EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 4, 2017, 730-738
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Modules that Have a δ-supplement in Every Extension

Esra Öztürk Sözen1,∗, Şenol Eren1

1 Ondokuz Mayıs University, Faculty of Science and Arts
Department of Mathematics, Turkey

Abstract. Let R be a ring and M be a left R-module. In this paper, we define modules with
the properties (δ-E) and (δ-EE), which are generalized version of Zöschinger’s modules with the
properties (E) and (EE), and provide various properties of these modules. We prove that the class
of modules with the property (δ-E) is closed under direct summands and finite direct sums. It is
shown that a module M has the property (δ-EE) if and only if every submodule of M has the
property (δ-E). It is a known fact that a ring R is perfect if and only if every left R-module has
the property (E). As a generalization of this, we prove that if R is a δ-perfect ring then every left
R-module has the property (δ-E). Moreover, the converse is also true on δ-semiperfect rings.

2010 Mathematics Subject Classifications: 16D10, 16D90

Key Words and Phrases: Supplement, δ-supplement, δ-perfect ring, δ-semiperfect ring, module
extension

1. Introduction

In this paper R is an associative ring with identity and all modules are unital left
R-modules. Let M be a module X ≤ M means that X is a submodule of M or M is an
extension of X. Recall that a submodule N ≤ M is called small, denoted by N � M,
if N + L 6= M, for all proper submodules L of M. We call T a supplement of N in M
if M = T + N and T ∩ N is small in T. A module M is called supplemented if every
submodule of M has a supplement in M [14]. L ≤M is said to be essential in M, denoted
by L EM , if L ∩ K 6= 0 for each nonzero submodule K ≤ M. The singular submodule
of a module M (denoted by Z(M)) is Z(M) = {x ∈M | Ix = 0 for some ideal I ER} . A
module M is called singular if Z(M) = M. Every submodule and every factor module of a
singular module is singular. We refer to [6] for the further properties of singular modules.

In [15], Zhou introduced the concept of δ-small submodules as a generalization of small
submodules. A submodule N of M is said to be δ-small in M (denoted by N �δ M) if
whenever M = N + K and M

K is singular, we have M = K. And we denote the sum of
all δ-small submodules of M by δ(M). A submodule L of M is called a δ-supplement of

∗Corresponding author.

Email addresses: esraozturk55@hotmail.com (E. Ö. Sözen), seren@omu.edu.tr (Ş. Eren)

http://www.ejpam.com 730 c© 2017 EJPAM All rights reserved.



E. Ö. Sözen, Ş. Eren / Eur. J. Pure Appl. Math, 10 (4) (2017), 730-738 731

N in M if M = N + L and N ∩ L �δ L and M is called δ-supplemented in case every
submodule of M has a δ-supplement in M [7].

For a module M consider the following conditions:
(E) : M has a supplement in every extension.

(EE) : M has ample supplements in every extension.
The concept of these modules with these properties was first introduced by Zöschinger

[16]. Adapting his concept in [4], Çalışıcı and Türkmen introduced modules with the prop-
erties (CE) and (CEE) as a generalization of the properties (E) and (EE). In addition,
in [9] the authors worked on modules that have a weak supplement in every extension and
in [5] Eryılmaz introduced modules that have a δ-supplement in every torsion extension.

In this paper we investigate the structure of modules with the properties (δ-E) and
(δ-EE) as a generalization of Zöschinger’s modules with the properties (E) and (EE).
We prove that a module has the property (δ-EE) if and only if every submodule has the
property (δ-E). We show that every direct summand and δ-small cover of M with the
property (δ-E) has the property (δ-E). Using the property (δ-E), we present a relation
between δ-perfect rings and modules with the property (δ-E), which are a generalization
of perfect rings, that is, R is a δ-perfect ring, then every left R-module has the property
(δ-E). Moreover we obtain that if every left R-module has the property (δ-E), then R is
a δ-semiperfect ring.

2. Preliminaries

In this section, we begin by stating the following lemmas and theorems for the com-
pleteness.

2.1. δ-Small Submodues

Lemma 1. ([15, Lemma 1.2]). Let N be a submodule of M . The following are equivalent:

1. N �δ M.

2. If X + N = M, then M = X ⊕ Y for a projective semisimple submodule Y with
Y ⊆ N.

3. If X +N = M with M
X Goldie torsion, then X = M .

Lemma 2. ([15, Lemma 1.3]). Let M be a module.

1. For submodules N , K, L of M with K ⊆ N , we have

(a) N �δ M if and only if K �δ M and N
K �δ

M
K .

(b) N + L�δ M if and only if N �δ M and L�δ M.

2. If K �δ M and f : M −→ N is a homomorphism, then f(K)�δ N.

In particular, if K <<δ M ⊆ N, then K �δ N.



E. Ö. Sözen, Ş. Eren / Eur. J. Pure Appl. Math, 10 (4) (2017), 730-738 732

3. Let K1 ⊆M1 ⊆M, K2 ⊆M2 ⊆M and M = M1⊕M2. Then K1⊕K2 �δ M1 ⊕M2

if and only if K1 �δ M1 and K2 �δ M2.

2.2. δ-Supplemented Modules

Lemma 3. ([7, P rop.2.7]). Let U and V be submodules of a module M. Assume that V
is a δ-supplement of U in M. Then

1. If W + V = M for some W ⊆ U, then V is a δ-supplement of W in M .

2. If K �δ M, then V is a δ-supplement of U +K in M.

3. For K �δ M we have K ∩ V �δ V and so δ(V ) = V ∩ δ(M).

4. For L ⊆ U, V+L
L is a δ-supplement of U

L in M
L .

5. If δ(M) �δ M, or δ(M) ⊆ U and if p : M −→ M
δ(M) is the canonical projection,

then M
δ(M) = p(U)⊕ p(V ).

In [7], a projective module P is called a projective δ-cover of a module M if there exists
an epimorphism f : P −→M with Ker(f)�δ M, and a ring R is called δ-perfect (resp.,
δ-semiperfect) if every R-module (resp., every simple R-module) has a projective δ-cover.
In addition, a module M is called δ-lifting if for any N ≤M, there exists a decomposition
M = A⊕ B such that A ≤ N and N ∩ B is δ-small in B since B is a direct summand of
M.

Theorem 4. [7, Theorem3.3].The following are equivalent for a ring R :

1. R is δ-semiperfect.

2. Every finitely generated module is δ-supplemented.

3. Every finitely generated projective module is δ-supplemented.

4. Every finitely generated projective module is δ-lifting.

5. Every left ideal of R has a δ-supplement in RR.

Theorem 5. [7, Theorem 3.4]. The following statements are equivqlent for a ring R :

1. R is δ-perfect.

2. Every module is δ-supplemented.

3. Every projective module is δ-supplemented.

4. Every projective module is δ-lifting.



E. Ö. Sözen, Ş. Eren / Eur. J. Pure Appl. Math, 10 (4) (2017), 730-738 733

3. Modules with the Properties (δ-E) and (δ-EE)

In this section, we define the concept of modules with the properties (δ-E) and (δ-EE).

Definition 1. A module M has the property (δ-E) if it has a δ-supplement in each module
in which it is contained as a submodule.

Definition 2. A module M has the property (δ-EE) if it has ample δ-supplements in each
module in which it is contained as a submodule, where U ≤ M has ample δ-supplements
in M if for every V ≤M with U + V = M, there is a δ-supplement V

′
of U with V

′ ≤ V.

It is clear that every module with the property (E) has the property (δ-E). Also there
exists the same relation between modules with the properties (EE) and (δ-EE). At the
end of this section, we shall give an example of a module which has the property (δ-E)
but not (E).

Zöschinger proved in [16] that a module has the property (EE) if and only if every
submodule has the property (E).We give an analogous characterization of our modules
with the following proposition.

Proposition 1. A module M has the property (δ-EE) if and only if every submodule of
M has the property (δ-E).

Proof. Let M be a module and N be any extension of M. Suppose that for a submodule
X ≤ N, X + M = N. By hypothesis, the submodule X ∩M of M has a δ-supplement
V in X, that is, (X ∩M) + V = X and (X ∩M) ∩ V �δ V . Then, N = M + X =
M + [(X ∩M) + V ] = M + V and M ∩ V = M ∩ (V ∩X) = (X ∩M) ∩ V �δ V. Hence,
V is a δ-supplement of M in N such that V ≤ X.

Conversely, let U be a submodule of M and N be any module containing U. Then we
can draw the following pushout:

i1 and i2 are inclusion homomorphisms in this diagram. Additionally α : M −→ F
and β : N −→ F are monomorphisms by the properties of push out (see, for example, [11,
Exercise 5.10]). Let α(M) = M

′ ⊆ F and β(N) = N
′ ⊆ F. Then it can be easily shown



E. Ö. Sözen, Ş. Eren / Eur. J. Pure Appl. Math, 10 (4) (2017), 730-738 734

that F = M
′
+N

′
. So by using hypothesis, M

′ ∼= M has a δ-supplement V in F such that
V ≤ N ′ , that is, M

′
+ V = F and M

′ ∩ V �δ V. Hence,

(M
′ ∩N ′) + V = (N

′ ∩M ′
) + V = N

′ ∩ (M
′
+ V ) = N

′ ∩ F = N
′
, and

(M
′ ∩N ′) ∩ V = M

′ ∩ (N
′ ∩ V ) = M

′ ∩ V �δ V.

So V is a δ-supplement ofM
′∩N ′ inN

′
. Now we will show that β−1(V ) is a δ-supplement of

U in N. We have an isomorphism
∼
β : N −→ N

′
defined as β(x) =

∼
β(x) for all x ∈ N, since

β is a monomorphism. Using this, we obtain
∼
β−1(V ) is a δ-supplement of β−1(M

′ ∩N ′) in
∼
β−1(N

′
) since V is a δ-supplement of M

′∩N ′ in N
′
. It can be seen that

∼
β−1(V ) = β−1(V ),

∼
β−1(N

′
) = N and

∼
β−1(M

′ ∩N ′) = U. Thus β−1(V ) is a δ-supplement of U in N.

Corollary 1. A module with the property (δ-EE) has the property (δ-E) and it is also
δ-supplemented.

Recall that R is a (right) δ-V ring if for any right R-module M, δ(M) = 0 (see, [13]).

Proposition 2. Let R be δ-V ring and M be an R-module. Then the following statements
are equivalent:

1. M has the property (δ-E).

2. M is injective.

Proof. (1) =⇒ (2) : Suppose that M has the property (δ-E). Let N be any extension
of M. So, there exists a δ-supplement V of M in N, that is, M +V = N and M ∩V �δ V
and so M ∩ V ≤ δ(V ). Since R is a δ-V ring, δ(V ) = 0. So, N = M ⊕ V. Therefore, M is
injective.

(2) =⇒ (1) : is clear.

Now we show that the property (δ-E) is preserved by direct summands in the following
proposition:

Proposition 3. Every direct summand of any module with the property (δ-E) has the
property (δ-E).

Proof. Let M be a module with the property (δ-E), U be a direct summand of M and
N be any extension of U. Then there exists a submodule A of M such that M = U ⊕ A.
By hypothesis, M has a δ-supplement V in A ⊕N such that (A ⊕ U) + V = A ⊕N and
(A⊕ U) ∩ V <<δ V. Let g : A⊕N −→ N be the projection onto N. Then

N = g(A⊕N) = g((A⊕ U) + V ) = g(A⊕ U) + g(V ) = U + g(V ), and

g((A⊕ U) ∩ V ) = U ∩ g(V )�δ g(V ).

Hence, g(V ) is a δ-supplement of U in N.



E. Ö. Sözen, Ş. Eren / Eur. J. Pure Appl. Math, 10 (4) (2017), 730-738 735

Proposition 4. Let A ≤ B. If A and B
A have the property (δ-E), so does B.

Proof. Let N be any extension of B. So, there is a δ-supplement V
A of B

A in N
A and a

δ-supplement T of A in V. We have δ-small epimorphisms f : T −→ V
A and g : VA −→

N
B

that
Ker f = T ∩ A �δ T and Ker g = V

A ∩
B
A �δ

V
A . Then, g ◦ f : T −→ N

B is a δ-small
epimorphism such that T ∩B = Ker (g ◦ f)�δ T. Moreover, we have

B + T = (B +A) + T = B + (A+ T ) = B + V = N

since V
A is a δ-supplement of B

A in N
A . This completes the proof.

Corollary 2. If M1 and M2 have the property (δ-E), so does M1 ⊕M2.

Proof. Let 0 −→ M1 −→ M1 ⊕M2 −→ M2 −→ 0 be a short exact sequence. Result
follows by Proposition 4.

Proposition 5. Let 0 −→ K −→ M −→ L −→ 0 be a short exact sequence. If K and L
have the property (δ-E), so does M. If the sequence splits the converse is also true.

Proof. Let N be any extension of M. So N
K is an extension of M

K and is is a well known
fact that M

K
∼= L. Then there exists a δ-supplement V

K for M
K in N

K , that means M
K + V

K = N
K

and M
K ∩

V
K �δ

V
K for some V

K ≤
N
K . Since K ≤ V and K has the property (δ-E),

K+K
′

= V, K∩K ′ �δ K
′
for some K

′ ≤ V. Hence, N = M+V = M+K+K
′

= M+K
′
.

Now we claim that M ∩ K ′ �δ K
′
. For this let M ∩ K ′ + T = K

′
with K

′

T is singular.

K + M ∩K ′ + T = K + K
′

and by the modular law (K + K
′
) ∩M + T = V. Following

this, V ∩M + T = V is obtained. It can be easily seen written that V ∩M
K + T+K

K = V
K ,

additionally, V
T+K is singular since,

V

T +K
=
K +K

′

T +K
=
K + (K

′
+ T )

T +K
=

(T +K) +K
′

T +K
∼=

K
′

(T +K) ∩K ′
=

K
′

T + (K ∩K ′)
≤ K

′

T

and M
K ∩

V
K �δ

V
K . So T+K

K = V
K and of course T +K = V . (T +K)∩K ′ = K

′
can be

seen and by the modular lae, T + (K ∩K ′) = K
′

is obtained. This provides T = K
′

since

K ∩K ′ <<δ K
′

and K
′

T is singular. Moreover, suppose that the sequence splits, Then K
and L have the property (δ-E) by corollary 2.

Corollary 3. Let Mi (i = 1, 2, ..., n) be any finite collection of modules and M = M1 ⊕
M2⊕ ... ⊕ Mn. Then M has the property (δ-E) if and only if Mi has the property (δ-E)
for each i = 1, 2, ..., n.

Proof. It can be proved easily for n = 2 by using the previous theorem and can be
generalized on n.

We give the following known lemma for the completeness.



E. Ö. Sözen, Ş. Eren / Eur. J. Pure Appl. Math, 10 (4) (2017), 730-738 736

Lemma 6. Every simple submodule S of a module M is either a direct summand of M
or small in M (see in [10])

Proposition 6. Every simple module has the property (δ-E).
Proof. Let S be a simple module and N be any extension of S. Then by Lemma 4,

S � N and so S �δ N or S ⊕ S′ = N for a submodule S
′ ≤ N. If S �δ N, then N is a

δ-supplement of S in N or if S is a direct summand of N then S
′

is a δ-supplement of S
in N . So in each case S has a δ-supplement in N . This means that S has the property
(δ-E).

Theorem 7. Every module with composition series has the property (δ-E).

Proof. Let 0 = M0 ≤M1 ≤M2 ≤ ... ≤Mn−1 ≤Mn = M be any composition series of
a module M. We shall prove the theorem by induction on nεN. If n = 1, then M = M1 is
simple, and so M has the property (δ-E) by Proposition 6. Assume that this is true for
each k ≤ n− 1.Then Mn−1 has the property (δ-E). Since Mn

Mn−1
has the property (δ-E) as

a simple module, M has the property (δ-E) by Proposition 4.

Corollary 4. A finitely generated semisimple module has the property (δ-E).

In the following proposition we will prove that modules with the property (δ-E) are
closed onder factor modules, under a special condition.

Proposition 7. Let A ≤ B ≤ C with C
A injective. If B has the property (δ-E), so does

B
A .

Proof. Let N be any extension of B
A . So we have the following commutative diagram

with exact rows since C
A is injective, (see in [10]).

Since h is monic and B has the property (δ-E), B ∼= h(B) has a δ-supplement V in P,
that is, h(B) + V = P and h(B)∩ V �δ V. We claim that g(V ) is a δ-supplement of B

A in
N.

B

A
+ g(V ) = (fσ)(B) + g(V ) = g(h(B)) + g(V ) = g(P ) = N, and

B

A
∩ g(V ) = f(σ(B)) ∩ g(V ) = g[h(B) ∩ V ]�δ g(V )



REFERENCES 737

since h(B) ∩ V �δ V and g is a homomorphism.

A ring R is left perfect if and only if every left R-module has the property (E) (see
[16]). Now we show only one side of this fact is valid for δ-perfect rings.

Proposition 8. If R is a δ-perfect ring, then every left R-module has the property (δ-E).
Proof. Suppose that a ring R is δ-perfect. Let M be an R-module and N be any

extension of M. N is δ-supplemented since R is δ-perfect. So M has a δ-supplemented in
N as a submodule of N. Hence, M has the property (δ-E).

Proposition 9. Let R be a ring. If every left R-module has the property (δ-E), then R
is a δ-semiperfect ring.

Proof. Since every left R-module has the property (δ-E), every ideal of R also has
the property (δ-E) as a submodule of RR. So every ideal of R has a δ-supplement in RR.
Hence R is δ-semiperfect by [6, Theorem 3.3].

Example 1. Let F be a field,

I =

(
F F
0 F

)
, R = {(x1, ..., xn, x, x, ...) | n ε N, xi ε M2(F ), x ε I}

with component-wise operations, R is a ring. By Example 4.3 in [15], R is a δ-perfect ring
that is not perfect. And so RR is an example of a module that has the property (δ-E) but
not have the property (E).

Acknowledgements

The authors sincerely thank the reviewers for the valuable suggestions and comments.

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