Idempotent elements in rings

Content created by Jonathan Prieto-Cubides, Fredrik Bakke and Egbert Rijke.

Created on 2022-05-24.
Last modified on 2025-10-31.

module ring-theory.idempotent-elements-rings where
Imports 
open import foundation.propositions
open import foundation.sets
open import foundation.subtypes
open import foundation.universe-levels

open import ring-theory.rings

Idea

An idempotent element in a ring is an element x such that x² = x.

Definition

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

  is-idempotent-element-ring-Prop : Prop l
  is-idempotent-element-ring-Prop = Id-Prop (set-Ring R) (mul-Ring R x x) x

  is-idempotent-element-Ring : UU l
  is-idempotent-element-Ring = type-Prop is-idempotent-element-ring-Prop

  is-prop-is-idempotent-element-Ring : is-prop is-idempotent-element-Ring
  is-prop-is-idempotent-element-Ring =
    is-prop-type-Prop is-idempotent-element-ring-Prop

idempotent-element-Ring :
  {l : Level} (R : Ring l) → UU l
idempotent-element-Ring R = type-subtype (is-idempotent-element-ring-Prop R)

module _
  {l : Level} (R : Ring l) (x : idempotent-element-Ring R)
  where

  element-idempotent-element-Ring : type-Ring R
  element-idempotent-element-Ring =
    inclusion-subtype (is-idempotent-element-ring-Prop R) x

  is-idempotent-element-idempotent-element-Ring :
    is-idempotent-element-Ring R element-idempotent-element-Ring
  is-idempotent-element-idempotent-element-Ring =
    is-in-subtype-inclusion-subtype (is-idempotent-element-ring-Prop R) x

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