Library HoTT.Algebra.Rings.QuotientRing

Require Import WildCat.Core WildCat.Equiv.
Require Import Algebra.Congruence.
Require Import Algebra.AbGroups.
Require Import Classes.interfaces.abstract_algebra.
Require Import Algebra.Rings.Ring.
Require Import Algebra.Rings.Ideal.

Quotient Rings

In this file we define the quotient of a ring by an ideal.

Import Ideal.Notation.
Local Open Scope ring_scope.
Local Open Scope wc_iso_scope.

Section QuotientRing.

  Context (R : Ring) (I : Ideal R).

  Instance plus_quotient_group : Plus (QuotientAbGroup R I) := group_sgop.

  Instance iscong_mult_incosetL
    : @IsCongruence R ring_mult (in_cosetL I).
  Proof.
    snrapply Build_IsCongruence.
    intros x x' y y' p q.
    change (I ( - (x × y) + (x' × y'))).
    rewrite <- (rng_plus_zero_l (x' × y')).
    rewrite <- (rng_plus_negate_r (x' × y)).
    rewrite 2 rng_plus_assoc.
    rewrite <- rng_mult_negate_l.
    rewrite <- rng_dist_r.
    rewrite <- rng_plus_assoc.
    rewrite <- rng_mult_negate_r.
    rewrite <- rng_dist_l.
    rapply subgroup_in_op.
    - by rapply isrightideal.
    - by rapply isleftideal.
  Defined.

  Instance mult_quotient_group : Mult (QuotientAbGroup R I).
  Proof.
    intro x.
    srapply Quotient_rec.
    { intro y; revert x.
      srapply Quotient_rec.
      { intro x.
        apply class_of.
        exact (x × y). }
      intros x x' p.
      apply qglue.
      by rapply iscong. }
    intros y y' q.
    revert x.
    srapply Quotient_ind_hprop.
    intro x.
    simpl.
    apply qglue.
    by rapply iscong.
  Defined.

  Instance one_quotient_abgroup : One (QuotientAbGroup R I) := class_of _ one.

  Instance isring_quotient_abgroup : IsRing (QuotientAbGroup R I).
  Proof.
    split.
    1: exact _.
    1: repeat split.
    1: exact _.
Associativity follows from the underlying operation
    { intros x y.
      snrapply Quotient_ind_hprop; [exact _ | intro z; revert y].
      snrapply Quotient_ind_hprop; [exact _ | intro y; revert x].
      snrapply Quotient_ind_hprop; [exact _ | intro x ].
      unfold sg_op, mult_is_sg_op, mult_quotient_group; simpl.
      apply ap.
      apply associativity. }
    (* Left and right identity follow from the underlying structure *)
    1,2: snrapply Quotient_ind_hprop; [exact _ | intro x].
    1-2: unfold sg_op, mult_is_sg_op, mult_quotient_group; simpl.
    1-2: apply ap.
    1: apply left_identity.
    1: apply right_identity.
Finally distributivity also follows
    { intros x y.
      snrapply Quotient_ind_hprop; [exact _ | intro z; revert y].
      snrapply Quotient_ind_hprop; [exact _ | intro y; revert x].
      snrapply Quotient_ind_hprop; [exact _ | intro x ].
      unfold sg_op, mult_is_sg_op, mult_quotient_group,
        plus, mult, plus_quotient_group; simpl.
      apply ap.
      apply simple_distribute_l. }
    { intros x y.
      snrapply Quotient_ind_hprop; [exact _ | intro z; revert y].
      snrapply Quotient_ind_hprop; [exact _ | intro y; revert x].
      snrapply Quotient_ind_hprop; [exact _ | intro x ].
      unfold sg_op, mult_is_sg_op, mult_quotient_group,
        plus, mult, plus_quotient_group; simpl.
      apply ap.
      apply simple_distribute_r. }
  Defined.

  Definition QuotientRing : Ring
    := Build_Ring (QuotientAbGroup R I) _ _ _ _ _.

End QuotientRing.

Infix "/" := QuotientRing : ring_scope.

Quotient map
Definition rng_quotient_map {R : Ring} (I : Ideal R)
  : RingHomomorphism R (R / I).
Proof.
  snrapply Build_RingHomomorphism'.
  1: rapply grp_quotient_map.
  repeat split.
Defined.

Global Instance issurj_rng_quotient_map {R : Ring} (I : Ideal R)
  : IsSurjection (rng_quotient_map I).
Proof.
  exact _.
Defined.

Specialized induction principles

We provide some specialized induction principes for QuotientRing that require cleaner hypotheses than the ones given by Quotient_ind.
Definition QuotientRing_ind {R : Ring} {I : Ideal R} (P : R / I → Type)
  `{∀ x, IsHSet (P x)}
  (c : ∀ (x : R), P (rng_quotient_map I x))
  (g : ∀ (x y : R) (h : I (- x + y)), qglue h # c x = c y)
  : ∀ (r : R / I), P r
  := Quotient_ind _ P c g.

And a version eliminating into hprops. This one is especially useful.
Definition QuotientRing_ind_hprop {R : Ring} {I : Ideal R} (P : R / I → Type)
  `{∀ x, IsHProp (P x)} (c : ∀ (x : R), P (rng_quotient_map I x))
  : ∀ (r : R / I), P r
  := Quotient_ind_hprop _ P c.

Quotient thoery

First isomorphism theorem for commutative rings
Definition rng_first_iso `{Funext} {A B : Ring} (f : A $-> B)
  : A / ideal_kernel f ≅ rng_image f.
Proof.
  snrapply Build_RingIsomorphism''.
  1: rapply abgroup_first_iso.
  split.
  { intros x y.
    revert y; srapply QuotientRing_ind_hprop; intros y.
    revert x; srapply QuotientRing_ind_hprop; intros x.
    srapply path_sigma_hprop.
    exact (rng_homo_mult _ _ _). }
  srapply path_sigma_hprop.
  exact (rng_homo_one _).
Defined.

Invariance of equal ideals
Lemma rng_quotient_invar {R : Ring} {I J : Ideal R} (p : (I ↔ J)%ideal)
  : R / I ≅ R / J.
Proof.
  snrapply Build_RingIsomorphism'.
  { srapply equiv_quotient_functor'.
    1: exact equiv_idmap.
    intros x y; cbn.
    apply p. }
  repeat split.
  1,2: intros x; simpl.
  1,2: srapply QuotientRing_ind_hprop.
  1,2: intros y; revert x.
  1,2: srapply QuotientRing_ind_hprop.
  1,2: intros x; rapply qglue.
  1: change (J ( - (x + y) + (x + y))).
  2: change (J (- ( x × y) + (x × y))).
  1,2: rewrite rng_plus_negate_l.
  1,2: apply ideal_in_zero.
Defined.

We phrase the first ring isomorphism theroem in a slightly differnt way so that it is easier to use. This form specifically asks for a surjective map
Definition rng_first_iso' `{Funext} {A B : Ring} (f : A $-> B)
  (issurj_f : IsSurjection f)
  (I : Ideal A) (p : (I ↔ ideal_kernel f)%ideal)
  : A / I ≅ B.
Proof.
  etransitivity.
  1: apply (rng_quotient_invar p).
  etransitivity.
  2: rapply (rng_image_issurj f).
  apply rng_first_iso.
Defined.