EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 478-485
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

e∗-Essential Small Submodules and e∗-Hollow Modules
Hiba R. Baanoon1,2,∗, Wasan Khalid2

1 Mathematics Department, College of Education, University of Misan, Iraq
2 Mathematics Department, College of Science, University of Baghdad, Iraq

Abstract. The purpose of this paper is to introduce the concepts of e∗-small essential submod-
ules, e∗-radical submodules, and e∗-hollow modules as a generalizations of the concepts of small
submodules, radical submodules, and hollow modules, respectively. We will prove some properties
of these concepts.
2020 Mathematics Subject Classifications: 16D10, 16D90, 16D99, 16s90
Key Words and Phrases: e∗-Small essential submodule, Small submodule, e∗-Radical submod-
ule, e∗-Hollow module, Hollow module

1. Introduction

Let R be a ring with identity, M is a right R-module and E(M) the injective hull of
M . A submodule N of M is called a small submodule of M denoted (N � M) if for any
submodule A of M such that M = N + A, we have A = M [6] Recall that a submodule
A of R-module B is called essential in B if every nonzero submodule of B has nonzero
intersection with A [6], [4] and [5].

Oscan in [2] introduced the concept of cosingular submodule as follows: Z∗(M) =
{m ∈ M |mR � E(M)}. An R-module M is called cosingular if Z∗(M) = M . Baanoon
and Khaild in [1] introduced a type of submodule which called e∗-essential as follows. A
submodule A of M is said to be e∗-essential in M if A∩B 6= 0 for each nonzero cosingular
submodule B of M . Denoted by A ≤e∗ M .

As in [7], we will used e∗-essential submodule that appeared in [1], to present a new
generalization of a small sumodule namely e∗-essential small submodule. e∗-essential small
submodules leads us to introduce e∗-hollow module as a generalization of hollow modules.
In this paper main properties of these concepts are proved.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4301

Email addresses: hibabaanoon@uomisan.edu.iq (H.R. Baanoon),
wasan.hasan@sc.uobaghdad.edu.iq (W. Khalid)

https://www.ejpam.com 478 © 2022 EJPAM All rights reserved.



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (2) (2022), 478-485 479

2. e∗-Essential Small Submodules

In this section, so one generalization of small submodules are introduced with some
properties. Recall that a submodule A of M is said to be e∗-essential denoted by A ≤e∗ M
if A ∩B 6= 0 for each nonzero cosingular submodule B of M [1].
The following gives some properties of e∗-essential submodules.

Lemma 1. [1] Let M be an R-module.

1. If A ≤ B ≤ M , then A ≤e∗ M if and only if A ≤e∗ B ≤e∗ M

2. Let f : M → M
′ be an R-homomorphism. If A ≤e∗ M

′, then f−1(A) ≤e∗ M .

3. If A ≤e∗ B ≤ M and A
′ ≤e∗ B

′ ≤ M , then A ∩A
′ ≤e∗ B ∩B

′.

Definition 1. Let M be an R-module, a submodule A of M is said to be e∗-essential
small in M denoted by A �e∗ M , if whenever M = A + B (where B is an e∗-essential
submodule of M) implies that M = B.

Examples and Remarks 1.

1. Every small submodule is e∗-essential small submodule, but the converse need not to
be true in general. For example, in Z6 as a Z-module, the only e∗-essential submodule
is Z6 [1]. So, every submodule of Z6 is e∗-essential small. while 〈2〉 is not a small
submodule since 〈2〉+ 〈3〉 = Z6 but 〈3〉 6= Z6.

2. Consider Z4 as a Z-module, the submodles Z4 and 〈2〉 are cosingular[2] and e∗-
essential, hence 〈2〉 is an e∗-essential small submodule.

3. Consider Z6 as a Z6-module. In this module every submodule is e∗-essential [1], so
〈2〉 + 〈3〉 = Z6 but 〈3〉 6= Z6. Therefore, 〈2〉 is not e∗-essential small submodule.
Thus, e∗-essential submodule need not to be e∗-essential small.

4. Let M be an R-module, then:

• The trivial submodule is always e∗-essential small in M .
• M �e∗ M if and only if M is a simple module.

In the following, we introduce the basic properties of e∗-essential small submodules.

Proposition 1. Let M be an R-module, N a submodule of M and K a submodule of N .

1. If N �e∗ M , then K �e∗ M and N
K �e∗

M
K .

2. If K �e∗ N , then K �e∗ M .

Proof.



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (2) (2022), 478-485 480

1. Let L be an e∗-essential submodule of M such that K + L = M . Since K ≤ N and
N �e∗ M , L = M . Thus K �e∗ M .
Now, to prove that N

K �e∗
M
K , let M

K = A
K + N

K where A
K is an e∗-essential submodule

of M
K , hence A is e∗-essential submodule of M by lemma 1, and M = A+N , since

N �e∗ M . Thus, M = A implies that A
K = M

K . Therefore, N
K �e∗

M
K .

2. Let L be an e∗-essential submodule of M such that K+L = M . Hence, L∩N ≤e∗ N
by lemma 1, and K+(L∩N) = N∩(K+L) = N , since K �e∗ N . Thus, L∩N = N ,
N ≤ L. So K ≤ L. Hence, L = K + L = M . Therefore, K �e∗ M .

Proposition 2. Let M be an R-module, K and N submodules of M such that K ≤ N .
If K �e∗ M and N is a direct summand e∗-essential submodule of M , then K �e∗ N .

Proof. Let L be an e∗-essential submodule of N such that K + L = N . Since N is
a direct summand of M , there exists a submodule N

′ of M such that M = N ⊕ N
′ and

M = (K + L) ⊕ N
′
= K + (L + N

′
). Since L ≤e∗ N ≤e∗ M , by lemma 1, this implies

that L ≤e∗ M and since L ≤ L + N
′ ≤ M also by the same lemma, this implies that

L + N
′
leqe∗M . K �e∗ N implies that L + N

′
= M . Now, for any n ∈ N , there exists

l ∈ L and n
′ ∈ N

′ such that n = l + n
′, so n − l = n

′ ∈ N ∩ N
′
= 0, hence n = l and

N ≤ L. Therefore, N = L and K �e∗ N

The following proposition shows that, the homorphic image of an e∗-essential small
submodule is e∗-essential small submodule.

Proposition 3. If K �e∗ M and f : M → N is an R-homomorphism, then f(K) �e∗ N .
Proof. Let L be an e∗-essential submodule of N such that f(K)+L = N . hence f−1(L)

is e∗-essential in M by lemma 1. Let m ∈ M , hence f(m) ∈ N = f(K)+L, so there exist
k ∈ K and l ∈ L such that f(m) = f(k) + l. Thus, l = f(m − k) so, m − k ∈ f−1(L)
and m = m − k + k ∈ K + f−1(L). Hence, K + f−1(L) = M since K �e∗ M . Thus
f−1(L) = M and f(M) = f(f−1(L)) = f(L) ∩ L, hence f(M) ⊆ L i.e. f(K) ⊆ L.
Therefore, L = f(K) + L = N and f(K) �e∗ N .

The sum of e∗-essential small submodules is e∗-essential small submodule as the fol-
lowing proposition shows.

Proposition 4. Let N and L be submodules of an R-module M . Then N + L �e∗ M if
and only if N �e∗ M and L �e∗ M .

Proof. ⇒) Let K be e∗-essential in M such that K +N = M . So, K +N + L = M .
By assumption, K = M and N �e∗ M . Similarly for L �e∗ M .
⇐) Let A be e∗-essential in M such that N + L+A = M , M = N + (L+A) = M , since
A ≤ A+L ≤ M and A ≤e∗ M by lemma 1, this implies that L+A = M . Now, N �e∗ M
implies that L+A+M and L �e∗ M implies that A = M . Therefore, N + L �e∗ M .

The following corollary follows from Proposition 3 and Proposition 4.

Corollary 1. Let M = M1 ⊕M2 and Ki a submodule of Mi, i = 1, 2. Then Ki �e∗ Mi,
i = 1, 2 if and only if K1 ⊕K2 �e∗ M1 ⊕M2.



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (2) (2022), 478-485 481

3. e∗- Radical Submodule

Recall that for an R-module M , if M has maximal submodule, then the radical of M
is the intersection of all maximal submodules of M dented by Rad(M) [6]. We generalize
this concept as the following:

Definition 2. Let M be R-module. Then the intersection of all e∗-essential maximal
submodule of M is called e∗- radical submodule denoted by Rad(M)

e∗
.

If M has no e∗-essential maximal submodule, then Rad(M)
e∗

= M .

The following proposition gives the relationship between e∗-essential small submodules
and e∗-essential maximal submodules.

Proposition 5. Let M be an R-module and m ∈ M , then 〈m〉 is not e∗-essential small if
and only if there exists an e∗-essential maximal submodule N of M with m /∈ N .

Proof. ⇒) Consider the set
Γ = {B|B is a proper e∗-essential submodule of M and 〈m〉 + B = M}. Since 〈m〉 is
not e∗-essential small, there exists B

′ ≤e∗ M such that 〈m〉 + B
′
= M and B

′ 6= M ,
hence Γ 6= φ. Let {Cα}α∈λ be a chain in Γ, hence ∪α∈λCα is a proper submodule and
since Cα ≤ ∪α∈λCα ≤ M for each α ∈ λ with Cα ≤e∗ M , then ∪α∈λCα ≤e∗ M with
〈m〉 + ∪α∈λCα = M . So, by Zorn’s lemma, Γ has a maximal element say B0. We claim
that B0 is maximal in M . Otherwise if B0 � C 6 M , then M = B0+〈m〉 ≤ C+〈m〉 ≤ M .
Thus, 〈m〉+C = M and since B0 6e∗ M , hence C 6e∗ M . Now, if C 6= M , hence C ∈ Γ
which is a contradiction. Thus, C = M . So B0 6e∗ M which is maximal in M . Now,
if m ∈ B0, then 〈m〉 ⊆ B0 and since 〈m〉 + B0 = M , we have B0 = M which is a
contradiction. So, m /∈ B0 i.e. there exists an e∗-essential maximal submodule of M that
does not contain m.

⇐) To show that 〈x〉 is not e∗-essential small in M . If not, then as x /∈ N and N is as
maximal submodule we have 〈x〉+N = M . Now, 〈x〉 �e∗ M and N ≤e∗ M implies that
N = M which is a contraindication. Therefore, 〈x〉 is not e∗-essential small submodule of
M .

Examples and Remarks 2.

1. Let M be an R-module, then Rad(M) ≤ Rad(M)
e∗

. But the converse need not to be

true in general. For example: Consider Z6 as a Z-module, Rad(Z6) = {0}. When
Rad(Z6)

e∗
= Z6, since the maximal submodules of Z6 are 〈2〉 and 〈3〉 while the only

e∗-essntial submodule is Z6 [1].

2. In Z4 as a Z-module Rad(Z4)
e∗

= {0, 2}. Since all submodules of Z4 are: {0}, {0, 2}

and Z4. Hence, the e∗-essential submodule of Z4 are: {0, 2} and Z4. Thus, the only
e∗-essential maximal submodule is {0, 2}.



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (2) (2022), 478-485 482

Theorem 1. Let M be an R-module, then Rad(M)
e∗

=
∑

N
N�e∗M

.

Proof. Let m /∈ Rad(M)
e∗

then there exists an e∗-essential maximal N of M such

that m /∈ N . Hence by proposition 5, we have that 〈m〉 is not e∗-essential small. Thus,
m /∈

∑
{N |N �e∗ M}. Therefore,

∑
{N |N �e∗ M} ⊆ Rad(M)

e∗
.

Now, let x ∈ Rad(M)
e∗

and x /∈
∑

{N |N �e∗ M}. Hence, 〈x〉 is not e∗-essential small

and by proposition 5, there exists an e∗-essential maximal submodule K of M such that
x /∈ K but Rad(M)

e∗
≤ K which is a contradiction. Thus, x ∈

∑
{N |N �e∗ M} and

Rad(M)
e∗

≤
∑

{N |N �e∗ M}. Therefore, Rad(M)
e∗

=
∑

{N |N �e∗ M}.

Proposition 6. If f : M → M
′ is an R-homomorphism, then f(Rad(M)

e∗
) ≤ Rad(M

′
)

e∗
.

In particular, Rad(M)
e∗

is a fully invariant submodule of M .

Proof. By Therorm 1, Rad(M)
e∗

=
∑

K
K�e∗M

. Hence, f(Rad(M)
e∗

) =
∑

f(K)
K�e∗M

. By

Proposition 3, Since K �e∗ M then f(K) �e∗ M
′. Thus,

∑
f(K)

K�e∗M
≤ Rad(M

′
)

e∗
and

f(Rad(M))
e∗

≤ Rad(M
′
)

e∗
.

Corollary 2. Let M be an R-module and N be a submodule of M , then:

1. Rad(N)
e∗

≤ Rad(M)
e∗

.

2.
Rad(M)

e∗
N ≤ Rad(MN )

e∗
.

4. e∗-Hollow Modules

Recall that a non-zeroR-moduleM is called a hollow module if every proper submodule
of M is small in M [3]. In this section we introduce e∗-hollow modules as a generalization
of hollow modules and investigate some of their properties.

Definition 3. A non zero R-module M is called e∗-hollow module if every proper sub-
module of M is e∗-essential small in M .

Examples and Remarks 3.

1. Every hollow module is e∗-hollow module. But the converse need not to be true in
general. For example: in Z6 as Z-module every proper submodule is e∗-essential
small, hence Z6 is e∗-hollow module, but it is not hollow, since 〈2〉 is not small
submodule.



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (2) (2022), 478-485 483

2. Consider Z6 as a Z6-module. since 〈2〉 is not an e∗-essential small submodule. Thus,
Z6 is not e∗-hollow module.

3. The direct sum of two e∗-hollow modules need not to be e∗-hollow. For example:
Z4 as a Z-module is e∗-hollow since 〈2〉 and Z4 are the only e∗-essntial submodules.
So all the proper submodules are e∗-essntial small. Also, Z3 as a Z-module is e∗-
hollow since the only e∗-essntial submodule is Z3 it self. But Z4 ⊕Z3 ' Z12 and Z12

is not e∗-hollow. since the only e∗-essntial submodule of Z12 are 〈2〉 and Z12 with
〈3〉+ 〈2〉 = Z12 but 〈2〉 6= Z12.

4. Any R-module which has no proper e∗-essential submodule is e∗-hollow.

Proposition 7. The epimorphic image of an e∗-hollow module is e∗-hollow.
Proof. Let f : M → M

′ be an R-epimorphism, with M an e∗-hollow module. Let
B be a proper submodule of M

′. Hence f−1(B) is a proper submodule of M . since if
not, f−1(B) = M implies that ff−1(B) = B = M

′ which is a contradiction. Since M
is e∗-hollow then f−1(B) is e∗-essential small. By proposition 3, ff−1(B) = B is an
e∗-essential small submodule. Therefore, M ′is e∗-hollow.

Corollary 3. If M is an e∗-hollow module, then M
N is e∗-hollow for any proper submodule

N of M .

Remark 1. The converse of the above corollary need not to be true in general. For
example: Consider Z24 as a Z-module which is not e∗-hollow. Since every submodule
of Z24 is cosingular then the only e∗-essntial submodule of Z24 are 〈0〉, 〈2〉, 〈4〉, and Z24.
Since 〈3〉+ 〈2〉 = Z24 and 〈2〉 6= Z24 we have that 〈3〉 is not e∗-essential small. But Z24

〈4〉 is
e∗-hollow module since Z24

〈4〉 ' Z4.

The following proposition shows that under certain conditions the converse of corollary
3 is true. Recall that a submodule A of a module M is called e∗-closed if A has no proper
e∗-essential extension inside M [1].

Lemma 2. [1] If B ≤ K are submodules of an R-module M such that B is e∗-closed in
M and K is e∗-essential in M , then K

B
≤e∗

M

B
.

Proposition 8. Let M be an R-module. If M
N is e∗-hollow with N is a proper small

e∗-closed submodule, then M is e∗-hollow.
Proof. Let L be a proper submodule of M and K an e∗-essential submodule of M such

that L+K = M . Then M
N = L+N

N + K+N
N implies that M 6= L+N . For if M = L+N

with N a small submodule of M i.e. M = L which is a contradiction. Thus, M
N 6= L+N

N .
Since, N ≤e∗ M then by lemma 1, N ≤ce∗ K +N ≤e∗ M , and by lemma 2, K+N

N ≤e∗
M
N .

Since M
N is e∗-hollow, then K+N

N = M
N , and M = K + N because N � M . Therefore,

K = M and M is e∗-hollow.



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (2) (2022), 478-485 484

Proposition 9. Let M be an e∗-hollow module, if M has proper a e∗-essential submodule
N and M

N is finitely generated then M is finitely generated.
Proof. Since M

N is finitely generated there are x1, x2, ..., xn ∈ M such that M
N =

〈x1 + N, x2 + N, ..., xn + N〉. We claim that M = 〈x1, x2, ..., xn〉. Let m ∈ M , hence
m + N ∈ M

N and m + N = (x1r1 + x2r2 + ... + xnrn) + N for some r1, r2, ..., rn ∈ R.
So, m − (x1r1 + x2r2 + ... + xnrn) ∈ N . Let n = m − (x1r1 + x2r2 + ... + xnrn) where
n ∈ N , hence m = (x1r1 + x2r2 + ... + xnrn) + n. Thus, M = 〈x1, x2, ..., xn〉 + N . If
〈x1, x2, ..., xn〉 6= M , then 〈x1, x2, ..., xn〉 �e∗ M , since N �e∗ M . Hence, M = N which
is a contradiction. Therefore, M = 〈x1, x2, ..., xn〉.

The following proposition is a characterizes e∗-hollow modules.

Proposition 10. An R-module M is e∗-hollow module if and only if every proper e∗-
essential submodule of M is small in M .

Proof. ⇒) Clear
⇐) Let A be a proper submodule of M and B an e∗-essential submodule of M such that
A+B = M . If B 6= M then B is a proper e∗-essential submodule of M and by assumption
B is small. Hence A = M which is a contradiction. Thus, B = M and A is e∗-essential
small in M . Therefore, M is e∗-hollow.

Definition 4. Let M be an R-module. A submodule A of M is called e∗-coclosed if
whenever B ≤ A, A

B �e∗
M
B , implies that A = B.

One may ask a question. Is any submodule of an e∗-hollow module e∗-hollow?
The following proportion gives a partial answer.

Proposition 11. Let M be an e∗-hollow R-module.

1. An e∗-essential direct summand of an e∗-hollow module is e∗-hollow.

2. An e∗-coclosed submodule of an e∗-hollow is e∗-hollow.

Proof.

1. Let A be an e∗-essential direct summand of M and B a proper submodule of A with
L ≤e∗ A such that B + L = A. Since L ≤e∗ A ≤e∗ M , then by lemma 1, L ≤e∗ M .
Also, since A is a direct summand of M , there is a submodule A

′ of M such that
A⊕ A

′
= M . Thus, M = B + L+ A

′ with L+ A
′ ≤e∗ M and hence B is a proper

submodule of M . This implies that B is e∗-essential small in M . Hence, M = L+A
′

and A = A ∩M = A ∩ (L + A
′
) = L + (A ∩ A

′
) = L. Therefore, B is e∗-essential

small in A and A is e∗-hollow.

2. Let A be a e∗-coclosed submodule of M and B a proper submodule of A with C an
e∗-essential submodule of A such that B + C = A. Since M is e∗-hollow then by
corollary 3, M

C is e∗-hollow. Now, A
C is a proper submodule of M

C implies that A
C is

e∗-essential small of M
C since A is e∗-coclosed. Thus A = C and B is e∗-essential

small of A.The case A
C = M

C , implies that A = M . Thus A is e∗-hollow.



REFERENCES 485

Acknowledgements

The authors would like to thank the reviewers for their invaluable comments and
suggestions that led to this improved version of the paper.

References

[1] H. Baanoon and W. Khalid. e*-essential submodule. European Journal of Pure and
Applied Mathematics, 15(1):224–228, 2022.

[2] A Çiğdem Özcan. Modules with small cyclic submodules in their injective hulls. 2002.

[3] Patrick Fleury. Hollow modules and local endomorphism rings. Pacific Journal of
Mathematics, 53(2):379–385, 1974.

[4] Kenneth Goodearl. Ring theory: Nonsingular rings and modules, volume 33. CRC
Press, 1976.

[5] Michiel Hazewinkel, Nadiya Gubareni, and Vladimir V Kirichenko. Algebras, rings
and modules, volume 1. Springer Science & Business Media, 2004.

[6] Friedrich Kasch. Modules and rings, volume 17. Academic press, 1982.

[7] DX Zhou and XR Zhang. Small-essential submodules and morita duality. Southeast
Asian Bulletin of Mathematics, 35(6), 2011.