Trivial rings

module ring-theory.trivial-rings where

open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.negation
open import foundation.propositions
open import foundation.sets
open import foundation.universe-levels

open import ring-theory.rings

Idea

Trivial rings are rings in which 0 = 1.

Definition

is-trivial-ring-Prop : {l : Level} → Ring l → Prop l
is-trivial-ring-Prop R =
  Id-Prop (set-Ring R) (zero-Ring R) (one-Ring R)

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

is-prop-is-trivial-Ring :
  {l : Level} (R : Ring l) → is-prop (is-trivial-Ring R)
is-prop-is-trivial-Ring R = is-prop-type-Prop (is-trivial-ring-Prop R)

is-nontrivial-ring-Prop : {l : Level} → Ring l → Prop l
is-nontrivial-ring-Prop R =
  neg-Prop (is-trivial-ring-Prop R)

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

is-prop-is-nontrivial-Ring :
  {l : Level} (R : Ring l) → is-prop (is-nontrivial-Ring R)
is-prop-is-nontrivial-Ring R = is-prop-type-Prop (is-nontrivial-ring-Prop R)

Properties

The carrier type of a trivial ring is contractible

is-contr-is-trivial-Ring :
  {l : Level} (R : Ring l) →
  is-trivial-Ring R →
  is-contr (type-Ring R)
pr1 (is-contr-is-trivial-Ring R p) = zero-Ring R
pr2 (is-contr-is-trivial-Ring R p) x =
  equational-reasoning
    zero-Ring R
      ＝ mul-Ring R (zero-Ring R) x
        by inv (left-zero-law-mul-Ring R x)
      ＝ mul-Ring R (one-Ring R) x
        by ap-binary (mul-Ring R) p refl
      ＝ x
        by left-unit-law-mul-Ring R x