Idempotent ring elements

theories/Algebra/Rings/Idempotent.v

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+Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids.
+Require Import Basics.Equivalences Basics.Trunc Types.Sigma WildCat.Core.
+Require Import Nat.Core Rings.Ring.
+
+Local Open Scope mc_scope.
+
+(** * Idempotent elements of rings *)
+
+(** ** Definition *)
+
+Class IsIdempotent (R : Ring) (e : R)
+  := rng_idem : e * e = e.
+
+Global Instance ishprop_isidempotent R e : IsHProp (IsIdempotent R e).
+Proof.
+  unfold IsIdempotent; exact _.
+Defined.
+
+(** *** Examples *)
+
+(** Zero is idempotent. *)
+Global Instance isidempotent_zero (R : Ring) : IsIdempotent R 0
+  := rng_mult_zero_r 0.
+
+(** One is idempotent. *)
+Global Instance isidempotent_one (R : Ring) : IsIdempotent R 1
+  := rng_mult_one_r 1.
+
+(** If [e] is idempotent, then [1 - e] is idempotent. *)
+Global Instance isidempotent_compliment (R : Ring) (e : R) `{IsIdempotent R e}
+  : IsIdempotent R (1 - e).
+Proof.
+  unfold IsIdempotent.
+  rewrite rng_dist_l_negate.
+  rewrite 2 rng_dist_r_negate.
+  rewrite 2 rng_mult_one_l.
+  rewrite rng_mult_one_r.
+  rewrite rng_idem.
+  rewrite rng_plus_negate_r.
+  rewrite rng_negate_zero.
+  nrapply rng_plus_zero_r.
+Defined.
+
+(** If [e] is idempotent, then it is also an idempotent element of the opposite ring. *)
+Global Instance isidempotent_op (R : Ring) (e : R) `{IsIdempotent R e}
+  : IsIdempotent (rng_op R) e
+  := rng_idem (R:=R).
+
+(** Any positive power of an idempotent element [e] is [e]. *)
+Definition rng_power_idem {R : Ring} (e : R) `{IsIdempotent R e} (n : nat)
+  : (1 <= n)%nat -> rng_power e n = e.
+Proof.
+  intros L; induction L.
+  - snrapply rng_mult_one_r.
+  - rhs_V rapply rng_idem.
+    exact (ap (e *.) IHL).
+Defined.
+
+(** Ring homomorphisms preserve idempotent elements. *)
+Global Instance isidempotent_rng_homo {R S : Ring} (f : R $-> S) (e : R)
+  : IsIdempotent R e -> IsIdempotent S (f e).
+Proof.
+  intros p.
+  lhs_V nrapply rng_homo_mult.
+  exact (ap f p).
+Defined.
+
+(** ** Orthogonal idempotent elements *)
+
+(** Two idempotent elements [e] and [f] are orthogonal if both [e * f = 0] and [f * e = 0]. *)
+Class IsOrthogonal (R : Ring) (e f : R)
+  `{!IsIdempotent R e, !IsIdempotent R f} := {
+  rng_idem_orth : e * f = 0 ;
+  rng_idem_orth' : f * e = 0 ;
+}.
+
+Definition issig_IsOrthogonal {R : Ring} {e f : R}
+  `{IsIdempotent R e, IsIdempotent R f}
+  : _ <~> IsOrthogonal R e f
+  := ltac:(issig).
+
+Global Instance ishprop_isorthogonal R e f
+  `{IsIdempotent R e, IsIdempotent R f}
+  : IsHProp (IsOrthogonal R e f).
+Proof.
+  exact (istrunc_equiv_istrunc _ issig_IsOrthogonal). 
+Defined.
+
+(** *** Properties *)
+
+(** Two idempotents being orthogonal is a symmetric relation. *)
+Definition isorthogonal_swap (R : Ring) (e f : R) `{IsOrthogonal R e f}
+  : IsOrthogonal R f e
+  := {| rng_idem_orth := rng_idem_orth' ; rng_idem_orth' := rng_idem_orth |}.
+Hint Immediate isorthogonal_swap : typeclass_instances.
+
+
+(** If [e] and [f] are orthogonal idempotents, then they are also orthogonal idempotents in the opposite ring. *)
+Global Instance isorthogonal_op {R : Ring} (e f : R) `{r : IsOrthogonal R e f}
+  : IsOrthogonal (rng_op R) e f.
+Proof.
+  snrapply Build_IsOrthogonal.
+  - exact (rng_idem_orth' (R:=R)).
+  - exact (rng_idem_orth (R:=R)).
+Defined.
+
+(** If [e] and [f] are orthogonal idempotents, then [e + f] is idempotent. *)
+Global Instance isidempotent_plus_orthogonal {R : Ring} (e f : R)
+  `{IsOrthogonal R e f}
+  : IsIdempotent R (e + f).
+Proof.
+  unfold IsIdempotent.
+  rewrite rng_dist_l.
+  rewrite 2 rng_dist_r.
+  rewrite 2 rng_idem.
+  rewrite 2 rng_idem_orth.
+  by rewrite rng_plus_zero_r, rng_plus_zero_l.
+Defined.
+
+(** An idempotent element [e] is orthogonal to its complement [1 - e]. *)
+Global Instance isorthogonal_compliment {R : Ring} (e : R) `{IsIdempotent R e}
+  : IsOrthogonal R e (1 - e).
+Proof.
+  snrapply Build_IsOrthogonal.
+  1: rewrite rng_dist_l_negate,rng_mult_one_r.
+  2: rewrite rng_dist_r_negate, rng_mult_one_l.
+  1,2: rewrite rng_idem.
+  1,2: apply rng_plus_negate_r.
+Defined.