Nilpotent elements in rings

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

Created on 2022-05-20.
Last modified on 2024-04-11.

module ring-theory.nilpotent-elements-rings where
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.existential-quantification
open import foundation.identity-types
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.universe-levels

open import ring-theory.nilpotent-elements-semirings
open import ring-theory.powers-of-elements-rings
open import ring-theory.rings
open import ring-theory.subsets-rings


A nilpotent element in a ring is an element x for which there is a natural number n such that x^n = 0.


module _
  {l : Level} (R : Ring l)

  is-nilpotent-element-ring-Prop : type-Ring R  Prop l
  is-nilpotent-element-ring-Prop =
    is-nilpotent-element-semiring-Prop (semiring-Ring R)

  is-nilpotent-element-Ring : type-Ring R  UU l
  is-nilpotent-element-Ring = is-nilpotent-element-Semiring (semiring-Ring R)

  is-prop-is-nilpotent-element-Ring :
    (x : type-Ring R)  is-prop (is-nilpotent-element-Ring x)
  is-prop-is-nilpotent-element-Ring =
    is-prop-is-nilpotent-element-Semiring (semiring-Ring R)


Zero is nilpotent

is-nilpotent-zero-Ring :
  {l : Level} (R : Ring l)  is-nilpotent-element-Ring R (zero-Ring R)
is-nilpotent-zero-Ring R = is-nilpotent-zero-Semiring (semiring-Ring R)

If x and y commute and are both nilpotent, then x + y is nilpotent

is-nilpotent-add-Ring :
  {l : Level} (R : Ring l) 
  (x y : type-Ring R)  (mul-Ring R x y  mul-Ring R y x) 
  is-nilpotent-element-Ring R x  is-nilpotent-element-Ring R y 
  is-nilpotent-element-Ring R (add-Ring R x y)
is-nilpotent-add-Ring R = is-nilpotent-add-Semiring (semiring-Ring R)

If x is nilpotent, then so is -x

is-nilpotent-element-neg-Ring :
  {l : Level} (R : Ring l) 
  is-closed-under-negatives-subset-Ring R (is-nilpotent-element-ring-Prop R)
is-nilpotent-element-neg-Ring R {x} H =
  apply-universal-property-trunc-Prop H
    ( is-nilpotent-element-ring-Prop R (neg-Ring R x))
    ( λ (n , p) 
      intro-exists n
        ( ( power-neg-Ring R n x) 
          ( ( ap (mul-Ring R (power-Ring R n (neg-one-Ring R))) p) 
            ( right-zero-law-mul-Ring R (power-Ring R n (neg-one-Ring R))))))

If x is nilpotent and y commutes with x, then xy is also nilpotent

module _
  {l : Level} (R : Ring l)

  is-nilpotent-element-mul-Ring :
    (x y : type-Ring R) 
    mul-Ring R x y  mul-Ring R y x 
    is-nilpotent-element-Ring R x 
    is-nilpotent-element-Ring R (mul-Ring R x y)
  is-nilpotent-element-mul-Ring =
    is-nilpotent-element-mul-Semiring (semiring-Ring R)

  is-nilpotent-element-mul-Ring' :
    (x y : type-Ring R) 
    mul-Ring R x y  mul-Ring R y x 
    is-nilpotent-element-Ring R x 
    is-nilpotent-element-Ring R (mul-Ring R y x)
  is-nilpotent-element-mul-Ring' =
    is-nilpotent-element-mul-Semiring' (semiring-Ring R)

Recent changes