{-# OPTIONS --safe --warning=error --without-K #-}

open import LogicalFormulae
open import Setoids.Setoids
open import Rings.Definition
open import Rings.IntegralDomains.Definition

open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)

module Rings.Irreducibles.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (intDom : IntegralDomain R) where

open Setoid S
open Ring R
open import Rings.Units.Definition R

record Irreducible (r : A) : Set (a ⊔ b) where
  field
    nonzero : (r ∼ 0R) → False
    nonunit : (Unit r) → False
    irreducible : (x y : A) → (x * y) ∼ r → (Unit x → False) → Unit y