Keskin and Harmanci defined the family
B
(
M,X
) = {
A
≤
M
 ∃
Y
≤
X
, ∃
f
∈ Hom
_{R}
(
M,X
/
Y
), Ker
f
/
A
≪
M
/
A
}. And Orhan and Keskin generalized projective modules via the class
B
(
M,X
). In this note we introduce
X
local summands and
X
hollow modules via the class
B
(
M,X
). Let
R
be a right perfect ring and let
M
be an
X
lifting module. We prove that if every coclosed submodule of any projective module contains its radical, then
M
has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang’s result
[9, Theorem 3.4]
. Let
X
be an
R
module. We also prove that for an
X
hollow module
H
such that every nonzero direct summand
K
of
H
with
K
∈
B
(
H,X
), if
H
⊕
H
has the internal exchange property, then
H
has a local endomorphism ring.
1. INTRODUCTION
Extending modules and lifting modules have been studied extensively in recent years by many ring theorists (see, for example,
[3]
,
[5]

[14]
).
Let
M
and
X
be
R
modules. In
[8]
, D. Keskin and A. Harmanci defined the family
B
(
M,X
) = {
A
≤
M
 ∃
Y
≤
X
, ∃
f
∈ Hom
_{R}
(
M
,
X
/
Y
), Ker
f
/
A
≪
M
/
A
}. They considered the following conditions:

B(M,X)(D1): For anyA∈B(M,X), there exists a direct summandA* ≤⊕Msuch thatA/A* ≪M/A*

B(M,X)(D2): For anyA∈B(M,X), ifB≤⊕M,M/A≃BimpliesA≤⊕M

B(M,X)(D3): For anyA∈B(M,X) andB≤⊕M, ifA≤⊕MandM=A+BthenA∩B≤⊕M
They defined that
M
is said to be
X

discrete
if
B
(
M,X
)(
D
_{1}
) and
B
(
M,X
)(
D
_{2}
) hold, and is said to be
Xquasidiscrete
if
B
(
M,X
)(
D
_{1}
) and
B
(
M,X
)(
D
_{3}
) hold. Furthermore,
M
is said to be
Xlifting
if
B
(
M,X
)(
D
_{1}
) holds. We have just seen that the following implications hold:
Xdiscrete
⟹
Xquasidiscrete
⟹
Xlifting
.
Throughout this paper, all rings
R
considered are associative rings with identity and all
R
modules are unital.
Let
M
be a right
R
module and
N
a submodule of
M
. The notation
N
≤
_{⊕}
M
means that
N
is a direct summand of
M
.
A submodule
K
of
M
is called a
small
submodule (or
superfluous
submodule) of
M
, abbreviated
K
≪
M
, in the case when, for every submodule
L
≤
M
,
K
+
L
=
M
implies
L
=
M
.
2. PRELIMINARIES
Let
A
and
P
be submodules of
M
with
P
∈
B
(
M,X
).
P
is called an
Xsupplement
of
A
if it is minimal with the property
A
+
P
=
M
equivalently, if
M
=
A
+
P
and
A
∩
P
≪
P
.
The module
M
is called
Xamply supplemented
if for any submodules
A, B
of
M
with
A
∈
B
(
M,X
) and
M
=
A
+
B
there exists an
X
supplement
P
of
A
such that
P
≤
B
.
Let
N
_{1}
≤
N
_{2}
≤
M
.
N
_{1}
is a
coessential
submodule of
N
_{2}
in
M
, abbreviated
N
_{1}
≤
_{c}
N
_{2}
in
M
, if the kernel of the canonical map
M
/
N
_{1}
→
M
/
N
_{2}
→ 0 is small in
M
/
N
_{1}
, or equivalently, if
M
=
N
_{2}
+
X
with
N
_{1}
≤
X
implies
M
=
X
.
A submodule
N
of
M
is said to be
coclosed
in
M
(or a
coclosed
submodule of
M
), if
N
has no proper coessential submodule in
M
. i.e.,
N
′ ≤
_{c}
N
in
M
implies
N
=
N
′. It is easy to see that any direct summand of a module
M
is coclosed in
M
. Note that every
X
supplement submodule of
M
is coclosed in
M
.
For
N
′ ≤
N
≤
M
,
N
′ is called a
coclosure
of
N
in
M
if
N
′ is a coclosed submodule of
M
with
N
′ ≤
_{c}
N
in
M
. Any submodule of a module has a closure, however, coclosure does not exist in general.
Lemma 2.1
(
[9, Lemma 1.4]
,
[5, 3.2, 3.7]
). Let
A
≤
B
≤
M
.
Then the following hold:

(1)A≤cB in M if and only if M=A+K for any submodule K of M with M=B+K.

(2) IfA≪M and B is coclosed in M, A≪B.
Lemma 2.2
(cf.,
[16, 41.14]
).
Any projective module satisfies the following condition:
(
D
)
If
M
_{1}
and M
_{2}
are direct summands of M such that
M
_{1}
∩
M
_{2}
≪
M and M
=
M
_{1}
+
M
_{2}
,
then M
=
M
_{1}
⊕
M
_{2}
.
Lemma 2.3
(
[13, Theorem 3.5]
).
If M is a lifting module with the condition (D), then M can be expressed as a direct sum of hollow modules
.
Lemma 2.4
(
[1, Lemma 17.17]
).
Suppose that M has a projective cover. If P is projective with an epimorphism
𝜑 :
P
→
M
,
then P has a decomposition P
=
P
_{1}
⊕
P
_{2}
such that P
_{1}
≤
Ker
𝜑
and
𝜑 
_{P2}
:
P
_{2}
→
M is a projective cover of M
.
Theorem 2.5
(
[3, Theorem 1.1.24]
).
For an Rmodule M, the following hold:

(1)If M is a quasiinjective module, then M is a fully invariant submodule of E(M).

(2)If M is a quasiinjective module, then any direct decomposition E(M) =E1⊕ ⋯ ⊕Eninduces M= (M∩E1) ⊕ ⋯ ⊕ (M∩En).

(3)If M is a quasiprojective module with a projective cover𝜑 :P→M,Ker𝜑is a fully invariant submodule of P; whence any endomorphism of P induces an endomorphism of M.

(4)If M is a quasiprojective module with a projective cover𝜑 :P→M,then any direct decomposition P=P1⊕⋯⊕Pninduces M= 𝜑(P1)⊕⋯⊕𝜑(Pn).
A ring
R
is called
right perfect
if every right
R
module has a projective cover.
Proposition 2.6.
The following statements are equivalent:

(i)Every cyclic right Rmodule has a projective cover;

(ii)RRis a lifting module.
Proof
. (i) ⟹ (ii) Let
A
be a submodule of
R_{R}
and let 𝜑 :
R
→
R
/
A
be the canonical epimorphism. Since
R
/
A
has a projective cover, by Lemma 2.4, there exists a decomposition
R_{R}
=
eR
⊕ (1 
e
)
R
such that (𝜑 
_{eR}
) :
eR
→
R
/
A
→ 0 a projective cover and (1 
e
)
R
≤
A
. This implies Ker (𝜑 
_{eR}
) =
A
∩
eR
≪
eR
. i.e.,
R
=
eR
⊕ (1 
e
)
R
such that
A
∩
eR
≪
eR
. Thus
R_{R}
is lifting.
(ii) ⟹ (i) Suppose that
R_{R}
is lifting. We claim that
R
/
A
has a projective cover. Since
R_{R}
is lifting, for any
A
≤
R
, there exists
A
* ≤
_{c}
A
such that
R
=
A
* ⊕
A
**. Then 𝜋 
_{A**}
:
A
** →
R
/
A
→ 0 is a projective cover of
R
/
A
, where 𝜋 :
R
→
R
/
A
→ 0 is the canonical epimorphism. □
As corollaries of Proposition 2.6, we obtain the following two results.
Corollary 2.7.
Let P be a projective module. Then the following statements are equivalent:

(i)Every factor module of P has a projective cover;

(ii)P is lifting.
Corollary 2.8.
The following statements are equivalent:

(i)Every simple right Rmodule has a projective cover;

(ii)RRsatisfies the lifting property for simple factor modules.
Lemma 2.9
(
[9, Lemma 3.1]
,
[5, 3.2]
).
Let f
:
M
→
N be an epimorphism
.
Suppose K
≤
_{c}
K
′
in M
.
Then f
(
K
) ≤
_{c}
f
(
K
′)
in N
.
Lemma 2.10
(
[8, Lemma 2.2]
).
Let M, N and X be Rmodules. Then the following hold:

(1)If A∈B(M,X)and B≤A with A/B≪M/B, then B∈B(M,X).

(2)Let h:M→N be an epimorphism and A∈B(M,X)with Ker h≤A.Then h(A) ∈B(N,X).Conversely, if h(A) ∈B(N,X)and Ker h≤A,then A∈B(M,X).

(3)Let B≤A≤M.Then A∈B(M,X)if and only if A/B∈B(M/B,X).

(4)Let h:N→M be an epimorphism and A∈B(M,X).Then h1(A) ∈B(N,X).
3. MAIN RESULTS
Theorem 3.1.
Let R be a ring. The following conditions are equivalent:

(1)R is right perfect;

(2)Every projective right Rmodule is lifting;

(3)Every quasiprojective right Rmodule is lifting;

(4)Every countably generated free right Rmodule is lifting.
Proof
. (1) ⟺ (2) This follows from Corollary 2.7.
(2) ⟹ (3) Let
Q_{R}
be a quasiprojective module and let
A
be a submodule of
Q
. Consider the canonical epimorphism
f
:
Q
→
Q
/
A
. We can take a projective module
P_{R}
such that
Q
is a homomorphic image of
P
, i.e., we have an epimorphism
g
:
P
→
Q
. Since P is a lifting module, by Lemma 2.4, there exists a decomposition
P
=
P
_{1}
⊕
P
_{2}
such that
P
_{1}
≤
g
^{1}
(
A
),
fg

_{P2}
:
P
_{2}
→
Q
/
A
is a projective cover. As
Q
is a quasiprojective module, the decomposition
P
=
P
_{1}
⊕
P
_{2}
induces a direct decomposition
Q
=
g
(
P
_{1}
)⊕
g
(
P
_{2}
) by Theorem 2.5. Then
g
(
P
_{1}
) ≤
A
and
g
(
P
_{2}
) ∩
A
≪
g
(
P
_{2}
) hold.
(3) ⟹ (2) Obvious.
(1) ⟹ (4) This follows from
[1, Theorem 28.4]
.
(4) ⟹) (1) By (4),
R
is semiperfect and
R
/
J
(
R
) is semisimple. Since
R
^{(}
^{)}
is lifting, there exists a decomposition
R
^{(}
^{)}
=
X
⊕
Y
such that
X
≤ Rad(
R
^{(}
^{)}
) and Rad(
R
^{(}
^{)}
) ∩
Y
≪
Y
. Because Rad(
R
^{(}
^{)}
) = Rad(
X
)⊕Rad(
Y
) and
X
≤ Rad(
R
^{(}
^{)}
), we see Rad(
X
) =
X
, which implies
X
= 0 and
R
^{(}
^{)}
J
(
R
) = Rad(
R
^{(}
^{)}
^{)}
≪
R
^{(}
^{)}
. Hence, by
[1, Lemma 28.3]
,
J
(
R
) is right
T
nilpotent. Thus
R
is right perfect. □
A family {
X
_{λ}
 λ ∈ Λ} of submodules of a module
M
with
X
_{λ}
∈
B
(
M,X
) is called an
Xlocal summand
of
M
, if ∑
_{λ∈Λ}
X
_{λ}
is direct and ∑ ⊕
_{λ∈F}
X
_{λ}
≤
_{⊕}
M
for every finite subset
F
⊆ Λ.
By anology with the proof of
[14, Lemma 2.4]
or
[11, Theorem 2.17]
, we have the following lemma.
Lemma 3.2.
If every Xlocal summand of a module M is a direct summand, then M has an indecomposable decomposition.
By Lemma 2.1(1), we have the following lemma.
Lemma 3.3.
Assume P_{i}
≤
_{c}
Q_{i}
in P for every i
∈
I
.
Then
∑⊕
_{i∈I}
P_{i}
≤
_{c}
∑⊕
_{i∈I}
Q_{i} in P.
Lemma 3.4.
Let
{
P_{i}
}
_{i∈I}
be a set of Rmodules. Assume P_{i}
∈
B
(
M,X
)
for every
i
∈
I
.
Then
∑⊕
_{i∈I}
P_{i}
∈
B
(
M,X
).
Proof
. Since
P_{i}
∈
B
(
M,X
), there exist a submodule
Y
of
X
and a homomorphism
f_{i}
:
M
→
X
/
Y
such that Ker
f
/
P_{i}
≪
M
/
P_{i}
. Put
f
= ∑⊕
_{i∈I}
f_{i}
. Then
f
:
M
→
X
/
Y
such that
Ker f/∑⊕_{i∈I} P_{i} ≪ M/∑⊕_{i∈I} P_{i}.
Thus ∑⊕
_{i∈I}
P_{i}
∈
B
(
M,X
). □
Lemma 3.5.
Let X be a right Rmodule. Suppose that R is a right perfect ring. Then every projective right Rmodule is Xlifting.
Proof
. Let
P
be a projective module. For any
A
∈
B
(
P,X
), consider the canonical epimorphism 𝜑 :
P
→
P
/
A
. Since
P
/
A
has a projective cover, by Lemma 2.4, there exists a decomposition
P
=
P
_{1}
⊕
P
_{2}
such that
P
_{1}
≤ Ker 𝜑 and 𝜑 
_{P2}
:
P
_{2}
→
P
/
A
is a projective cover of
P
/
A
. Hence
P
is
X
lifting. □
Proposition 3.6.
Let R be a right perfect ring and let M be an Xlifting module. Then M is Xamply supplemented.
Proof
. Let
A
,
B
≤
M
such that
B
∈
B
(
M,X
) and
M
=
A
+
B
. Since
M
=
A
+
B
and
B
∈
B
(
M,X
), there exist
Y
≤
X
and
f
:
M
→
X
/
Y
such that Ker
f
/
B
≪
M
/
B
.
Consider the isomorphism 𝛼 :
M
/
B
→
A
/
A
∩
B
. Then 𝛼(Ker f/B) = Ker
f
/
A
∩
B
. Hence Ker
f
/
A
∩
B
≪
M
/
A
∩
B
. Therefore
A
∩
B
∈
B
(
M,X
). As
M
is
X
lifting, there exists a direct summand
K
of
M
such that
K
≤
_{c}
A
∩
B
in
M
. Then
A
∩
B
=
K
⊕ [
K
* ∩ (
A
∩
B
)],
M
= (
A
∩
B
) +
K
* and (
A
∩
B
) ∩
K
* ≪
K
*. Thus
M
=
B
+ (
A
∩
K
*).
Let
D
be a coclosure of
A
∩
K
* in
M
. Then
M
=
B
+
D
and
B
∩
D
≤
B
∩ (
A
∩
K
*) ≪
K
*. Hence
B
∩
D
≪
K
*. Since
D
is coclosed in
M
,
B
∩
D
≤
D
and
B
∩
D
≪
M
,
B
∩
D
≪
D
. Thus
D
is an
X
supplement of
B
in
M
such that
D
≤
A
. □
Lemma 3.7
(
[8, Lemma 3.2]
).
Every epimorphic image of an Xamply supplemented Rmodule is Xamply supplemented.
Lemma 3.8.
Let M be an Xamply supplemented module and let Ker f
≪
M
N
→ 0.
Suppose K is coclosed in M with Ker f
≤
K
.
Then f
(
K
)
is coclosed in N
.
Proof
. By Lemma 3.7,
N
is
X
amply supplemented. Let
L
≤
_{c}
f
(
K
) in
N
. We claim that
L
=
f
(
K
). Since
f
is an epimorphism, there exists a submodule
T
of
K
in
M
with
f
(
T
) =
L
. Since
N
is
X
amply supplemented, there exists an
X
supplement
P
of
f
(
K
) such that
P
≤
N
. i.e.,
N
=
f
(
K
) +
P
and
f
(
K
) ∩
P
≪
P
. Since
f
is an epimorphism, there exists a submodule
Q
of
M
with
f
(
Q
) =
P
. Then
M
=
K
+
Q
+Ker
f
. As Ker
f
≪
M, M
=
K
+
Q
. This implies
N
=
f
(
K
)+
f
(
Q
) =
f
(
K
)+
P
=
L
+
P
=
f
(
T
)+
f
(
Q
). Then
M
=
T
+
Q
+Ker
f
=
T
+
Q
. Thus
T
≤
_{c}
K
in
M
by Lemma 2.1(1). As
K
is coclosed in
M, T
=
K
. Hence
L
=
f
(
T
) =
f
(
K
). Therefore
f
(
K
) is coclosed in
N
. □
Proposition 3.9.
Suppose that M is an Xlifting module. Then every coclosed submodule K of M with K
∈
B
(
M,X
)
is a direct summand
.
Proof
. Since
M
is
X
lifting, there exists a direct summand
K
* such that
K
* ≤
_{c}
K
in
M
. As
K
is coclosed in
M, K
=
K
* ≤
_{⊕}
M
. □
Theorem 3.10.
Let R be a right perfect ring and let M be an Xlifting module. Assume that every coclosed submodule of any projective module contains its radical. Then every Xlocal summand of M is a direct summand
.
Proof
. Let
M
be an
X
lifting module and let ∑
_{i∈I}
X_{i}
be an
X
local summand of
M
with
X_{i}
∈
B
(
M,X
). Since
R
is right perfect,
M
has a projective cover, say Ker
f
≪
P
M
→ 0. By Lemma 3.5,
P
is projective
X
lifting. Since
X_{i}
∈
B
(
M,X
),
f
^{1}
(
X_{i}
) ∈
B
(
P,X
) by Lemma 2.10(4). So there exists a decomposition
P
=
P_{i}
⊕
(
i
∈
I
) such that
P_{i}
≤
_{c}
f
^{1}
(
X_{i}
) in
P
. By Lemma 2.9,
f
(
P_{i}
) ≤
_{c}
f
(
f
^{1}
(
X_{i}
)) =
X_{i}
in
M
. As
X_{i}
is coclosed in
M
,
f
(
P_{i}
) =
X_{i}
. First we prove that ∑
_{i∈I}
P_{i}
is direct. Let
F
be a finite subset of
I
 {
i
}. Since ∑⊕
_{i∈I}
X_{i}
is an
X
local summand of
M
, we see
f(P_{i} + ∑_{j∈F}P_{j}) = X_{i} ⊕ (∑ ⊕_{j∈F} X_{j}) ≤_{⊕} M.
So there exists a direct summand
Y
of
M
such that
M
=
X_{i}
⊕ (∑ ⊕
_{j∈F}
X_{j}
) ⊕
Y
. As
P
is lifting, there exists a decomposition
P
=
Q
⊕
Q
* such that
Q
≤
_{c}
f
^{1}
(
Y
) in
P
. Then
f
(
Q
) =
Y
. Thus we see
P = P_{i} + ∑_{j∈F}P_{j} + Q + Ker f = P_{i} + ∑_{j∈F}P_{j} + Q.
Then
P_{i}
∩(∑
_{j∈F}
P_{j}
+
Q
) ⊆ Ker
f
≪
P
. Similarly, we see
Q
∩(
P_{i}
+∑
_{j∈F}
P_{j}
≪
P
and
P_{j}
∩ (
P_{i}
+∑
_{l∈F{j}}
P_{l}
+
Q
) ≪
P
. By Lemma 2.2, we obtain
P
=
P_{i}
⊕ (∑
_{j∈F}
P_{j}
) ⊕
Q
. Hence ∑
_{i∈I}
P_{i}
is direct. By the same argument, we see ∑ ⊕
_{i∈I}
P_{i}
is an
X
local summand of
P
. By Lemma 2.3, ∑ ⊕
_{i∈I}
P_{i}
≤
_{⊕}
P
. So
f
(∑ ⊕
_{i∈I}
P_{i}
) is coclosed in
M
by Lemma 3.8. Since
M
is
X
lifting, we see
∑ ⊕_{i∈I} X_{i} = f(∑ ⊕_{i∈I} P_{i}) ≤_{⊕} M.
Thus any
X
local summand of
M
is a direct summand. □
By Lemma 3.2. and Theorem 3.10, we obtain the first main theorem.
Theorem 3.11.
Suppose that every coclosed submodule of any projective module contains its radical. Then every Xlifting module over right perfect rings has an indecomposable decomposition.
Let
X
be an
R
module. A nonzero
R
module
H
is
Xhollow
if for any proper submodule
K
of
H
with
K
∈
B
(
H,X
),
K
≪
H
.
Proposition 3.12.
Let H and X be Rmodules. Assume that every nonzero direct summand K of H with K
∈
B
(
H,X
).
Then H is Xhollow if and only if H is indecomposable Xlifting.
Proof. (⟹) Assume
H
is
X
hollow. Let
K
∈
B
(
H,X
) with
K
⪇
H
. Since
H
is
X
hollow,
K
≪
H
. So there exists a decomposition
H
= 0 ⊕
H
such that 0 ≤
_{c}
K
in
H
. Thus
H
is
X
lifting. Now, assume that
H
=
H
_{1}
⊕
H
_{2}
,
H_{i}
≠ 0,
i
= 1, 2. Since
H
is
X
hollow,
H_{i}
≪
H
,
i
= 1, 2. Hence
H_{i}
= 0. This is a contradiction. Therefore
H
is indecomposable.
(⟸) Suppose that
H
is indecomposable
X
lifting. Let
K
∈
B
(
H,X
) with
K
⪇
H
. By hypothesis, there exists a decomposition
H
=
K
* ⊕
K
** such that
K
* ≤
_{c}
K
in
H
. As
H
is indecomposable, we have either
K
* = 0 or
K
** = 0. If
K
* = 0, then
K
≪
H
. In the second case,
H
=
K
. This is a contradiction. □
A module
M
is said to have the (
finite
)
exchange property
if, for any (finite) index set
I
, whenever
M
⊕
N
= ⊕
_{i∈I}
A_{i}
for modules
N
and
A_{i}
, then
M
⊕
N
=
M
⊕(⊕
_{i∈I}
B_{i}
) for some submodules
B_{i}
≤
A_{i}
. A module
M
has the (
finite
)
internal exchange property
if, for any (finite) direct sum decomposition
M
= ⊕
_{i∈I}
M_{i}
and any direct summand
X
of
M
, there exist submodules
≤
M_{i}
such that
M
=
X
⊕ (⊕
_{i∈I}
).
By Proposition 3.12, we obtain the second main theorem.
Theorem 3.13
(cf.,
[15, Proposition 1]
).
Let X be an Rmodule and let H be an Xhollow module. Assume that every nonzero direct summand K of H with K
∈
B
(
H,X
).
If H
⊕
H has the internal exchange property, then H has a local endomorphism ring.
Corollary 3.14
(cf.,
[5, 12.2]
).
Let X be an Rmodule and let H be an Xhollow module. Assume that every nonzero direct summand K of H with K
∈
B
(
H,X
).
Then the following conditions are equivalent:

(1)H has a local endomorphism ring;

(2)H has the finite exchange property;

(3)H has the exchange property.
Anderson F.W.
,
Fuller K.R.
1992
Rings and Categories of Modules
SpringerVerlag
BerlinHeidelbergNew York
Azumaya G.
,
Mbuntum F.
,
Varadarajan K.
1975
On Mprojective and Minjective modules
Pacific J. Math.
95
9 
16
Baba Y.
,
Oshiro K.
Artinian Rings and Related Topics, Lecture Note
Clark J.
,
Lomp C.
,
Vanaja N.
,
Wisbauer R.
2007
Lifting modules
Birkhauser Boston
Boston
Dung N.V.
,
Huynh N.V.
,
Smith P.F.
,
Wisbauer R.
1994
Extending modules
Longman Group Limited
Harada M.
1983
Factor categories with applications to direct decomposition of modules
Marcel Dekker
New York
Keskin D.
,
Harmanci A.
2004
A relative version of the lifting property of modules
Algebra Colloquium
11
(3)
361 
370
Kuratomi Y.
,
Chang C.
Lifting modules over right perfect rings
Comm. Algebra
LópezPermouth S.R.
,
Oshiro K.
,
Rizvi S.T.
1998
On the relative (quasi) continuity of modules
Comm. Algebra
26
(11)
3497 
3510
DOI : 10.1080/00927879808826355
Mohamed S.H.
,
Müller B.J.
1990
Continuous and Discrete modules
London Math. Soc. Lect. Notes
Cambrige Univ. Press
147
Orhan N.
,
Keskin D.
2005
Characterizations of lifting modules in terms of cojective modules and the class of B(M,X)
Int. J. Math.
16
(6)
647 
660
DOI : 10.1142/S0129167X05003041
Oshiro K.
1983
Semiperfect modules and quasisemiperfect modules
Osaka J. Math.
20
337 
372
Wisbauer R.
1991
Foundations of Modules and Ring Theory
Gordon and Breach Science Publishers