# Library UniMath.Algebra.Modules.Core

Authors Anthony Bordg and Floris van Doorn, February-December 2017

Require Import UniMath.MoreFoundations.All.
Require Import UniMath.Algebra.RigsAndRings.
Require Import UniMath.Algebra.Groups.
Require Import UniMath.Algebra.Monoids.

## Contents

• The ring of endomorphisms of an abelian group
• Modules (the definition of the small type of R-modules over a ring R)
• R-module morphisms

Local Open Scope addmonoid_scope.

## The ring of endomorphisms of an abelian group

Two binary operations on the set of endomorphisms of an abelian group

Definition monoidfun_to_isbinopfun {G : abgr} (f : monoidfun G G) : isbinopfun f := pr1 (pr2 f).

Definition ringofendabgr_op1 {G: abgr} : binop (monoidfun G G).
Proof.
intros f g.
apply (@monoidfunconstr _ _ (λ x : G, f x + g x)).
apply tpair.
- intros x x'.
rewrite (monoidfun_to_isbinopfun f).
rewrite (monoidfun_to_isbinopfun g).
apply (abmonoidrer G).
- rewrite (monoidfununel f).
rewrite (monoidfununel g).
rewrite (lunax G).
reflexivity.
Defined.

Definition ringofendabgr_op2 {G : abgr} : binop (monoidfun G G).
Proof.
intros f g.
apply (monoidfuncomp g f).
Defined.

Declare Scope abgr_scope.
Notation "f + g" := (ringofendabgr_op1 f g) : abgr_scope.

The underlying set of the ring of endomorphisms of an abelian group

Definition setofendabgr (G : abgr) : hSet :=
make_hSet (monoidfun G G) (isasetmonoidfun G G).

A few access functions

Definition pr1setofendabgr {G : abgr} (f : setofendabgr G) : G G := pr1 f.

Definition pr2setofendabgr {G : abgr} (f : setofendabgr G) : ismonoidfun (pr1 f) := pr2 f.

Definition setofendabgr_to_isbinopfun {G : abgr} (f : setofendabgr G) :
isbinopfun (pr1setofendabgr f) := pr1 (pr2 f).

Definition setofendabgr_to_unel {G : abgr} (f : setofendabgr G) : pr1setofendabgr f 0 = 0 :=
pr2 (pr2setofendabgr f).

We endow setofendabgr with the two binary operations defined above
ringofendabgr_op1 G and ringofendabgr_op2 G are ring operations
ringofendabgr_op1 is a monoid operation

Local Open Scope abgr_scope.

Lemma isassoc_ringofendabgr_op1 {G : abgr} : isassoc (@ringofendabgr_op1 G).
Proof.
intros f g h.
use total2_paths_f.
- apply funextfun.
intro.
apply (pr2 G).
- apply isapropismonoidfun.
Defined.

Lemma setofendabgr_un0 {G: abgr} : monoidfun G G.
Proof.
apply (@monoidfunconstr _ _ (λ x : G, 0)).
apply make_dirprod.
- intros x x'.
rewrite (lunax G).
reflexivity.
- reflexivity.
Defined.

Lemma islunit_setofendabgr_un0 {G : abgr} : islunit (@ringofendabgr_op1 G) setofendabgr_un0.
Proof.
intro f.
use total2_paths_f.
- apply funextfun. intro x.
apply (lunax G (pr1setofendabgr f x)).
- apply isapropismonoidfun.
Defined.

Lemma isrunit_setofendabgr_un0 {G : abgr} : isrunit (@ringofendabgr_op1 G) setofendabgr_un0.
Proof.
intros f.
use total2_paths_f.
- apply funextfun. intro x.
apply (runax G (pr1setofendabgr f x)).
- apply isapropismonoidfun.
Defined.

Lemma isunit_setofendabgr_un0 {G : abgr} : isunit (@ringofendabgr_op1 G) setofendabgr_un0.
Proof. exact (make_isunit islunit_setofendabgr_un0 isrunit_setofendabgr_un0). Defined.

Lemma isunital_ringofendabgr_op1 {G : abgr} : isunital (@ringofendabgr_op1 G).
Proof. exact (make_isunital setofendabgr_un0 isunit_setofendabgr_un0). Defined.

Lemma ismonoidop_ringofendabgr_op1 {G : abgr} : ismonoidop (@ringofendabgr_op1 G).
Proof. exact (make_ismonoidop isassoc_ringofendabgr_op1 isunital_ringofendabgr_op1). Defined.

Local Close Scope abgr_scope.

ringofendabgr_op1 is a group operation

Definition setofendabgr_inv {G : abgr} : monoidfun G G monoidfun G G.
Proof.
intro f.
apply (@monoidfunconstr G G (λ x : G, grinv G (pr1setofendabgr f x))).
apply make_dirprod.
- intros x x'.
rewrite (setofendabgr_to_isbinopfun f).
rewrite (grinvop G).
apply (commax G).
- rewrite (setofendabgr_to_unel f).
apply (grinvunel G).
Defined.

Local Open Scope abgr_scope.

Lemma islinv_setofendabgr_inv {G : abgr} :
islinv (@ringofendabgr_op1 G) setofendabgr_un0 setofendabgr_inv.
Proof.
intro f.
use total2_paths_f.
- apply funextfun. intro x.
apply (grlinvax G).
- apply isapropismonoidfun.
Defined.

Lemma isrinv_setofendabgr_inv {G : abgr} :
isrinv (@ringofendabgr_op1 G) setofendabgr_un0 setofendabgr_inv.
Proof.
intro f.
use total2_paths_f.
- apply funextfun. intro x.
apply (grrinvax G).
- apply isapropismonoidfun.
Defined.

Lemma isinv_setofendabgr_inv {G : abgr} :
isinv (@ringofendabgr_op1 G) (unel_is (@ismonoidop_ringofendabgr_op1 G)) setofendabgr_inv.
Proof. exact (make_isinv islinv_setofendabgr_inv isrinv_setofendabgr_inv). Defined.

Definition invstruct_setofendabgr_inv {G : abgr} :
invstruct (@ringofendabgr_op1 G) ismonoidop_ringofendabgr_op1.
Proof. exact (make_invstruct (@setofendabgr_inv G) (@isinv_setofendabgr_inv G)). Defined.

Lemma isgrop_ringofendabgr_op1 {G : abgr} : isgrop (@ringofendabgr_op1 G).
Proof. exact (make_isgrop ismonoidop_ringofendabgr_op1 invstruct_setofendabgr_inv). Defined.

Lemma iscomm_ringofendabgr_op1 {G : abgr} : iscomm (@ringofendabgr_op1 G).
Proof.
intros f g.
use total2_paths_f.
- apply funextfun. intro x.
apply (commax G).
- apply (isapropismonoidfun).
Defined.

Lemma isabgrop_ringofendabgr_op1 {G : abgr} : isabgrop (@ringofendabgr_op1 G).
Proof. exact (make_isabgrop isgrop_ringofendabgr_op1 iscomm_ringofendabgr_op1). Defined.

ringofendabgr_op2 is a monoid operation

Lemma isassoc_ringofendabgr_op2 {G : abgr} : isassoc (@ringofendabgr_op2 G).
Proof.
intros f g h.
use total2_paths_f.
- apply funcomp_assoc.
- apply isapropismonoidfun.
Defined.

Definition setofendabgr_un1 {G: abgr} : monoidfun G G.
Proof.
apply (@monoidfunconstr _ _ (idfun G)).
apply make_dirprod.
- intros x x'. reflexivity.
- reflexivity.
Defined.

Lemma islunit_setofendabgr_un1 {G : abgr} : islunit (@ringofendabgr_op2 G) setofendabgr_un1.
Proof.
intro f.
use total2_paths_f.
- apply funextfun. intro x. reflexivity.
- apply isapropismonoidfun.
Defined.

Lemma isrunit_setofendabgr_un1 {G : abgr} : isrunit (@ringofendabgr_op2 G) setofendabgr_un1.
Proof.
intros f.
use total2_paths_f.
- apply funextfun. intro x. reflexivity.
- apply isapropismonoidfun.
Defined.

Lemma isunit_setofendabgr_un1 {G : abgr} : isunit (@ringofendabgr_op2 G) setofendabgr_un1.
Proof. exact (make_isunit islunit_setofendabgr_un1 isrunit_setofendabgr_un1). Defined.

Lemma isunital_ringofendabgr_op2 {G : abgr} : isunital (@ringofendabgr_op2 G).
Proof. exact (make_isunital setofendabgr_un1 isunit_setofendabgr_un1). Defined.

Lemma ismonoidop_ringofendabgr_op2 {G : abgr} : ismonoidop (@ringofendabgr_op2 G).
Proof. exact (make_ismonoidop isassoc_ringofendabgr_op2 isunital_ringofendabgr_op2). Defined.

ringofendabgr_op2 is distributive over ringofendabgr_op1

Lemma isldistr_setofendabgr_op {G : abgr} :
isldistr (@ringofendabgr_op1 G) (@ringofendabgr_op2 G).
Proof.
intros f g h.
use total2_paths_f.
- apply funextfun. intro x.
apply (setofendabgr_to_isbinopfun h).
- apply isapropismonoidfun.
Defined.

Lemma isrdistr_setofendabgr_op {G : abgr} :
isrdistr (@ringofendabgr_op1 G) (@ringofendabgr_op2 G).
Proof.
intros f g h.
use total2_paths_f.
- apply funextfun. intro x. reflexivity.
- apply isapropismonoidfun.
Defined.

Lemma isdistr_setofendabgr_op {G : abgr} :
isdistr (@ringofendabgr_op1 G) (@ringofendabgr_op2 G).
Proof. exact (make_dirprod isldistr_setofendabgr_op isrdistr_setofendabgr_op). Defined.

Lemma isringops_setofendabgr_op {G : abgr} :
isringops (@ringofendabgr_op1 G) (@ringofendabgr_op2 G).
Proof.
exact (make_isringops isabgrop_ringofendabgr_op1 ismonoidop_ringofendabgr_op2 isdistr_setofendabgr_op).
Defined.

The set of endomorphisms of an abelian group is a ring

## Modules: the definition of the small type of R-modules over a ring R

A module over R may be defined as a ring homomorphism from R to the ring of endomorphisms of an Abelian group (in other words, a ring action on the abelian group). An equivalence with the more common axiomatic definition is established below.
In this development, we concern ourselves with left modules. Recall that a right module is equivalent to a left module over the opposite ring.

Definition module_struct (R : ring) (G : abgr) : UU := ringfun R (ringofendabgr G).

Definition module (R : ring) : UU := G, module_struct R G.

Definition pr1module {R : ring} (M : module R) : abgr := pr1 M.

Coercion pr1module : module >-> abgr.

Definition pr2module {R : ring} (M : module R) : module_struct R (pr1module M) := pr2 M.

Identity Coercion id_module_struct : module_struct >-> ringfun.

Definition make_module {R : ring} (G : abgr) (f : module_struct R G) : module R := tpair _ G f.

Lemma isasetmodule {R : ring} (M : module R) : isaset M.
Proof.
exact (pr2 (pr1 (pr1 (pr1 M)))).
Defined.

The ring action gives rise to a notion of multiplication.

Definition module_mult {R : ring} (M : module R) : R M M :=
λ r : R, λ x : M, (pr1setofendabgr (pr2module M r) x).

Declare Scope module_scope.
Notation "r * x" := (module_mult _ r x) : module_scope.

Delimit Scope module_scope with module.

Local Open Scope module.

Lemma module_mult_0_to_0 {R : ring} {M : module R} (x : M) : ringunel1 × x = @unel M.
Proof.
unfold module_mult. cbn.
assert (pr2module M ringunel1 = @ringunel1 (ringofendabgr M)).
- exact (rigfun_to_unel_rigaddmonoid (pr2module M)).
- rewrite X.
reflexivity.
Defined.

Local Open Scope addmonoid.

Lemma module_mult_is_ldistr {R : ring} {M : module R} (r : R) (x y : M) :
r × (x + y) = r × x + r × y.
Proof. exact (pr1 (pr2 (pr2module M r)) x y). Defined.

Lemma module_mult_is_rdistr {R : ring} {M : module R} (r s : R) (x : M) :
(op1 r s) × x = r × x + s × x.
Proof. exact (maponpaths (λ r, pr1setofendabgr r x) (pr1 (pr1 (pr2 (pr2module M))) r s)). Defined.

Lemma module_mult_assoc {R : ring} {M : module R} (r s : R) (x : M) :
(op2 r s) × x = r × (s × x).
Proof. exact (maponpaths (λ r, pr1setofendabgr r x) (pr1 (pr2 (pr2 (pr2module M))) r s)). Defined.

Lemma module_mult_1 {R : ring} {M : module R} (r : R) : r × unel M = unel M.
Proof. exact (pr2 (pr2 (pr2module M r))). Defined.

Lemma module_mult_unel2 {R : ring} {M : module R} (m : M) : ringunel2 × m = m.
Proof. exact (maponpaths (λ r, pr1setofendabgr r m) (pr2 (pr2 (pr2 (pr2module M))))). Defined.

Lemma module_inv_mult {R : ring} {M : module R} (r : R) (x : M) :
@grinv _ (r × x) = r × @grinv _ x.
Proof.
apply grinv_path_from_op_path. now rewrite <- module_mult_is_ldistr, grrinvax, module_mult_1.
Defined.

Lemma module_mult_neg1 {R : ring} {M : module R} (x : M) : ringminus1 × x = @grinv _ x.
Proof.
apply pathsinv0. apply grinv_path_from_op_path.
refine (maponpaths (λ y, y × ((_ × x)%module))%multmonoid (!(module_mult_unel2 x)) @ _).
now rewrite <- module_mult_is_rdistr, ringrinvax1, module_mult_0_to_0.
Defined.

Lemma module_inv_mult_to_inv1 {R : ring} {M : module R} (r : R) (x : M) :
@grinv _ (r × x) = ringinv1 r × x.
Proof.
apply grinv_path_from_op_path.
now rewrite <- module_mult_is_rdistr, ringrinvax1, module_mult_0_to_0.
Defined.

Lemma module_mult_inv_to_inv1 {R : ring} {M : module R} (r : R) (x : M) :
r × @grinv _ x = ringinv1 r × x.
Proof.
now rewrite <- module_inv_mult_to_inv1, module_inv_mult.
Defined.

To construct a module from a left action satisfying four axioms

Definition mult_isldistr_wrt_grop {R : ring} {G : abgr} (m : R G G) : UU :=
r : R, x y : G, m r (x + y) = (m r x) + (m r y).

Definition mult_isrdistr_wrt_ringop1 {R : ring} {G : abgr} (m : R G G) : UU :=
r s : R, x : G, m (op1 r s) x = (m r x) + (m s x).

Definition mult_isrdistr_wrt_ringop2 {R : ring} {G : abgr} (m : R G G) : UU :=
r s : R, x : G, m (op2 r s) x = m r (m s x).

Definition mult_unel {R : ring} {G : abgr} (m : R G G) : UU := x : G, m ringunel2 x = x.

Local Close Scope addmonoid.

Definition mult_to_ringofendabgr {R : ring} {G : abgr} {m : R G G}
(ax1 : mult_isldistr_wrt_grop m) (r : R) : ringofendabgr G.
Proof.
use monoidfunconstr.
intro x. exact (m r x).
apply make_dirprod.
+ intros x y. apply ax1.
+ apply (grlcan G (m r (unel G))).
rewrite runax.
rewrite <- (ax1 r (unel G) (unel G)).
rewrite runax.
apply idpath.
Defined.

Definition mult_to_module_struct {R : ring} {G : abgr} {m : R G G}
(ax1 : mult_isldistr_wrt_grop m) (ax2 : mult_isrdistr_wrt_ringop1 m)
(ax3 : mult_isrdistr_wrt_ringop2 m) (ax4 : mult_unel m) : module_struct R G.
Proof.
split with (λ r : R, mult_to_ringofendabgr ax1 r).
apply make_dirprod.
- apply make_dirprod.
+ intros r s.
use total2_paths2_f.
× apply funextfun. intro x. apply ax2.
× apply isapropismonoidfun.
+ use total2_paths2_f.
× apply funextfun. intro x. change (m ringunel1 x = unel G). apply (grlcan G (m (ringunel1) x)).
rewrite runax. rewrite <- (ax2 ringunel1 ringunel1 x). rewrite ringrunax1. apply idpath.
× apply isapropismonoidfun.
- apply make_dirprod.
+ intros r s.
use total2_paths2_f.
× apply funextfun. intro x. apply ax3.
× apply isapropismonoidfun.
+ use total2_paths2_f.
× apply funextfun. intro x. apply ax4.
× apply isapropismonoidfun.
Defined.

Definition mult_to_module {R : ring} {G : abgr} {m : R G G} (ax1 : mult_isldistr_wrt_grop m)
(ax2 : mult_isrdistr_wrt_ringop1 m) (ax3 : mult_isrdistr_wrt_ringop2 m)
(ax4 : mult_unel m) : module R := make_module G (mult_to_module_struct ax1 ax2 ax3 ax4).

### R-module morphisms

#### Linearity

Definition islinear {R : ring} {M N : module R} (f : M N) :=
r : R, x : M, f (r × x) = r × (f x).

Definition linearfun {R : ring} (M N : module R) : UU := f : M N, islinear f.

Definition make_linearfun {R : ring} {M N : module R} (f : M N) (is : islinear f) :
linearfun M N := tpair _ f is.

Definition pr1linearfun {R : ring} {M N : module R} (f : linearfun M N) : M N := pr1 f.

Coercion pr1linearfun : linearfun >-> Funclass.

Definition linearfun_islinear {R} {M N : module R} (f : linearfun M N) :
islinear f := pr2 f.

Lemma islinearfuncomp {R : ring} {M N P : module R} (f : linearfun M N) (g : linearfun N P) :
islinear (funcomp f g).
Proof.
intros r x.
unfold funcomp.
rewrite (linearfun_islinear f).
rewrite (linearfun_islinear g).
apply idpath.
Defined.

Definition linearfuncomp {R : ring} {M N P : module R} (f : linearfun M N) (g : linearfun N P) :
linearfun M P := tpair _ (funcomp f g) (islinearfuncomp f g).

Lemma isapropislinear {R : ring} {M N : module R} (f : M N) :
isaprop (islinear f).
Proof.
apply (impred 1 _). intro r.
apply (impred 1 _). intro x.
apply (setproperty N).
Defined.

The type of linear functions M -> N is a set.
Lemma isasetlinearfun {R : ring} (M N : module R) : isaset (linearfun M N).
Proof.
intros. apply (isasetsubset (@pr1linearfun R M N)).
- change (isofhlevel 2 (M N)).
apply impred.
exact (fun xsetproperty N).
- refine (isinclpr1 _ _).
intro.
apply isapropislinear.
Defined.

#### modulefun

Definition ismodulefun {R : ring} {M N : module R} (f : M N) : UU :=
(isbinopfun f) × (islinear f).

Definition make_ismodulefun {R : ring} {M N : module R} {f : M N}
(H1 : isbinopfun f) (H2 : islinear f) : ismodulefun f :=
tpair _ H1 H2.

Lemma isapropismodulefun {R : ring} {M N : module R} (f : M N) :
isaprop (ismodulefun f).
Proof.
exact (@isofhleveldirprod 1 (isbinopfun f) (islinear f)
(isapropisbinopfun f) (isapropislinear f)).
Defined.

Definition modulefun {R : ring} (M N : module R) : UU := f : M N, ismodulefun f.

Definition make_modulefun {R : ring} {M N : module R} (f : M N) (is : ismodulefun f) :
modulefun M N := tpair _ f is.

Local Notation "R-mod( M , N )" := (modulefun M N) : module_scope.

Section accessors.
Context {R : ring} {M N : module R} (f : R-mod(M, N)).

Definition pr1modulefun : M N := pr1 f.

Definition modulefun_ismodulefun : ismodulefun pr1modulefun := pr2 f.

Definition modulefun_to_isbinopfun : isbinopfun pr1modulefun :=
pr1 modulefun_ismodulefun.

Definition modulefun_to_binopfun : binopfun M N :=
make_binopfun pr1modulefun modulefun_to_isbinopfun.

Definition modulefun_to_islinear : islinear pr1modulefun := pr2 modulefun_ismodulefun.

Definition modulefun_to_linearfun : linearfun M N :=
make_linearfun pr1modulefun modulefun_to_islinear.
End accessors.

Coercion pr1modulefun : modulefun >-> Funclass.

Lemma ismodulefun_comp {R} {M N P : module R} (f : R-mod(M,N)) (g : R-mod(N,P)) :
ismodulefun (g f)%functions.
Proof.
exact (make_dirprod (isbinopfuncomp (modulefun_to_binopfun f)
(modulefun_to_binopfun g))
(islinearfuncomp (modulefun_to_linearfun f)
(modulefun_to_linearfun g))).
Defined.

Definition modulefun_comp
{R} {M N P : module R} (f : R-mod(M, N)) (g : R-mod(N,P)) : R-mod(M,P) :=
(funcomp f g,, ismodulefun_comp f g).

Lemma modulefun_unel {R : ring} {M N : module R} (f : R-mod(M, N)) : f (unel M) = unel N.
Proof.
rewrite <- (module_mult_0_to_0 (unel M)).
rewrite ((modulefun_to_islinear f) ringunel1 (unel M)).
rewrite (module_mult_0_to_0 _).
reflexivity.
Defined.

Definition moduletomonoid {R : ring} (M : module R) : abmonoid := abgrtoabmonoid (pr1module M).

Definition modulefun_to_monoidfun {R : ring} {M N : module R} (f : R-mod(M, N)) :
monoidfun (moduletomonoid M) (moduletomonoid N) :=
tpair _ (pr1 f) (tpair _ (pr1 (pr2 f)) (modulefun_unel f)).

Definition modulefun_from_monoidfun {R : ring} {M N : module R} (f : R-mod(M, N))
(H : islinear (pr1 f)) : R-mod(M, N) :=
make_modulefun (pr1 f) (make_ismodulefun (pr1 (pr2 f)) H).

Lemma modulefun_paths {R : ring} {M N : module R} {f g : R-mod(M, N)} (p : pr1 f = pr1 g) :
f = g.
Proof.
use total2_paths_f.
- exact p.
- use proofirrelevance. use isapropismodulefun.
Defined.

Lemma modulefun_paths2 {R : ring} {M N : module R} {f g : modulefun M N} (p : pr1 f ¬ pr1 g) :
f = g.
Proof. exact (modulefun_paths (funextfun _ _ p)). Defined.

Lemma isasetmodulefun {R : ring} (M N : module R) : isaset (R-mod(M, N)).
Proof.
intros. apply (isasetsubset (@pr1modulefun R M N)).
- change (isofhlevel 2 (M N)).
apply impred. intro.
apply (setproperty N).
- refine (isinclpr1 _ _). intro.
apply isapropismodulefun.
Defined.

Lemma modulehombinop_ismodulefun {R : ring} {M N : module R} (f g : R-mod(M, N)) :
@ismodulefun R M N (λ x : pr1 M, (pr1 f x × pr1 g x)%multmonoid).
Proof.
use make_ismodulefun.
- exact (pr1 (abmonoidshombinop_ismonoidfun (modulefun_to_monoidfun f)
(modulefun_to_monoidfun g))).
- intros r m. rewrite (modulefun_to_islinear f). rewrite (modulefun_to_islinear g).
rewrite <- module_mult_is_ldistr. reflexivity.
Defined.

Definition modulehombinop {R : ring} {M N : module R} : binop (R-mod(M, N)) :=
(λ f g, make_modulefun _ (modulehombinop_ismodulefun f g)).

Lemma unelmodulefun_ismodulefun {R : ring} (M N : module R) : ismodulefun (λ x : M, (unel N)).
Proof.
use make_ismodulefun.
- use make_isbinopfun. intros m m'. use pathsinv0. use lunax.
- intros r m. rewrite module_mult_1. reflexivity.
Qed.

Definition unelmodulefun {R : ring} (M N : module R) : R-mod(M, N) :=
make_modulefun _ (unelmodulefun_ismodulefun M N).

Lemma modulebinop_runax {R : ring} {M N : module R} (f : R-mod(M, N)) :
modulehombinop f (unelmodulefun M N) = f.
Proof.
use modulefun_paths2. intros x. use (runax N).
Qed.

Lemma modulebinop_lunax {R : ring} {M N : module R} (f : R-mod(M, N)) :
modulehombinop (unelmodulefun M N) f = f.
Proof.
use modulefun_paths2. intros x. use (lunax N).
Qed.

Lemma modulehombinop_assoc {R : ring} {M N : module R} (f g h : R-mod(M, N)) :
modulehombinop (modulehombinop f g) h = modulehombinop f (modulehombinop g h).
Proof.
use modulefun_paths2. intros x. use assocax.
Qed.

Lemma modulehombinop_comm {R : ring} {M N : module R} (f g : R-mod(M, N)) :
modulehombinop f g = modulehombinop g f.
Proof.
use modulefun_paths2. intros x. use (commax N).
Qed.

Lemma modulehomabmodule_ismoduleop {R : ring} {M N : module R} :
ismonoidop (λ f g : R-mod(M, N), modulehombinop f g).
Proof.
use make_ismonoidop.
- intros f g h. exact (modulehombinop_assoc f g h).
- use make_isunital.
+ exact (unelmodulefun M N).
+ use make_isunit.
× intros f. exact (modulebinop_lunax f).
× intros f. exact (modulebinop_runax f).
Defined.

Lemma modulehombinop_inv_ismodulefun {R : ring} {M N : module R} (f : R-mod(M, N)) :
ismodulefun (λ m : M, grinv N (pr1 f m)).
Proof.
use tpair.
- use make_isbinopfun. intros x x'. cbn.
rewrite (pr1 (pr2 f)). rewrite (pr2 (pr2 (pr1module N))). use (grinvop N).
- intros r m. rewrite <- module_inv_mult. apply maponpaths.
apply (pr2 f).
Qed.

Definition modulehombinop_inv {R : ring} {M N : module R} (f : R-mod(M, N)) : R-mod(M, N) :=
tpair _ _ (modulehombinop_inv_ismodulefun f).

Lemma modulehombinop_linvax {R : ring} {M N : module R} (f : R-mod(M, N)) :
modulehombinop (modulehombinop_inv f) f = unelmodulefun M N.
Proof.
use modulefun_paths2. intros x. use (@grlinvax N).
Qed.

Lemma modulehombinop_rinvax {R : ring} {M N : module R} (f : R-mod(M, N)) :
modulehombinop f (modulehombinop_inv f) = unelmodulefun M N.
Proof.
use modulefun_paths2. intros x. use (grrinvax N).
Qed.

Lemma modulehomabgr_isabgrop {R : ring} (M N : module R) :
isabgrop (λ f g : R-mod(M, N), modulehombinop f g).
Proof.
use make_isabgrop.
use make_isgrop.
- use modulehomabmodule_ismoduleop.
- use make_invstruct.
+ intros f. exact (modulehombinop_inv f).
+ use make_isinv.
× intros f. exact (modulehombinop_linvax f).
× intros f. exact (modulehombinop_rinvax f).
- intros f g. exact (modulehombinop_comm f g).
Defined.

Definition modulehomabgr {R : ring} (M N : module R) : abgr.
Proof.
use make_abgr.
use make_setwithbinop.
use make_hSet.
- exact (R-mod(M, N)).
- exact (isasetmodulefun M N).
- exact (@modulehombinop R M N).
- exact (modulehomabgr_isabgrop M N).
Defined.

Definition modulehombinop_scalar_ismodulefun {R : commring} {M N : module R} (r : R)
(f : R-mod(M, N)) : ismodulefun (λ m : M, r × (pr1 f m)).
Proof.
use tpair.
- use make_isbinopfun. intros x x'. cbn.
rewrite (pr1 (pr2 f)). rewrite module_mult_is_ldistr. reflexivity.
- intros r0 m. rewrite (modulefun_to_islinear f). do 2 rewrite <- module_mult_assoc.
rewrite ringcomm2. reflexivity.
Qed.

Definition modulehombinop_smul {R : commring} {M N : module R} (r : R) (f : R-mod(M, N)) :
modulefun M N :=
make_modulefun _ (modulehombinop_scalar_ismodulefun r f).

Definition modulehommodule {R : commring} (M N : module R) : module R.
Proof.
use make_module.
use (modulehomabgr M N).
use mult_to_module_struct.
exact modulehombinop_smul.
- intros r f g. use modulefun_paths2. intros x. apply module_mult_is_ldistr.
- intros r r0 f. use modulefun_paths2. intros x. apply module_mult_is_rdistr.
- intros r r0 f. use modulefun_paths2. intros x. apply module_mult_assoc.
- intros f. use modulefun_paths2. intros x. cbn. apply module_mult_unel2.
Defined.

#### Isomorphisms (moduleiso)

Definition moduleiso {R : ring} (M N : module R) : UU :=
w : pr1module M pr1module N, ismodulefun w.

Section accessors_moduleiso.
Context {R : ring} {M N : module R} (f : moduleiso M N).

Definition moduleiso_to_weq : (pr1module M) (pr1module N) := pr1 f.
Definition moduleiso_ismodulefun : ismodulefun moduleiso_to_weq := pr2 f.
Definition moduleiso_to_modulefun : R-mod(M, N) :=
(tpair _ (pr1weq moduleiso_to_weq) (pr2 f)).
End accessors_moduleiso.

Coercion moduleiso_to_weq : moduleiso >-> weq.
Coercion moduleiso_to_modulefun : moduleiso >-> modulefun.

Definition make_moduleiso {R} {M N : module R} f is : moduleiso M N := tpair _ f is.

Lemma isbinopfuninvmap {R} {M N : module R} (f : moduleiso M N) :
isbinopfun (invmap f).
Proof.
intros x y.
apply (invmaponpathsweq f).
rewrite (homotweqinvweq f (op x y)).
apply pathsinv0.
transitivity (op ((moduleiso_to_weq f) (invmap f x)) ((moduleiso_to_weq f) (invmap f y))).
apply (modulefun_to_isbinopfun f (invmap f x) (invmap f y)).
rewrite 2 (homotweqinvweq f).
apply idpath.
Defined.

Lemma islinearinvmap {R} {M N : module R} (f : moduleiso M N) : islinear (invmap f).
Proof.
intros r x.
apply (invmaponpathsweq f).
transitivity (module_mult N r x).
exact (homotweqinvweq f (module_mult N r x)).
transitivity (module_mult N r (moduleiso_to_weq f (invmap (moduleiso_to_weq f) x))).
rewrite (homotweqinvweq (moduleiso_to_weq f) x).
apply idpath.
apply pathsinv0.
apply (pr2 (moduleiso_ismodulefun f) r (invmap f x)).
Defined.

Definition invmoduleiso {R} {M N : module R} (f : moduleiso M N) : moduleiso N M.
Proof.
use make_moduleiso.
- exact (invweq f).
- apply make_dirprod.
+ exact (isbinopfuninvmap f).
+ exact (islinearinvmap f).
Defined.

Definition moduleiso' {R} (M N : module R) : UU :=
w : monoidiso (pr1module M) (pr1module N), islinear w.

Lemma moduleiso_to_moduleiso' {R} (M N : module R) :
moduleiso M N moduleiso' M N.
Proof.
intro w.
use tpair.
- use tpair.
+ exact w.
+ use tpair.
× exact (modulefun_to_isbinopfun w).
× apply (modulefun_unel w).
- exact (modulefun_to_islinear w).
Defined.

Lemma moduleiso'_to_moduleiso {R} (M N : module R) :
moduleiso' M N moduleiso M N.
Proof.
intro w.
use tpair.
- exact (pr1 w).
- apply make_dirprod.
+ exact (pr1 (pr2 (pr1 w))).
+ exact (pr2 w).
Defined.

Lemma moduleiso'_to_moduleisweq_iso {R} (M N : module R) :
isweq (moduleiso'_to_moduleiso M N).
Proof.
use (isweq_iso _ (moduleiso_to_moduleiso' M N)).
- intro w.
unfold moduleiso'_to_moduleiso, moduleiso_to_moduleiso'. cbn.
induction w as [w1 w2]; cbn.
use total2_paths_f; cbn.
× use total2_paths_f; cbn.
+ reflexivity.
+ apply isapropismonoidfun.
× apply isapropislinear.
- reflexivity.
Defined.

Definition moduleiso'_to_moduleweq_iso {R} (M N : module R) : (moduleiso' M N) (moduleiso M N) :=
make_weq (moduleiso'_to_moduleiso M N) (moduleiso'_to_moduleisweq_iso M N).