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

open import LogicalFormulae
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Ideals.Definition

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

module Rings.Ideals.Maximal.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {pred : A → Set c} (i : Ideal R pred) where

record MaximalIdeal {d : _} : Set (a ⊔ b ⊔ c ⊔ lsuc d) where
  field
    notContained : A
    notContainedIsNotContained : (pred notContained) → False
    isMaximal : {bigger : A → Set d} → Ideal R bigger → ({a : A} → pred a → bigger a) → (Sg A (λ a → bigger a && (pred a → False))) → ({a : A} → bigger a)