Radical ideals of rings

Content created by Jonathan Prieto-Cubides, Egbert Rijke, Fredrik Bakke, Maša Žaucer and Gregor Perčič.

Created on 2022-05-20.
Last modified on 2023-09-21.

module ring-theory.radical-ideals-rings where
open import foundation.propositions
open import foundation.universe-levels

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


A radical ideal in a ring R is an ideal I such that 1 + x is a multiplicative unit for every x ∈ I.


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

  is-radical-ideal-prop-Ring : Prop (l1  l2)
  is-radical-ideal-prop-Ring =
      ( type-ideal-Ring R I)
      ( λ x 
        is-invertible-element-prop-Ring R
          ( add-Ring R (one-Ring R) (inclusion-ideal-Ring R I x)))

  is-radical-ideal-Ring : UU (l1  l2)
  is-radical-ideal-Ring =
    type-Prop is-radical-ideal-prop-Ring

  is-prop-is-radical-ideal-Ring :
    is-prop is-radical-ideal-Ring
  is-prop-is-radical-ideal-Ring =
    is-prop-type-Prop is-radical-ideal-prop-Ring

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