EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 1, 2022, 224-228
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

e∗-Essential submodule

Hiba R. Baanoon1,∗, Wasan Khalid 2

College of Science, University of Baghdad, Baghdad, Iraq

Abstract. The purpose of this paper is to introduce a new concept in a module M over a ring R,
this concept is called e∗-essential submodule, which is a generalization of an essential submodule.
We will introduce some examples and properties about this concept such that, what is the in-
verse image of e∗-essential submodule, the intersection of e∗-essential submodules and direct sum
of e∗-essential submodules. We will show the relationship between e∗-essential submodule and
Noetherian R-module. Also we will define e∗-closed submodule with some properties

2020 Mathematics Subject Classifications: 16D90, 16D99, 16P40

Key Words and Phrases: Essential submodule, Small submodule, e∗-Essential submodule, e∗-
Closed submodule, Noetherian R-module

1. Introduction

Let R be a ring with identity, M be a right R-module and E(M) be the injective hull
of M . A submodule N of an R-module M is called a small submodule of M (N ≪ M)
if for any submodule A of M such that M = N + A, then A = M [5]. Leonard defines
a module M to be small if it is a small submodule of some R-module and he shows that
M is small if and only if M is small in its injective hull [1]. 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 [5], [3] and [4].

Oscan in [2], introduced the concept of cosingular submodule as the following: Z∗(M) =
{m ∈ M |mR ≪ E(M)}. An R-module M is called cosingular Z∗(M) = M .

As in [6], we will used the Oscan presented to generalize the essential submodule, to
introduce the concepte e∗-essential and investigate some properties.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4215

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

http://www.ejpam.com 224 © 2022 EJPAM All rights reserved.



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (1) (2022), 224-228 225

2. e∗-Essential submodule

Definition 1.

• Let M be R-module, a submodule A of M is said to be e∗-essential if A ∩B ̸= 0 for
each nonzero cosingular submodule B of M . denoted by A ≤e∗ B.

• A right ideal B of a ring R is e∗-essential in R if and only if B is e∗-essential
submodule of RR.

• An R-homomorphism f : A → B is said e∗-essential if and only if, Im(f) is e∗-
essential submodule in B.

• We may deduce the following from the definition:

1. A ≤e∗ M if A ∩K = 0, then K = 0 where K is cosingular submodule in M .

2. If M ̸= 0 and L ≤e∗ M then L ̸= 0.

Examples and Remarks 1.

1. Every essential submodule is e∗-essential, but the converse need not to be true in
general. For example, in Z6 as Z6-module, the only cosingular submodle of Z6 is
{0}. Hence every submodule K of Z6 is e∗-essential, since K ∩ {0} = 0. Therefore,
{0, 2, 4} is e∗-essential which is not essential submodule in Z6 as Z6-module since
there is a nonzero submodule {0, 3} but {0, 2, 4} ∩ {0, 3} = 0.

2. For any R-module M , we have M ≤e∗ M .

3. Every nonzero submodule of Z as Z-module is cosingular [2]. Hence, nZ ∩ mZ =
nmZ ̸= 0 for each n ̸= 0 and m ̸= 0. So that every submodule of Z is e∗-essential.

4. In Z6 as Z-module every submodule is cosingular [2], but {0, 2, 4} is not e∗-essential
since {0, 2, 4} ∩ {0, 3} = 0 where {0, 3} a nonzero cosingular submodule.

5. The image of e∗-essential need not be e∗-essential for example. Let f : Z → Z2 be a

Z-homomorphism defined by f(x) =

{
0 if xeven
1 if xodd

So f(2Z) = {0}. Hence, 2Z is e∗-essential in Z but {0} is not e∗-essential in Z2,
since {0} ∩ Z2 = 0 where Z2 is nonzero cosingular.

6. The quotient submodule of e∗-essential submodule need not to be e∗-essential, for
example: 2ZZ is e∗-essential submodule of ZZ, but

2Z
2Z = 0 not e∗-essential submodule

of Z
2Z

∼= Z2.

In the following lemma, gives a property of cosingular submodule



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (1) (2022), 224-228 226

Lemma 1. If K is cosingular submodule of B and B ≤ A ≤ M , then K is cosingular in
A.

Proof. Since K is cosingular submodule of B by Lemma 2.2 in [2] Z∗(K) = K∩Z∗(B)
and again since B is a submodule of A. So that, Z∗(K) = K∩(B∩Z∗(A)), from hypothesis
Z∗(K) = K. Hence, K ≤ Z∗(A). Therefore, Z∗(K) = K ∩ Z∗(A) = K, i.e. K is
cosingular in A.

Now, we will prove some properties which e∗-essential submodule satisfied:

Proposition 1. Let A ≤ B ≤ M , then A ≤e∗ M if and only if A ≤e∗ B ≤e∗ M
Proof. ⇒) Let K ̸= 0 be a cosingular submodule of B, hence K ≤ M since A ≤e∗ M .

Therefore A ∩ K ̸= 0. Hence, A ≤e∗ B. Now, for B ≤e∗ M , let 0 ̸= L be a consigular
submodule of M . Hence, A ∩ L ̸= 0 and since A ≤ B so that, B ∩ L ̸= 0.
⇐) Let N be a nonzero cosingular submodule of M . Since B ≤e∗ M . Hence, B ∩N ̸= 0,
so that B ∩N is a nonzero cosingular submodule of B ( since B ∩N ≤ N and by Lemma
2.2 in [2] Z∗(B ∩N) = (B ∩N) ∩ Z∗(N) = (B ∩N) ∩N = B ∩N . Since A ≤e∗ B then
A ∩B ∩N ̸= 0 and A ∩N ̸= 0. Therefore, A ≤e∗ M .

Corollary 1. If A1 ≤ A2 ≤ A3 ≤ M and A1 ≤e∗ M , then A2 ≤e∗ A3.
Proof. Let L be a nonzero cosingular in A3. By Lemma 1, we have that L is cosingular

in M and since A1 ≤e∗ M . Thus, A1 ∩L ̸= 0 and since A1 ≤ A2. Therefore, A2 ∩L ̸= 0,
i.e. A2 ≤e∗ A3.

Proposition 2. Let f : M → M
′
be R-homomorphism, if A ≤e∗ M

′
, then f−1(A) ≤e∗ M .

Proof. Let A ≤e∗ M
′
. Hence, f−1(A) ≤ M , suppose that f−1(A) is not e∗-essential

submodule of M , i.e. there exists a nonzero cosingular submodule B of M such that
f−1(A) ∩ B = 0. Since ker(f |B) = f−1(A) ∩ B = 0. Thus, B ∼= f(B). Also, we have
that A ∩ f(B) = 0 since if not, i.e. there exists 0 ̸= x = f(b) ∈ A ∩ f(b). Hence,
0 ̸= b ∈ f−1(A) ∩ B which is contradiction. Since B is cosingular by lemma 2.6 in [2],
f(B) is cosingular also A ≤e∗ M

′
. Hence, f(B) = 0 which is contradiction. Therefore,

f−1(A) ≤e∗ M .

Proposition 3. If A ≤e∗ B ≤ M and A
′ ≤e∗ B

′ ≤ M , then A ∩A
′ ≤e∗ B ∩B

′
.

Proof. Let K be a nonzero cosingular submodule of B ∩ B
′
. By Lemma 1 K be a

nonzero cosingular submodule of B and B
′
. Since A ≤e∗ B. So that, A ∩ K ̸= 0 and

since A ∩K is a nonzero submodule of cosingular K. Hence, A ∩K is cosingular. But,
A

′ ≤e∗ B
′
. Hence, A

′ ∩ (A ∩K) ̸= 0. Therefore, A ∩A
′ ≤e∗ B ∩B

′
.

Corollary 2. Let Bj ≤e∗ M for each j = 1, ..., n, then ∩n
i=1 ≤e∗ M

Proof. The prove by induction on n.

Proposition 4. Let M = M1 ⊕M2 with K1 ≤ M1 and K2 ≤ M2, then K1 ≤e∗ M1 and
K2 ≤e∗ M2 if and only if, K1 ⊕ k2 ≤e∗ M



H.R. Baanoon, W. Khalid / Eur. J. Pure Appl. Math, 15 (1) (2022), 224-228 227

Proof. ⇒) There exists an R-homomorphism ρ1 : M1 ⊕ M2 → M1 and ρ2 : M1 ⊕
M2 → M2 which define by ρ1(m1,m2) = m1 and ρ2(m1,m2) = m2. By Proposition 2
ρ−1
1 (K1) = K1 ⊕ M2 ≤e∗ M1 ⊕ M2 and ρ−1

2 (K2) = M1 ⊕ K2 ≤e∗ M1 ⊕ M2. Hence, by
Proposition 3 K1 ⊕M2 ∩M1 ⊕K2 = K1 ⊕ k2 ≤e∗ M

⇐) There exists an R-homomorphism J1 : M1 → M1 ⊕M2 and J2 : M2 → M1 ⊕M2

which define by J1(m1) = (m1, 0) and J2(m2) = (0,m2). By Proposition 2 J−1
1 (K1 ⊕

k2) = K1 ≤e∗ M1 and J−1
2 (K1 ⊕ k2) = K2 ≤e∗ M2.

In the following proposition we will give a characterization of e∗-essential submodule.

Proposition 5. Let M be R-module and N ≤ M , then N is e∗-essential submodule of M
if and only if N ∩ xR ̸= 0 for each nonzero cyclic cosingular submodule of M .

Proof. ⇒) Clear.
⇐) Let N be a submodule of M and K be a nonzero cosingular submodule of M . Hence,
there exists 0 ̸= x ∈ K with xR ≤ K, also Z∗(xR) = xR. So by hypothesis N ∩ xR ̸= 0.
Hence, N ∩K ̸= 0. Therefore, N ≤e∗ M .

In the following proposition shows that, the composition of e∗-essentialR-monomorphism
is also e∗-essential R-monomorphism.

Proposition 6. Let f : A → B and g : B → C are e∗-essential R-monomorphism. Then,
g ◦ f : A → C is also e∗-essential R-monomorphism.

Proof. Let L be cosingular submodule of C such that Im(g ◦ f) ∩ L = 0. Since g
is monomorphism 0 = kerg = g−1(0) = g−1 (Im(g ◦ f) ∩ L). Hence g−1 (Im(g ◦ f)) ∩
g−1(L) = Im(f) ∩ g−1(L) = 0. Since g−1 is R-homomorphism and L is cosingular sub-
module of C. Hence g−1(L) is cosingular submodule of B and since Im(f) ≤e∗ B. Thus,
g−1(L) = 0 and Im(g) ∩ L = 0. Since g : B → C is e∗-essential. Therefore, L = 0 i.e.
g ◦ f is e∗-essential R-monomorphism.

In the following proposition we will give another characterization of Noetherian R-
module. Also, it is show the relationship between e∗-essential submodule and Noetherian
R-module.

Proposition 7. An R-module M is Noetherian if and only if, every e∗-essential submod-
ule of M is finitely generated.

Proof. ⇒) Clear.
⇐) Let A be an essential submodule of M . Hence, A is e∗-essential and by the hypothsis
A is a finitely generated. Hence, every essential submodule is finitely generated by [3].
Therefore, M is Noetherian.



REFERENCES 228

3. e∗-Closed submodule

Definition 2. A submodule A of R-module C is said to be e∗-closed submodule of C, if
A has no proper e∗-essential extension inside C. denoted by A ≤e∗C C.

Examples and Remarks 2.

1. For any module M . 0 and M always e∗-closed.

2. In Z6 as Z6-module, {0, 2, 4} is not e∗-closed submodule since {0, 2, 4} is e∗-essential
in Z6.

In the following proposition shows that when the quotient submodule of e∗-essential
submodule is e∗-essential:

Proposition 8. Let M be R-module, If B ≤e∗C M and B ≤ K ≤e∗ M then
K

B
≤e∗

M

B
.

Proof. Let
L

B
be cosingular submodule of

M

B
with

K

B
∩ L

B
= 0. Hence, K ∩ L = B

since K ≤e∗ M . Thus, K ∩ L ≤e∗ M ∩ L = L. Hence B ≤e∗ L ≤ M but B ≤e∗C M ,

B = L. Hence,
L

B
= 0. Therefore,

K

B
≤e∗

M

B
.

Acknowledgements

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

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