subset-left-ideal-Semiring

  is-left-ideal-subset-left-ideal-Semiring :
    is-left-ideal-subset-Semiring R subset-left-ideal-Semiring
  is-left-ideal-subset-left-ideal-Semiring = pr2 I

  is-additive-submonoid-left-ideal-Semiring :
    is-additive-submonoid-Semiring R subset-left-ideal-Semiring
  is-additive-submonoid-left-ideal-Semiring =
    pr1 is-left-ideal-subset-left-ideal-Semiring

  contains-zero-left-ideal-Semiring :
    is-in-left-ideal-Semiring (zero-Semiring R)
  contains-zero-left-ideal-Semiring =
    pr1 is-additive-submonoid-left-ideal-Semiring

  is-closed-under-addition-left-ideal-Semiring :
    is-closed-under-addition-subset-Semiring R subset-left-ideal-Semiring
  is-closed-under-addition-left-ideal-Semiring =
    pr2 is-additive-submonoid-left-ideal-Semiring

  is-closed-under-left-multiplication-left-ideal-Semiring :
    is-closed-under-left-multiplication-subset-Semiring R
      subset-left-ideal-Semiring
  is-closed-under-left-multiplication-left-ideal-Semiring =
    pr2 is-left-ideal-subset-left-ideal-Semiring

Right ideals

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

  is-right-ideal-subset-Semiring :
    {l2 : Level}  subset-Semiring l2 R  UU (l1  l2)
  is-right-ideal-subset-Semiring P =
    is-additive-submonoid-Semiring R P ×
    is-closed-under-right-multiplication-subset-Semiring R P

right-ideal-Semiring :
  (l : Level) {l1 : Level} (R : Semiring l1)  UU ((lsuc l)  l1)
right-ideal-Semiring l R =
  Σ (subset-Semiring l R) (is-right-ideal-subset-Semiring R)

module _
  {l1 l2 : Level} (R : Semiring l1) (I : right-ideal-Semiring l2 R)
  where

  subset-right-ideal-Semiring : subset-Semiring l2 R
  subset-right-ideal-Semiring = pr1 I

  is-in-right-ideal-Semiring : type-Semiring R  UU l2
  is-in-right-ideal-Semiring x = type-Prop (subset-right-ideal-Semiring x)

  type-right-ideal-Semiring : UU (l1  l2)
  type-right-ideal-Semiring =
    type-subset-Semiring R subset-right-ideal-Semiring

  inclusion-right-ideal-Semiring : type-right-ideal-Semiring  type-Semiring R
  inclusion-right-ideal-Semiring =
    inclusion-subset-Semiring R subset-right-ideal-Semiring

  ap-inclusion-right-ideal-Semiring :
    (x y : type-right-ideal-Semiring)  x  y 
    inclusion-right-ideal-Semiring x  inclusion-right-ideal-Semiring y
  ap-inclusion-right-ideal-Semiring =
    ap-inclusion-subset-Semiring R subset-right-ideal-Semiring

  is-in-subset-inclusion-right-ideal-Semiring :
    (x : type-right-ideal-Semiring) 
    is-in-right-ideal-Semiring (inclusion-right-ideal-Semiring x)
  is-in-subset-inclusion-right-ideal-Semiring =
    is-in-subset-inclusion-subset-Semiring R subset-right-ideal-Semiring

  is-closed-under-eq-right-ideal-Semiring :
    {x y : type-Semiring R}  is-in-right-ideal-Semiring x 
    (x  y)  is-in-right-ideal-Semiring y
  is-closed-under-eq-right-ideal-Semiring =
    is-closed-under-eq-subset-Semiring R subset-right-ideal-Semiring

  is-closed-under-eq-right-ideal-Semiring' :
    {x y : type-Semiring R}  is-in-right-ideal-Semiring y 
    (x  y)  is-in-right-ideal-Semiring x
  is-closed-under-eq-right-ideal-Semiring' =
    is-closed-under-eq-subset-Semiring' R subset-right-ideal-Semiring

  is-right-ideal-subset-right-ideal-Semiring :
    is-right-ideal-subset-Semiring R subset-right-ideal-Semiring
  is-right-ideal-subset-right-ideal-Semiring = pr2 I

  is-additive-submonoid-right-ideal-Semiring :
    is-additive-submonoid-Semiring R subset-right-ideal-Semiring
  is-additive-submonoid-right-ideal-Semiring =
    pr1 is-right-ideal-subset-right-ideal-Semiring

  contains-zero-right-ideal-Semiring :
    is-in-right-ideal-Semiring (zero-Semiring R)
  contains-zero-right-ideal-Semiring =
    pr1 is-additive-submonoid-right-ideal-Semiring

  is-closed-under-addition-right-ideal-Semiring :
    is-closed-under-addition-subset-Semiring R subset-right-ideal-Semiring
  is-closed-under-addition-right-ideal-Semiring =
    pr2 is-additive-submonoid-right-ideal-Semiring

  is-closed-under-right-multiplication-right-ideal-Semiring :
    is-closed-under-right-multiplication-subset-Semiring R
      subset-right-ideal-Semiring
  is-closed-under-right-multiplication-right-ideal-Semiring =
    pr2 is-right-ideal-subset-right-ideal-Semiring

Two-sided ideals

is-ideal-subset-Semiring :
  {l1 l2 : Level} (R : Semiring l1) (P : subset-Semiring l2 R)  UU (l1  l2)
is-ideal-subset-Semiring R P =
  is-additive-submonoid-Semiring R P ×
  ( is-closed-under-left-multiplication-subset-Semiring R P ×
    is-closed-under-right-multiplication-subset-Semiring R P)

ideal-Semiring :
  (l : Level) {l1 : Level} (R : Semiring l1)  UU ((lsuc l)  l1)
ideal-Semiring l R =
  Σ (subset-Semiring l R) (is-ideal-subset-Semiring R)

module _
  {l1 l2 : Level} (R : Semiring l1) (I : ideal-Semiring l2 R)
  where

  subset-ideal-Semiring : subset-Semiring l2 R
  subset-ideal-Semiring = pr1 I

  is-in-ideal-Semiring : type-Semiring R  UU l2
  is-in-ideal-Semiring =
    is-in-subset-Semiring R subset-ideal-Semiring

  is-prop-is-in-ideal-Semiring :
    (x : type-Semiring R)  is-prop (is-in-ideal-Semiring x)
  is-prop-is-in-ideal-Semiring =
    is-prop-is-in-subset-Semiring R subset-ideal-Semiring

  type-ideal-Semiring : UU (l1  l2)
  type-ideal-Semiring =
    type-subset-Semiring R subset-ideal-Semiring

  inclusion-ideal-Semiring :
    type-ideal-Semiring  type-Semiring R
  inclusion-ideal-Semiring =
    inclusion-subset-Semiring R subset-ideal-Semiring

  ap-inclusion-ideal-Semiring :
    (x y : type-ideal-Semiring)  x  y 
    inclusion-ideal-Semiring x  inclusion-ideal-Semiring y
  ap-inclusion-ideal-Semiring =
    ap-inclusion-subset-Semiring R subset-ideal-Semiring

  is-in-subset-inclusion-ideal-Semiring :
    (x : type-ideal-Semiring) 
    is-in-ideal-Semiring (inclusion-ideal-Semiring x)
  is-in-subset-inclusion-ideal-Semiring =
    is-in-subset-inclusion-subset-Semiring R subset-ideal-Semiring

  is-closed-under-eq-ideal-Semiring :
    {x y : type-Semiring R}  is-in-ideal-Semiring x 
    (x  y)  is-in-ideal-Semiring y
  is-closed-under-eq-ideal-Semiring =
    is-closed-under-eq-subset-Semiring R subset-ideal-Semiring

  is-closed-under-eq-ideal-Semiring' :
    {x y : type-Semiring R}  is-in-ideal-Semiring y 
    (x  y)  is-in-ideal-Semiring x
  is-closed-under-eq-ideal-Semiring' =
    is-closed-under-eq-subset-Semiring' R subset-ideal-Semiring

  is-ideal-subset-ideal-Semiring :
    is-ideal-subset-Semiring R subset-ideal-Semiring
  is-ideal-subset-ideal-Semiring = pr2 I

  is-additive-submonoid-ideal-Semiring :
    is-additive-submonoid-Semiring R subset-ideal-Semiring
  is-additive-submonoid-ideal-Semiring =
    pr1 is-ideal-subset-ideal-Semiring

  contains-zero-ideal-Semiring :
    is-in-ideal-Semiring (zero-Semiring R)
  contains-zero-ideal-Semiring =
    pr1 is-additive-submonoid-ideal-Semiring

  is-closed-under-addition-ideal-Semiring :
    is-closed-under-addition-subset-Semiring R subset-ideal-Semiring
  is-closed-under-addition-ideal-Semiring =
    pr2 is-additive-submonoid-ideal-Semiring

  is-closed-under-left-multiplication-ideal-Semiring :
    is-closed-under-left-multiplication-subset-Semiring R subset-ideal-Semiring
  is-closed-under-left-multiplication-ideal-Semiring =
    pr1 (pr2 is-ideal-subset-ideal-Semiring)

  is-closed-under-right-multiplication-ideal-Semiring :
    is-closed-under-right-multiplication-subset-Semiring R subset-ideal-Semiring
  is-closed-under-right-multiplication-ideal-Semiring =
    pr2 (pr2 is-ideal-subset-ideal-Semiring)

  submonoid-ideal-Semiring : Submonoid l2 (additive-monoid-Semiring R)
  pr1 submonoid-ideal-Semiring = subset-ideal-Semiring
  pr1 (pr2 submonoid-ideal-Semiring) = contains-zero-ideal-Semiring
  pr2 (pr2 submonoid-ideal-Semiring) = is-closed-under-addition-ideal-Semiring

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