EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 2, 2021, 404-422
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Localization of Hopfian and Cohopfian Objects in the
Categories of A−Mod, AGr(A−Mod) and

COMP (AGr(A−Mod))

Seydina Ababacar Balde1,∗, Mohamed Ben Faraj Ben Maaouia2,
Ahmed Ould Chbih3

1 Applied Mathematics, UFR-SAT/Gaston BERGER, University, Saint-Louis, Senegal

Abstract. The aim of this paper is to study the localization of hopfian and cohopfian objects
in the categories A − Mod of left A-modules, AGr(A − Mod) of graded left A-modules and
COMP (AGr(A−Mod)) of complex sequences associated to graded left A-modules.
We have among others the main following results

(i) Let M be a noetherian graded left A-module, S a saturated multiplicative part formed by the
non-zero homogeneous elements of A verifying the left Ore conditions, N a submodule of M ,
M∗ is a noetherian quasi-injective complex sequence associated with M and N∗ is an essential
and completely invariant complex sub-sequence of M∗. Then, S−1(N∗) the complex sequence
of morphisms of left S−1A-modules is cohopfian if, and only, if S−1(M∗) is cohopfian ;

(ii) let M be a graded left A-module and S a saturated multiplicative part formed by the non-zero
homogeneous elements of A verifying the left Ore conditions. If M∗ is a hopfian, noetherian
and quasi-injective complex sequence associated with M , then the complex sequence of mor-
phisms of left S−1(A)-modules S−1(M∗) has the following property : �any epimorphism of
sub-complex S−1(N∗) of S−1(M∗) is an isomorphism � ;

(iii) let M be a graded left A-module, N a graded submodule of M , S a saturated multiplicative
part formed by the non-zero homogeneous elements of A verifying the left Ore conditions.
M∗ the quasi-projective complex sequence associated with M and N∗ a superfluous and
completely invariant complex sub-sequence of M∗. Then the complex morphism sequence of
left S−1(A)-modules S−1(N∗) is hopfian if, and only if, S−1(M∗/N∗) the complex sequence
associated with S−1(M/N) is hopfian.

2020 Mathematics Subject Classifications: Mathematics Subject classification codes

Key Words and Phrases: graded ring, a saturated multiplicative part formed by the non-zero
homogeneous elements of A,Ore conditions, hopfian, cohopfian, sequence complex, chain complex,
quasi-injective and quasi-projective.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.3889

Email addresses: sbalde878@gmail.com (S.A. Balde)
maaouiaalg@hotmail.com (M. BEN Maaouia)
achbih@gmail.com ( A. O. Chbih)

http://www.ejpam.com 404 c© 2021 EJPAM All rights reserved.



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 405

1. Introduction

In this paper, the ring A is supposed to be associative, unitary, not necessairly com-
mutative, and any left A-modules is unifary.
In this article, we study the localization of hopfian and cohopfian objects in the categories
A−Mod of left A-modules, AGr(A−Mod) of graded left A-modules and COMP (AGr(A-
−Mod)) of complex sequences associated to graded left A-modules. We rely on the ar-
ticles, �Graduation of Module of Fraction on a Graded Domain Ring not Necessarily
Commutative�[2], � Factorization of Graded Modules of Fractions �[3], �Module de
Fractions, Sous-modules S−saturée et Foncteur S−1 � [18] and�Hopfian and Cohopfian
Objects in the Categories of Gr(A−Mod) and COMP (Gr(A−Mod))�[23], which are
used as a basis for studying the notions of localization, hopficity and cohopficity. The tran-
sition to localization and the study of hopficity and cohopficity from the category of left
A−Mod whose objects are the left A-modules and the morphisms are the left A-module
morphisms to the category AGr(A −Mod) whose objects are the graded left A-module
and the morphisms are the graded left A-module graded morphisms and AGr(A−Mod)
to the category of COMP (AGr(A−Mod)) whose the objects are the complex sequences
of graded left A-modules and the morphisms are the chain complexes associated to the
graded morphims of graded left A-modules is not easy. In our opinion, these reasons jus-
tify our work. Thus, the paper is organized as follows

In the section 2, we study the localization of hopfian and cohopfian objects in the category
A−Mod and we show the following results :

(i) Let A be a ring, S a saturated multiplicative part of A verifying the left Ore condi-
tions, M a left A-module. If S−1M is hopfian, then M is hopfian ;

(ii) Let A be a ring, S a saturated multiplicative part of A verifying the left Ore condi-
tions, M a left A-module. If M is a cohopfian and completely invariant submodule
of left A-module S−1(M), then S−1(M) is a cohopfian left S−1(A)(respectively A)-
module ;

(iii) Let M be a noetherian quasi-injective left A-module, N be an essential and com-
pletely invariant submodule of M and S a saturated multiplicative part of A verifying
the left Ore conditions. Then, the left S−1(A)-modules S−1(N) is cohopfian if, and
only, if S−1(M) is cohopfian left S−1(A)-module ;

(iv) Let M be a quasi-projective left A-module, N a superfluous and completely invariant
sub-module of M , S a saturated multiplicative part of A verifying the left Ore con-
ditions. Then the left S−1(A)-modules S−1(N) is hopfian if, and only if, S−1(M/N)
is hopfian.

In section 3, we study the localization of hopfian and cohopfian objects in the category
AGr(A−Mod) and we prove the following results :



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 406

(i) Let A =
⊕
n∈Z

An be a graded ring, S a saturated multiplicative part formed by the

non-zero homogeneous elements of A verifying the left Ore conditions and M =⊕
n∈Z

Mn a graded left A-module, then S−1(M) =
⊕
i∈Z

(S−1M)i is hopfian(respectively

cohopfian) if, and only, if (S−1M)i is a hopfian(respectiveley cohopfien) group ;

(ii) let A be a graded ring, S a saturated multiplicative part formed by the non-zero
homogeneous elements of A verifying the left Ore conditions, M a graded left A-
-module. Then, if S−1(M) is a hopfian left graded S−1(A)-module, implies that M
is a hopfian left graded A-module ;

(iii) if S−1(M) is a left graded hopfian S−1(A)-module, then Mn for all n ∈ Z is a hopfian
group ;

(iv) let A be a graded ring, S a saturated multiplicative part formed by the non-zero
homogeneous elements of A verifying the left Ore conditions, M left graded A-
-module. Then, if M is a cohopfian and completely invariant submodule of left
A-module S−1(M), implies that, S−1(M) is a cohopfian graded left S−1(A)-module;

(v) ifM is a cohopfian and completely invariant submodule of the leftA-module S−1(M),
then (S−1M)i is a cohopfian group ;

(vi) Let M be a noetherian quasi-injective graded left A-module, N be an essential and
completely invariant graded submodule of M and S a saturated multiplicative part
formed by the non-zero homogeneous elements of A verifying the left Ore conditions.
Then, the graded left S−1(A)-modules S−1(M) is cohopfian if, and only, if S−1(N)
is cohopfian graded left S−1(A)-module ;

(vii) Let M be a graded quasi-projective left A-module, N a superfluous and completely
invariant graded sub-module of M , S a saturated multiplicative part formed by the
non-zero homogeneous elements of A verifying the left Ore conditions. Then the left
S−1(A)-modules S−1(N) is hopfian if, and only if, S−1(M/N) is hopfian.

In the section 4, we study the localization of hopfian and cohopfian objects in the category
COMP (AGr(A−Mod)) and we show the following results :

(i) Let A be a graded ring, S a saturated multiplicative part formed by the non-zero ho-
mogeneous elements of A verifying the left Ore conditions, M a graded left A-module
and M∗ the complex sequence of morphisms of graded left A-modules associated with
M . Then, if the complexe S−1(M∗) of morphisms of graded left S−1(A)-module is
hopfian, implies that M∗ is hopfian ;

(ii) Let A be a graded ring, S a saturated multiplicative part formed by the non-zero
homogeneous elements of A verifying the left Ore conditions, M a graded left A-
module. If M∗ the complex sequence associated with M , is cohopfian and completely
invariant, then S−1(M∗), the complex sequence associated with S−1(M) is cohopfian;



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 407

(iii) let M be a noetherian graded left A-module, S a saturated multiplicative part formed
by the non-zero homogeneous elements of A verifying the left Ore conditions, N a
submodule of M , M∗ is a noetherian quasi-injective complex sequence associated
with M and N∗ is an essential and completely invariant complex sub-sequence of
M∗. Then, S−1(N∗) the complex sequence of morphisms of left S−1(A)-modules is
cohopfian if, and only, if S−1(M∗) is cohopfian ;

(iv) let M be a graded left A-module and S a saturated multiplicative part formed
by the non-zero homogeneous elements of A verifying the left Ore conditions. If
M∗ is a hopfian, noetherian and quasi-injective complex sequence associated with
M , then the complex sequence of morphisms of left S−1A-modules S−1M∗ has the
following property : �any epimorphism of sub-complex S−1(N∗) of S−1(M∗) is an
isomorphism � ;

(v) let M be a graded left A-module, N a graded submodule of M , S a saturated
multiplicative part formed by the non-zero homogeneous elements of A verifying the
left Ore conditions. M∗ the quasi-projective complex sequence associated with M
and N∗ a superfluous and completely invariant complex sub-sequence of M∗. Then
the complex morphism sequence of left S−1(A)-modules S−1(N∗) is hopfian if, and
only if, S−1(M∗/N∗) the complex sequence associated with S−1(M/N) is hopfian.

2. Preliminaries

Definition 1.
Let A be a ring and {An}n∈Z a family of sub-group of A. If

(i) A =
⊕
n∈Z

An;

(ii) An ·Am ⊂ An+m, ∀ n, m ∈ Z.

Then we say that A is a graded ring. Else, if An = 0,∀n < 0. Then A is called positively
graded ring.

Definition 2.
Let A a graded ring, x be a non-zero element of A, we say that x is homogeneous of degree
n, if there exists n such that x ∈ An and we note deg(x) = n.

such that x ∈ An and we note that deg(x) = n.

Definition 3.
Let A =

⊕
n∈Z

An be a graded ring and M be a left A−module, then M is called a graded left

A−module if there exists a sequence (Mn)n∈Z of sub-group of M such that

(i) M =
⊕
n∈Z

Mn;



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 408

(ii) An ·Md ⊂Mn+d, ∀ n, d ∈ Z.

Definition 4.
Let A =

⊕
n∈Z

An be a graded ring, M =
⊕
n∈Z

Mn be a graded left A−module and N is a

sub-module of M , then N is called a graded sub-module of M , if ∀x =
∑

n∈Z xn ∈ N , with
xn ∈Mn, then xn ∈ N , ∀n ∈ Z.

Definition 5.
Let A =

⊕
n∈ZAn be a graded ring, M =

⊕
n∈Z

Mn, N =
⊕
n∈Z

Nn are two graded left A−modules

and f : M −→ N is a morphism of left A−modules, then f is called a graded morphism if for
any ms ∈Ms then f(ms) ∈ Ns.

Theorem 1.
Let A be a graded ring, the category of graded left A−module is the category denoted by
AGr(A−Mod) whose

(i) The objects are the graded left A−modules;

(ii) The morphisms are the graded morphisms.

Proof.
See [23]

Theorem 2.
Let A be a graded ring, M a graded left A-module and S S a saturated multiplicative part
formed by the non-zero homogeneous elements of A verifying the left Ore conditions, then :

(i) S−1A =
⊕
i∈Z

(S−1A)i is a graded ring, where

(S−1A)i = {as ∈ S
−1A,∃k, a ∈ Ak and deg(s) = k − i}.

(ii) S−1M =
⊕
i∈Z

(S−1M)i is a graded left S−1A-module, where

(S−1M)i = {ms ∈ S
−1M,∃p,m ∈Mp and deg(s) = p− i}.

Proof.
See [1]

Proposition 1.

Let A =
⊕
n∈N

An be a graded ring, M =
⊕
n∈Z

Mn and N =
⊕
n∈Z

Nn are two graded left

A-modules, f : M −→ N is graded morphism and S the set of non-zero homogeneous
elements of A, then we have :

(i) the following complex sequences :

S−1(M∗) : · · · −→ S−1(M(n+1))
S−1(dn+1)−→ S−1(M(n))

S−1(dn)−→ S−1(M(n−1)) −→ · · ·



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 409

(ii) the following chain complexes :

S−1(M∗) : · · · //

S−1(f∗)
��

// S−1(M(n+ 1))
S−1(dn+1)//

S−1(f(n+1))
��

S−1(M(n))
S−1(dn)//

S−1(f(n))
��

S−1(M(n− 1)) //

S−1(f(n−1))
��

· · ·

S−1(N∗) : · · · // S−1(N(n+ 1))
S−1(d′n+1)// S−1(N(n))

S−1(d′n)// S−1(N(n− 1)) // · · ·

Proof.
See [3]

Theorem 3.

Let M∗ : . . .M(n + 1)
dn+1−−−→ M(n)

dn−→ M(n − 1) −→ . . . be an object of COMP (AGr(A-
−Mod)). M∗ is quasi-injective in COMP (Gr(A−Mod)) if, and only, if for all n ∈ Z, M(n)
is a quasi-injective in COMP (AGr(A−Mod)).

Proof.
See [23]

Theorem 4.

Let M∗ : . . .M(n + 1)
dn+1−−−→ M(n)

dn−→ M(n − 1) −→ . . . be an object of COMP (Gr(A-
−Mod)). M∗ is quasi-projective if, and only, if for all n ∈ Z, M(n) is a quasi-projective left
A-modules .

Proof.
See [23]

3. Localization of hopfian and cohopfian objects in the category of
A−Mod

Definition 6.
Let M a left A-module. Then M is said hopfian (respectively cohopfian), if any epimorphism
(respectively monomorphism) of M is an automorphism of M .

Theorem 5.
Let A be a ring, S a saturated multiplicative part of A verifying the left Ore conditions, M a
left A-module. If S−1(M) is a hopfian left S−1(A)-module, then M is a hopfian left A-module.

Proof.
Let f : M −→M an epimorphism of left A-module. Then S−1(f) : S−1(M) −→ S−1(M)

define by [S−1(f)](ms ) = f(m)
s is an endomorphisme of S−1(A)-Mod. Let m′

s ∈ S
−1(M),

as f is an epimorphism, then there exists m ∈M,f(m) = m′.

So [S−1(f)](ms ) = f(m)
s = m′

s , thus S−1(f) is an epimorphism, since S−1(M) is hopfian,
so S−1(f) is an automorphism of S−1M .



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 410

Let m1 and m2 ∈M such that
f(m1) = f(m2)

=⇒ f(m1)
1 = m2

1 =⇒ [S−1(f)](m1) = [S−1(f)](m2)
=⇒ m1 = m2

Thus f is an automorphism of M , so M is hopfian.

Theorem 6.
Let A be a ring, S a saturated multiplicative part of A verifying the left Ore conditions, M
a left A-module. If M is a cohopfian and completely invariant submodule of left A-module
S−1(M), then S−1M is a cohopfian left S−1(A)(respectively A)-module.

Proof.
Let g : S−1(M) −→ S−1(M) be a S−1A-morphism, we remark also that g is a A-morphism.
As M is a completely invariant submodule of left A-module S−1(M), so g(M) ⊂M .
Suppose that g is a monomorphism.
Thus the induce morphism gind : M −→ M is a monomorphisme of M , since M is
cohopfian, then gind is an automorphism of M .
Let’s consider
S−1(gind) : S−1(M) −→ S−1(M)
m
s 7−→

gind(m)
s

So gind(m)
s = 1

s .
g(m)
1 .

Or g is a S−1(A)-morphisme, so 1
s .
gind(m)

1 = gind(
1
s
m
1 ) = gind(

m
s ) =⇒ S−1(gind) = g

g is a monomorphism, then gind is a monomorphism.
As M is cohopfian, then gind is an automorphism of M .
Let m′

s ∈ S
−1(M) =⇒ m′ ∈M , so there exists m ∈M : gind(m) = m′

thus m
s ∈ S

−1M , [S−1(gind)](
m
s ) = g(m)

s = m′

s

hence (g(ms )) = m′

s .
Thus S−1M is a cohopfian left S−1A-module.
Let’s prove now that S−1(M) is a cohopfian left A-module

Let f : S−1(M) −→ S−1(M) be a monomorphism of leftA-module, then f : S−1(M) −→
S−1(M) is a monomorphism, indeed,

let m
s and m′

s′ ∈ S
−1M : [S−1(f)](ms ) = [S−1(f)](m

′

s′ ) =⇒ f(m)
s = f(m′)

s′ =⇒ ∃x, y ∈ S such
that :{
x.f(m) = y.f(m′)

xs = ys′
=⇒

{
f(x.m) = f(y.m′)

xs = ys′
=⇒

{
x.m = y.m′

xs = ys′
=⇒ m

s = m′

s =⇒

S−1(f) is a monomorphisme, or S−1(M) is cohopfian, then S−1(f) is an automorphism.

Let m′

s′ ∈ S−1(M), since S−1(f) is an automorphism, then there exists m
s ∈ S−1M

such that
[S−1(f)](ms ) = m′

s′ =⇒ f(m)
s = m′

s′ =⇒
s
s
.m

s = m′

s′ =⇒ s.f(m
s
)

s = m′

s′ =⇒ f(m)
s = m′

s′ ,

thus there exists m
s ∈ S−1M such that f(ms ) = m′

s′ =⇒ f is an epimorphism, hence
S−1(M) is cohopfian.



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 411

Theorem 7.
Let M be a noetherian quasi-injective left A-module, N be an essential and completely invari-
ant submodule of M and S a saturated multiplicative of A verifying the left Ore conditions.
Then, the left S−1(A)-modules S−1(N) is cohopfian if, and only, if S−1(M) is cohopfian left
S−1(A)-module.

Proof.
Suppose that S−1(M) is cohopfian and left S−1(f) : S−1(N) −→ S−1(N) be a monomor-
phism. As S−1(M) is quasi-injective because M is noetherian. Then, there exists S−1(g) ∈
End(S−1(M)) such that S−1(g|S−1N ) = S−1(f).
S−1(g) is injective since S−1(N) is essential in S−1(M), and as S−1(M) is cohopfian,
S−1(g) is invertible. Let x

s ∈ S
−1(N), there exists y

t ∈ S
−1(M) such that x

s = [S−1(g)](yt ).
Or S−1(g−1) ∈ End(S−1()N)) and S−1(N) is completely invariant, so y

t = [S−1(g−1)](xs ) ∈
S−1(N), thus S−1(f) is an automorphisme, consequentely S−1(N) is cohopfian.
Reciprocally, suppose that S−1(N) is cohopfian and let S−1(f) : S−1(M) −→ S−1(M)
a monomorphism. Then S−1(f|S−1(N)) is a monomorphism of S−1(N). Thus, S−1(f) ∈
Aut(S−1(N)), hence [S−1(f)](S−1(N)) = S−1(N). As S−1(M) is quasi-injective, then
there exists S−1(L) a submodule of S−1(M) such that S−1(M) = [S−1(f)](S−1(M)) ⊕
S−1(L). Thus, we have 0 = [S−1(f)](S−1N) ∩ S−1(L) = S−1(N) ∩ S−1(L), since S−1(N)
is essential, then S−1(L) = 0, hence S−1(M) = [S−1(f)](S−1M), thus S−1(f) is an epi-
morphism, so S−1(M) is cohopfian.

Theorem 8.
Let M be a quasi-projective left A-module, N an superfluous and completely invariant sub-
module of M , S a saturated multiplicative part of A verifying the left Ore conditions. Then
the left S−1(A)-modules S−1(N) is hopfian if, and only if, S−1(M/N) is hopfian.

Proof.
Suppose that S−1(M/N) is hopfian and let S−1(f) : S−1(M) −→ S−1(M) an epimor-
phism.
As S−1(N) is completely invariant, then [S−1(f)](S−1(N)) ⊂ S−1(N), implies S−1(f)
induces an epimorphism S−1(f) : S−1(M/N) −→ S−1(M/N), since S−1(M/N) is hop-
fian, then S−1(f) is a graded automorphism. Put S−1(K) = ker(S−1(f)) and S−1(π) :
S−1(M) −→ S−1(M/N) the canonical projection, we have :
S−1(f) ◦ [S−1(π)](S−1(K)) = S−1(π ◦ f(K)) = 0
Indeed, ∀xs ∈ S

−1(K), we have :
S−1[(f ◦ π)](xs ) = [S−1(π ◦ f)](xs ), so

[S−1(π ◦ f)](xs ) = [S−1(π)](f(x)s ) = [S−1(π)](0s ) = 0, thus [S−1(f ◦ π)](K) = 0.
we have :
[S−1(f ◦ π)](K) = [S−1(π ◦ f)](K) = 0 =⇒ [S−1(f)](π(K)) = 0 =⇒ [S−1(π)](K) ⊂
S−1(N) =⇒ S−1(K) ⊂ S−1(N)
Since M is quasi- projective =⇒ S−1(M) is quasi-projective, there exists an endomor-
phism S−1(s) : S−1(M) −→ S−1(M) such that S−1(f ◦ s) = S−1(idS−1(M)), this implies
S−1(M) = S−1(K ⊕ Im(s)), or K = N and S−1(N) is superfluous in S−1(M), then



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 412

S−1(M) = S−1(Im(s)), so S−1(K) = ker[S−1(f)] = 0, thus S−1(f) is a monomorphism,
finally, S−1(f) is a monomorphism, so S−1(M) is a hopfian left S−1(A)-module.
Reciprocally,
let S−1(M) be hopfian, show that S−1(M/N) is hopfian.
Let S−1(ϕ) : S−1(M/N) −→ S−1(M/N) be an epimorphism of left S−1A-module, as
M is quasi-projective, then S−1(M) is quasi-projective. Consider S−1(π) : S−1(M) −→
S−1(M/N), then there exists S−1(f) ∈ End(S−1(M)) such that S−1(π ◦ f) = S−1(ϕ ◦ π).
Since S−1(ϕ) is an epimorphism, ∀xs ∈ S

−1(M/N),∃(yt ) ∈ S
−1(M/N) such that

[S−1(ϕ)](yt ) = x
s = [S−1(ϕ)](π(y)t )

[S−1(π ◦ f)](yt ) = [S−1(ϕ ◦ π)](yt )

[S−1(π)](f(x)s ) = [S−1(ϕ)](yt )

=⇒ [S−1(ϕ)](yt ) = [S−1(f)](yt ) = x
s

=⇒ f(y)
t −

x
s = 0

=⇒ f(y)
t −

x
s ∈ S

−1(N),
then S−1(M) = Im(S−1(f)) + S−1(N), as S−1(N) is superfluous, then Im(S−1(f)) =
S−1(M) =⇒ S−(f) is an epimorphism. So, S−1f is an automorphism, because S−1(M)
is hopfian.
So the restriction of S−1(f) over S−1(N) is a automorphism of S−1(N).
If [S−1(ϕ)](xs ) = [S−1(f)](xs ) = 0, then [S−1(f)](xs ) ∈ S−1(N), or S−1(N) is completely

invariant, then x
s ∈ S

−1(N), so x
s = 0 =⇒ ker(S−1(ϕ)) = S−1(N) = 0 =⇒ S−1(ϕ) is a

monomorphism =⇒ ϕ is an automorphism, lastly S−1(M/N) is hopfian, hence S−1(M/N)
is a hopfian left S−1(A)-module.

4. Localization of hopfian and cohopfian objects in the category of
AGr(A−Mod)

Definition 7.
Let M a graded left A-module. Then M is said hopfian (respectively cohopfian), if any
epimorphism (respectively monomorphism) of M is an automorphism of M .

Lemma 1.
Let A =

⊕
n∈Z

An be a graded ring, M =
⊕
n∈Z

Mn a graded left A-module, then M is hop-

fian(respectively cohopfian) if, and only if, Mn is a hopfian(respectiveley cohopfien) groupe.

Proof.

Suppose that M =
⊕
n∈Z

Mn is hopfian, show that Mn is a hopfian group.

Let f : M =
⊕
n∈Z

Mn −→ M =
⊕
n∈Z

Mn be a graded morphism, so we have the induce

morphism of groups : fn : Mn −→ Mn, for all x ∈ Mn, we have fn(x) = f(xn). We see
that fn is well defined because f is graded, moreover, for all xn and yn ∈ Mn, we have
fn(xn + yn) = f(xn + yn) = f(xn) + f(yn) = fn(xn) + fn(yn) =⇒ fn is a morphism of



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 413

groups.
let h : Mn −→Mn be an epimorphism of groups.
Put for all xi, i 6= n, f(xi) = xi and for all xn ∈ Mn, f(xn) = h(xn), it is easy to prove
that f is a graded epimorphism of graded left A-modules.
Since f is an automorphism because M hopfian by hypothesis, so h is an automorphism,
thus Mn is hopfien.

Suppose that Mn is hopfian, show that M =
⊕
n∈Z

Mn is hopfian.

Let f : M =
⊕
n∈Z

Mn −→M =
⊕
n∈Z

Mn be an epimorphism of left A-module. Show that f

is an automorphism. Prove that fn : Mn −→Mn is epimorphism of groups for all n ∈ Z
Let yn ∈Mn, then there exists x ∈M =

⊕
n∈Z

An.

Suppose that x =
∑

finie xt, or for all t, f(xt) ∈Mt

if t 6= n, f(xt) = 0, since f(x) = yn ∈ Mn, so for all yn ∈ Mn, there exists xn ∈ Mn such
that f(xn) = yn, or f(xn) = fn(xn) =⇒ fn(xn) = yn, so fn is an epimorphism, as Mn is
hopfian, so fn is an automorphism =⇒ f is also an automorphism, consequentely, M is
hopfian.
For the hopfian case, the proof is similary.

Theorem 9.
Let A =

⊕
n∈Z

An be a graded ring, S a saturated multiplicative part formed by the non-zero

homogeneous elements of A verifying the left Ore conditions and M =
⊕
n∈Z

Mn a graded left

A-module, then S−1(M) =
⊕
i∈Z

(S−1M)i is hopfian(respectively cohopfian) if, and only, if

(S−1M)i is a hopfian(respectiveley cohopfien) group.

Proof.

It suffices to prove that (S−1f) : S−1(M) =
⊕
i∈Z

(S−1M)i −→ S−1(M) =
⊕
i∈Z

(S−1M)i is

graded.
Since (S−1M)i = {ms ∈ S

−1M,∃p,m ∈ Mp and deg(s) = p− i} , we have [S−1(f)](ms ) =
f(m)
s or f is graded =⇒ f(m) ∈Mp =⇒ f(m)

s ∈ (S−1M)i, so S−1(f) is a graded morphism.
Then, by Lemma(1), we obtain the result.

Theorem 10.
Let A =

⊕
n∈Z

An be a graded ring, S a saturated multiplicative part formed by the non-zero

homogeneous elements of A verifying the left Ore conditions, M =
⊕
n∈Z

Mn a graded left

A-module. Then, if S−1(M) is a hopfian left graded S−1(A)-module, implies that M is a
hopfian left graded A-module.



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 414

Proof.

Let f : M −→M be a graded epimorphism ofM . Then S−1f : S−1(M) =
⊕
i∈Z

(S−1M)i −→

S−1(M) =
⊕
i∈Z

(S−1M)i defined by [S−1(f)](ms ) = f(m)
s is a graded endomorphism of

graded left S−1(A)-module, because [S−1(f)](mi
s ) = f(mi)

s , since f is graded, then f(mi) ∈
Mi, thus f(xi)

s ∈ (S−1M)i. Let m′

s ∈ S
−1(M), as f is a graded epimorphism, then there

exists m ∈M,f(m) = m′.

So [S−1(f)](ms ) = f(m)
s = m′

s , hence S−1(f) is a graded epimorphism, since S−1(M) is
hopfian, then S−1(f) is a graded automorphism of S−1(M).
Let m1 and m2 ∈M such that
f(m1) = f(m2)

=⇒ f(m1)
1 = f(m2)

1 =⇒ [S−1(f)](m1) = [S−1(f)](m2)
=⇒ m1 = m2

Thus f is a graded automorphism of M , so M is a hopfian graded left A-module.

Corollary 1.
Under the same conditions of the previous theorem. If S−1(M) is a hopfian graded left
S−1(A)-module, then Mn for all n ∈ Z is a hopfian group.

Proof.
It’s obvious by Theorem(10)

Theorem 11.
Let A graded ring, S a saturated multiplicative part formed by the non-zero homogeneous
elements of A verifying the left Ore conditions, M left graded A-module. Then, if M is a
cohopfian and completely invariant submodule of the left A-module S−1(M), implies that,
S−1(M) is a left cohopfian graded S−1(A)-module.

Proof.
Let g : S−1(M) −→ S−1(M) a graded S−1(A)-morphisme, we remark also that g is a
graded A-morphisme. As M is completely invariant as graded submodule of graded left
A-module S−1M , so g(M) ⊂M .
Suppose that g is a graded monomorphism.
So the induce graded morphism gind : M −→M is a graded monomorphism of M and, as
M is cohopfian, then gind is a graded automorphism of M .
Consider
S−1(gind) : S−1(M) −→ S−1(M)
m
s 7−→

g(m)
s

Thus g(m)
s = 1

s
g(m)
1 .

Or g is a graded S−1(A)-morphism, so 1
s = g(m)

1 = g(1s
m
1 ) = g(ms ) =⇒ S−1(gind) = g

g is a graded monomorphism, then gind is a graded monomorphism.
Since M is cohopfian, then gind is a graded automorphism of M .



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 415

Let m′

s ∈ S
−1(M) =⇒ m′ ∈M , so there exists m ∈M : gind(m) = m′

hence m
s ∈ S

−1(A), [S−1(gind)](
m
s ) = g(m)

s = m′

s

so (g(ms )) = m′

s .
Thus S−1(M) is a cohopfian left S−1A-module.

Corollary 2.
Under the same conditions of the previous theorem If M is a cohopfian and completely invariant
submodule of the left A-module S−1M , then (S−1M)i is a cohopfian group.

Proof.
By Theorem(11)

Theorem 12.
Let M be a noetherian quasi-injective graded left A-module, N be an essential and completely
invariant graded submodule of M and S a saturated multiplicative part formed by the non-zero
homogeneous elements of A verifying the left Ore conditions. Then, the garded left S−1A-
modules S−1(M) is cohopfian if, and only, if S−1(N) is cohopfian garded left S−1(A)-module.

Proof.
Suppose that S−1(M) is cohopfian and let S−1(f) : S−1(N) −→ S−1(N) be a graded
monomorphism. As S−1(M) is quasi-injective because M is noetherian. Then, there ex-
ists a graded morphism S−1(g) ∈ End(S−1(M)) such that S−1(g|S−1(N)) = S−1(f).
S−1(g) is injective since S−1(N) is essential in S−1(M), and as S−1(M) is cohopfian,
S−1(g) is invertible. Let x

s ∈ S
−1(N), there exists y

t ∈ S
−1(M) such that x

s = [S−1(g)](yt ).
Or S−1(g−1) ∈ End(S−1(N)) and S−1(N) is completely invariant, so y

t = [S−1(g−1)](xs ) ∈
S−1(N), thus S−1(f) is an automorphisme, consequentely S−1(N) is cohopfian.
Reciprocally, suppose that S−1(N) is cohopfian and let S−1(f) : S−1(M) −→ S−1(M)
a graded monomorphism. Then S−1(f|S−1(N)) is a graded monomorphism of S−1(N).
Thus, S−1(f) ∈ Aut(S−1(N)), hence [S−1(f)](S−1(N)) = S−1(N). As S−1(M) is quasi-
injective, then there exists S−1(L) a submodule of S−1(M) such that S−1(M) = [S−1(f)](S−1(M))⊕
S−1(L). Thus, we have 0 = [S−1(f)](S−1(N))∩S−1(L) = S−1(N)∩S−1(L), since S−1(N)
is essential, then S−1(L) = 0, hence S−1(M) = S−1[(f)](S−1(M)), thus S−1(f) is an epi-
morphism, so S−1(M) is cohopfian left S−1A-module.

Theorem 13.
Let M be a graded quasi-projective left A-module, N a superfluous and completely invariant
graded sub-module of M and S a saturated multiplicative part formed by the non-zero ho-
mogeneous elements of A verifying the left Ore conditions. Then the left S−1(A)-modules
S−1(N) is hopfian if, and only if, S−1(M/N) is hopfian.

Proof.
Suppose that S−1(M/N) is hopfian and let S−1(f) : S−1(M) −→ S−1(M) a graded epi-
morphism.
As S−1(N) is completely invariant, then [S−1(f)](N) ⊂ S−1(S−1(N)), implies S−1(f)



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 416

induces a graded epimorphism S−1((f)) : S−1(M/N) −→ S−1(M/N), since S−1(M/N)
is hopfian, then S−1((f)) is a graded automorphism. Put S−1(K) = ker(S−1(f)) and
S−1(π) : S−1(M) −→ S−1(M/N) the canonical projection, we have :
S−1((f)) ◦ [S−1(π)](S−1(K)) = S−1(π ◦ f(K)) = 0
Indeed, ∀xs ∈ S

−1(K), we have :
[S−1(f ◦ π)](xs ) = [S−1(π ◦ f)](xs ), so

[S−1(π ◦ f)](xs ) = [S−1(π)](f(x)s ) = [S−1(π)](0s ) = 0, thus [S−1(f ◦ π)](K) = 0.
we have :
[S−1(f ◦ π)](K) = S−1[π ◦ f ](K) = 0 =⇒ [S−1(f)](π(K)) = 0 =⇒ [S−1(π)](K) ⊂
S−1(N) =⇒ S−1(K) ⊂ S−1(N)
Since M is graded quasi- projective =⇒ S−1(M) is graded quasi-projective, there ex-
ists a graded endomorphism S−1(s) : S−1(M) −→ S−1(M) such that S−1(f ◦ s) =
S−1(idS−1(M)), this implies S−1(M) = S−1(K ⊕ Im(s)), or K = N and S−1(N) is su-
perfluous in S−1(M), then S−1(M) = S−1(Im(s)), so S−1(K) = ker(S−1(f)) = 0, thus
S−1(f) is a monomorphism,
finally, S−1(f) is a monomorphism, so S−1(M) is a graded hopfian left S−1(A)-module.
Reciprocally, suppose that S−1(M) is hopfian, show that S−1(M/N) is hopfian.
Let S−1(ϕ) : S−1(M/N) −→ S−1(M/N) a graded epimorphism of left S−1(A)-module, as
M is quasi-projective, then , S−1(M) is quasi-projective. Consider S−1(π) : S−1(M) −→
S−1(M/N), then there exists S−1(f) ∈ End(S−1(M)) such that S−1(π ◦ f) = S−1(ϕ ◦ π).
Since S−1(ϕ) is an epimorphism, ∀ (xs ) ∈ S−1(M/N),∃ (yt ) ∈ S

−1(M/N) such that

[S−1(ϕ)](yt ) = x
s = [S−1(ϕ)](π(y)t )

[S−1(π ◦ f)](yt ) = [S−1(ϕ ◦ π)](yt )

[S−1(π)]( (f(x))s ) = [S−1(ϕ)](yt )

=⇒ [S−1(ϕ)](yt ) = [S−1(f)](yt ) = x
s

=⇒ (f(y)t −
x
s = 0

=⇒ f(y)
t −

x
s ∈ S

−1(N),
then S−1(M) = Im(S−1(f)) + S−1(N), as S−1(N) is superfluous, then Im(S−1(f)) =
S−1(M) =⇒ S−1(f) is a graded epimorphism. So, S−1(f) is a graded automorphism,
because S−1(M) is hopfian.
So the restriction of S−1(f) over S−1(N) is a graded automorphism of S−1(N).
If [S−1(ϕ)](xs ) = [S−1(f)](xs ) = 0, then [S−1(f)](xs ) ∈ S−1(N), or S−1(N) is completely

invariant, then x
s ∈ S−1(N), so x

s = 0 =⇒ ker(S−1(ϕ)) = S−1(N) = 0 =⇒ S−1(ϕ) is
a monomorphism =⇒ S−1(ϕ) is an automorphism, lastly S−1(M/N) is hopfian, hence
S−1(M/N) is a hopfian left S−1A-module.

5. Localization of hopfian and cohopfian objects in the
COMP (AGr(A−Mod)) category

Definition 8.
Let M∗ an object of COMP (AGr(A−Mod)). Then M∗ is said to be hopfian (resp. cohopfian)



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 417

if any epimorphism (resp. monomorphism) f∗ of M∗ is an automorphism.

Lemma 2.
Let M be a graded left A-module, f : M −→ M a graded morphism and S a saturated
multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore
conditions. If f is an epimorphism (respectively a monomorphism), then S−1(f(n)) is an
epimorphism (respectively monomorphism).

Proof.
Since f(n) is the induce of f , then S−1(f(n)) is an epimorphism (respectively a monomor-
phisme).

Lemma 3.
Let M be a graded left A-module and S a saturated multiplicative part formed by the non-zero
homogeneous elements of A verifying the left Ore conditions. Then S−1(M) is a hopfian (re-
spectively cohopfian) garded left A-module if any n ∈ Z, S−1(M(n)) is a hopfian (respectively
cohopfian) graded left A-module.

Proof.
Let S−1(g) : S−1(M(n)) −→ S−1(M(n)) be a graded epimorphism (respectively a monomor-

phism), since S−1(M) = S−1(M(n)
⊕
k>n

Mn+k) ∼= S−1(M(n)
⊕
k>n

Mn+k) . Put S−1(f) =

S−1(g) + S−1(idMn+k
), where S−1(f) is an epimorphism (respectively a monomorphism)

of S−1(M).
As M is hopfian (respectively cohopfian) this implies that f is an isomorphism, i.e S−1(f)
is an isomorphism of S−1(M), thus S−1(g) is an isomorphism of S−1M(n)), so S−1(M(n))
is hopfian (respectivement cohopfian).

Theorem 14.
Let A be a graded ring, S a saturated multiplicative part formed by the non-zero homogeneous
elements of A verifying the left Ore conditions, M a graded left A-module and M∗ the complex
sequence of morphisms of graded left A-modules associated with M . Then, if the complexe
complex S−1(M∗) of morphisms of graded left S−1(A)-modules is hopfian, implies that M∗ is
hopfian.

Proof.
Let f∗ : M∗ −→ M∗ be an epimorphism of M∗. Then S−1(f(n)) : S−1(M(n)) −→
S−1(M(n)) is a graded epimorphism of graded left S−1(A)-module for all n ∈ Z. as
S−1M(n) is hopfian for all n ∈ Z, so S−1(f(n)) is a graded automorphism of S−1(M(n)).
Let m1 and m2 ∈M(n) such that
f(n)(m1) = f(n)(m2)

=⇒ f(n)(m1)
1 = f(n)(m2)

1 =⇒ [S−1(f(n))](m1) = [S−1(f(n))](m2)
=⇒ m1 = m2

Thus f(n) is a graded automorphism of M(n) for any n ∈ Z, so M is hopfian.



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 418

Theorem 15.
Let A be a graded ring, S a saturated multiplicative part formed by the non-zero homoge-
neous elements of A verifying the left Ore conditions, M a graded left A-module. If M∗ the
complex sequence associated with M , is cohopfian and completely invariant, then S−1(M∗),
the complex sequence associated with S−1(M) is cohopfian.

Proof.
Let g : S−1(M∗) −→ S−1(M∗) be a graded S−1(A)-morphism, we remark also that g is
a graded A-morphism. Since M∗ is completely invariant complex sequence of left S−1A-
modules , so g(M∗) ⊂M∗.
Suppose that g is an monomorphism.
Thus the induce morphism gind : M∗ −→ M∗ is a monomorphism of M∗ and since M∗ is
cohopfian, then gind is an automorphism graded of M∗.
Consider
S−1(gind(n)) : S−1(M(n)) −→ S−1(M(n))
m
s 7−→

g(n)(m)
s

So g(n)(m)
s = 1

s
g(n)(m)

1 .

Or g(n) is a graded S−1(A)-morphism , thus 1
s = g(n)(m)

1 = g(n)(1s
m
1 ) = g(n)(ms ) =⇒

S−1(gind(n)) = g(n)
g(n) is a graded monomorphism, then gind(n) is a graded monomorphism.
As M(n) is cohopfian for all n ∈ Z, then gind(n) is a graded automorphism of M(n).
Let m′

s ∈ S
−1(M(n)) =⇒ m′ ∈M(n), so there exists m ∈M(n) : gind(n)(m) = m′

hence m
s ∈ S

−1(M(n)), [S−1(gind(n))](ms ) = g(n)(m)
s = m′

s

thus (g(n)(ms )) = m′

s .
hence S−1(M(n)) for all n ∈ Z is cohopfian, consequentely S−1(M∗) is cohopfian.

Theorem 16.
Let M be a noetherian graded left A-module, S a saturated multiplicative part formed by the
non-zero homogeneous elements of A verifying the left Ore conditions, N a submodule of M ,
M∗ is a noetherian quasi-injective complex sequence associated with M and N∗ is an essential
and completely invariant complex sub-sequence of M∗. Then, S−1(N∗) the complex sequence
of morphisms of left S−1(A)-modules is cohopfian if, and only, if S−1(M∗) is cohopfian.

Proof.
Suppose that S−1(M∗) is cohopfian and let S−1(f∗) : S−1(N∗) −→ S−1(N∗) be a monomor-
phism. As M∗ is noetherian, then S−1(M∗) is noetherian quasi-injective, so for all
n ∈ Z, S−1(M(n)) is noetherian and quasi-injective. Then, there exists S−1(g(n)) ∈
End(S−1(M(n))) ∀n ∈ Z such that S−1(g|S−1(N(n))) = S−1(f(n)).
S−1(g(n)) is injective since S−1(N∗) is essential in S−1(M∗), hence ∀n ∈ Z, S−1(N(n))
is essential, and as S−1(M∗) is cohopfian, S−1(g) is invertible. Let x

s ∈ S
−1(N(n)), there

exists y
t ∈ S

−1(M(n)) such that x
s = [S−1(g(n))](yt ). Or S−1(g−1(n)) ∈ End(S−1(N(n)))

and S−1(N(n)) is completely invariant, so y
t = S−1(g−1)(xs ) ∈ S−1(N), thus S−1(f(n)) is

an epimorphism for all n ∈ Z. Thus S−1(f∗) is an automorphisme, consequentely S−1(N∗)



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 419

is cohopfian.
Reciprocally, suppose that S−1(N∗) is cohopfian and let S−1(f∗) : S−1(M∗) −→ S−1(M∗)
a monomorphism. Then S−1(f∗|S−1N∗) is a monomorphism of S−1(N∗), so for all n ∈
Z, S−1(f(n)|S−1N(n)) is a monomorphism of S−1(N(n)). Thus, S−1(f(n)) ∈ Aut(S−1(N(n))),
hence [S−1(f(n))](S−1(N(n))) = S−1(N(n)). As S−1(M(n)) is quasi-injective, then there
exists S−1(L(n)) a submodule of S−1(M(n)) such that S−1(M(n)) = [S−1(f(n))](S−1M(n))⊕
S−1(L(n)). Thus, we have 0 = [S−1(f(n))](S−1N)∩S−1(L(n)) = S−1(N(n))∩S−1(L(n)),
since S−1(N(n)) is essential, then S−1(L(n)) = 0, hence S−1(M(n)) = [S−1(f(n))](S−1M(n)),
thus S−1(f(n)) is an epimorphism for all n ∈ Z =⇒ S−1(f∗) is an epimorphism of chain
complex, so S−1(M∗) is cohopfian complex sequence.

Proposition 2.
let M be a graded left A-module and S a saturated multiplicative part formed by the non-zero
homogeneous elements of A verifying the left Ore conditions. If M∗ is a hopfian, noetherian and
quasi-injective complex sequence associated with M , then the complex sequence of morphisms
of left S−1(A)-modules S−1(M∗) has the following property : �any epimorphism of sub-
complex S−1(N∗) of S−1(M∗) is an isomorphism �.

Proof.
Let N∗ be a subcomplex of M∗ and S−1(f∗) : S−1N∗ −→ S−1M∗ an epimorphism of chain
complex. Since M∗ is quasi-injective, then for all n ∈ Z, M(n) is quasi-injective =⇒
S−1(M(n)) is quasi-injective for all n ∈ Z, so, there exists S−1(f̃(n)) ∈ End(S−1(M(n)))
such that S−1(f̃(n))|S−1(N(n)) = S−1(f(n)). Or S−1(f(n)) is surjective, then for all x

s ∈
S−1(M(n)), there exists y

t ∈ S
−1(N(n)) such that x

s = [S−1(f(n))](yt ), hence S−1(f̃(n)) is

surjective, as S−1(M(n)) is hopfian, thus ˜S−1(f)(n) ∈ Aut(S−1(N(n))) for all n ∈ Z. We
deduce that S−1(f(n)) is a monomorphism of S−1(N(n)) into S−1(M(n)), for all n ∈ Z.
Consequentely, S−1(f∗) is an isomorphism of chain complex.

Theorem 17.
Let M be a graded left A-module, N a graded submodule of M , S a saturated multiplicative
part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. M∗
the quasi-projective complex sequence associated with M and N∗ a superfluous and completely
invariant complex sub-sequence of M∗. Then the complex morphism sequence of left S−1(A)-
-modules S−1(N∗) is hopfian if, and only if, S−1(M∗/N∗) the complex sequence associated
with S−1(M/N) is hopfian.

Proof.
Suppose that S−1(M∗/N∗) is hopfian and let f∗ : M∗ −→M∗ an epimorphism.
As S−1(N∗) is completely invariant, then ∀n ∈ Z, [S−1(f(n))](S−1(N(n))) ⊂ S−1(N(n)),
implies that S−1(f(n)) induces an epimorphism S−1(f)(n) : S−1(M(n)/N(n)) −→ S−1(M(n)/N(n)),
since S−1(M∗/N∗) is hopfian, then S−1(f)(n) is an automorphism. Put S−1(K(n)) =
ker(S−1f(n)) and S−1(π(n)) : S−1(M(n)) −→ S−1(M(n)/N(n)) the canonical projec-
tion, we have :
S−1(f)(n) ◦ [S−1(π(n))](S−1(K(n))) = [S−1(π(n) ◦ f(n))](K) = 0



S. A. Balde, M. B. Maaouia, A. O. Chbih / Eur. J. Pure Appl. Math, 14 (2) (2021), 404-422 420

Indeed, ∀xs ∈ S
−1(K(n)), we have :

[S−1(f(n) ◦ π(n))](xs ) = [S−1(π(n) ◦ f(n))](xs ), so

[S−1(π(n) ◦ f(n))](xs ) = [S−1(π(n))](f(n)(x)s ) = [S−1(π(n))](0s ) = 0, thus [S−1(S−1(f)(n) ◦
π(n))](K) = 0.
we have :
[S−1(f(n)◦π(n))](K(n)) = [S−1(π(n)◦f(n))](K(n)) = 0 =⇒ [S−1(f(n))](π(n)(K(n))) =
0 =⇒ [S−1(π(n))](K(n)) ⊂ S−1(N(n)) =⇒ S−1(K(n)) ⊂ S−1(N(n))
Since M∗ is quasi- projective =⇒ S−1M∗ quasi-projective, there exists an endomorphism
S−(s(n)) : S−1(M(n)) −→ S−1(M(n)) such that S−1(f(n) ◦ s(n)) = S−1(id(n)S−1(M(n))),
this implies S−1(M(n)) = S−1(K(n) ⊕ Ims(n)), or K(n) = N(n) and S−1(N(n)) is
superfluous in S−1(M(n)), ∀n ∈ Z, then S−1(M(n)) = S−1(Ims(n)), so S−1(K(n)) =
ker(S−1(f)(n)) = 0,∀n ∈ Z, thus S−1(f(n)) is a monomorphism, ∀n ∈ Z
finally, S−1(f∗) is a monomorphism, so S−1(M∗) is hopfian.
Reciprocally,
If S−1(M∗) is hopfian, show that S−1(M∗/N∗) is hopfian.
Let S−1(ϕ(n)) : S−1(M∗/N∗) −→ S−1(M∗/N∗) an epimorphism of chain complex, as
S−1(M∗) is quasi-projective, then ∀n ∈ Z, S−(M(n)) is quasi-projective. Consider S−1(π(n)) :
S−1(M(n)) −→ S−1(M(n)/N(n)), then there exists S−1(f(n)) ∈ End(S−1(M(n))) such
that S−1(π(n) ◦ f(n)) = S−1(ϕ(n) ◦ π(n).
Since S−1(ϕ(n)) is an epimorphism, ∀xs ∈ S−1(M(n)/N(n)),∃(yt ) ∈ S−1(M(n)/N(n))
such that
[S−1(ϕ(n))](yt ) = x

s = [S−1(ϕ(n))](π(n)(y)t )
[S−1(π(n) ◦ f(n))](yt ) = [S−1(ϕ(n) ◦ π(n))](yt )

[S−1(π(n))]( (f(n)(x))s ) = [S−1(ϕ)](yt )

=⇒ [S−1(ϕ(n))](yt ) = [S−1(f(n))](yt ) = x
s

=⇒ (f(n))(y)t − x
s = 0

=⇒ f(n)(y)
t − x

s ∈ S
−1(N(n)),

then S−1(M) = S−1(Im(f)(n))+S−1(N(n), as S−1(N(n)) is superfluous, then S−1(Im(f)(n)) =
S−1(M(n)) =⇒ S−(f(n)) is an epimorphism. So, S−1f(n) is an automorphism, because
S−1(M(n)) is hopfian for all n ∈ Z.
So the restriction of S−1(f(n)) over S−1(N(n)) is an automorphisme of S−1(N(n)).
If [S−1(ϕ)(n)](xs ) = [S−1(f(n))](xs ) = 0, then S−1[(f(n))](xs ) ∈ S−1(N(n)), or S−1(N(n))

is completely invariant, then x
s ∈ S

−1(N(n)), so x
s = 0 =⇒ ker(S−1(ϕ(n))) = S−1(N(n)) =

0 =⇒ S−1(ϕ(n)) is a monomorphism =⇒ ϕ(n) is an automorphism for all n ∈ Z, lastly
S−1(M(n)/N(n)) is hopfian for all n ∈ Z, hence S−1(M∗/N∗) is a hopfian complex se-
quence.

Acknowledgements

I am grateful to Professor Mohamed BEN MAAOUIA, Director of Laboratory of Alge-
bra, Cryptography, Codes and Applications (LACCA) UFR-SAT, for helpful discussions



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