_ : left_id one mul;
  _ : right_id one mul;
  _ : left_distributive mul +%R;
  _ : right_distributive mul +%R;
  _ : one != 0
}.in R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 :=
  let _ := @Mixin R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 in
  @Mixin (Zmodule.Pack (Zmodule.class R)) _ _
     mulA mul1x mulx1 mul_addl mul_addr nz1.

Sein R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 :=
  let _ := @Mixin R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 in
  @Mixin (Zmodule.Pack (Zmodule.class R)) _ _
     mulA mul1x mulx1 mul_addl mul_addr nz1.

Sein R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 :=
  let _ := @Mixin R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 in
  @Mixin (Zmodule.Pack (Zmodule.class R)) _ _
     mulA mul1x mulx1 mul_addl mul_addr nz1.

Sein R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 :=
  let _ := @Mixin R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 in
  @Mixin (Zmodule.Pack (Zmodule.class R)) _ _
     mulA mul1x mulx1 mul_addl mul_addr nz1.

SeRecord class_of (R : Type) : Type := Class {
  base : Zmodule.class_of R;Record class_of (R : Type) : Type := Class {
  base : Zmodule.class_of R;Record class_of (R : Type) : Type := Class {
  base : Zmodule.class_of R;}.move=> q; apply/andP/eqP=> [[q_pr q0Notation xclass := (class : class_of xT).Definition pack b0 (m0 : mixin_of (@Zmodule.PackT b0)) :=
  fun bT b & phant_id (Zmodule.class bT) b =>e' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'ringType'  'of'  T ]") : form_scope.
End Exports.End Ring.Definition one (R : ringType) : R := Ring.one (Ring.class R).
Definition mul (R : ringType) : R -> R -> R := Rautomorphism *)
(* results for a non commutative ring in the proof of Gorenstein 2.6.3.      *)
Definition char (R : Ring.type) of phant R : nat_pred :=
  [pred p | prime p & p%:R == 0 :>Lemma subring_closed_semi : subring_closed -> semiring_closed.
Proof.

Section RingTheory.