EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 1, 2020, 158-169
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Some Results on C-retractable Modules

Abdoul Djibril Diallo1, Papa Cheikhou Diop2,∗, Mamadou Barry1

1 Département de Mathématiques et Informatique, Faculté des Sciences et Techniques,
Université Cheikh Anta Diop, Dakar, Sénégal
2 Département de Mathématiques , UFR Sciences et Technologies, Université de Thiès,
Thiès, Sénégal

Abstract. An R-module M is called c-retractable if there exists a nonzero homomorphism from
M to any of its nonzero complement submodules. In this paper, we provide some new results of c-
retractable modules. It is shown that every projective module over a right SI-ring is c-retractable.
A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which
is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi-
local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-
continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable
module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-
retractable is a C4-module are determined.

2020 Mathematics Subject Classifications: 13B10,13C05,13C13

Key Words and Phrases: Retractable modules, complement submodules, c-retractable modules,
projective modules

1. Introduction

Throughout all rings are associative with identity and all modules are unitary right
module. Let R be a ring. Following [19], we say that an R-module M is retractable if
HomR(M,N) 6= {0} for any nonzero submodules N of M . It is shown in [19] that every
projective module over a right V -ring is retractable. In [19] again, the semisimplicity of
retractable modules is studied. M. R. Vedadi [23], introduced the concept of essentially
retractable modules and proved that over semiprime right nonsingular rings, a nonsingular
essentially retractable module is precisely a module with non-zero dual. In [7], A. Ghor-
bani and M. R. Vedadi introduced and studied the notion of epi-retractable module, where
a module M is called epi-retractable if every submodule of M is a homomorphic image
of M . They reveal some applications of projective, nonsingular, injective epi-retractable

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i1.3588

Email addresses: cheikpapa@yahoo.fr (P. C. Diop),
dialloabdoulaziz58@yahoo.fr (A. D. Diallo ), mansabadion1@hotmail.com (M. Barry )

http://www.ejpam.com 158 c© 2020 EJPAM All rights reserved.



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 159

modules regarding the characterization of Bezout, pri, quasi-Frobenius rings. Note that
epi-retractable modules are retractable. Earlier, P. F. Smith and A. Tercan [20] intro-
duced C11-module as a generalization of extending modules, where a module M is said
to be satisfy C11-condition if every submodule of M has a complement which is a direct
summand. It is shown in ([20], Theorem 2.7) that a module satisfies (C11 if and only if
M = Z2(M) ⊕K for some (nonsingular) K of M and Z2(M) and K both satisfy (C11).
Later, the same authors investigated when a direct summand of a C11-module inherits the
property [21]. Recently, t-closed submodules of a module M are defined in [2] as closed
submodules of M which contain Z2(M). In [3], S. H. Asgari, A. Haghany and A.R. Rezaei
studied the modules M for which C11-condition holds for t-closed submodules (T11-type,
for short). They showed among others the following results: i) A T11-type module is ex-
actly a direct sum of a Z2-torsion module and a nonsingular C11-modules.
(ii) A T+

11-module (modules for which direct summands are T11-type) is precisely a di-
rect sum of Z2-torsion and nonsingular C+

11-module (modules for which direct summands
satisly C11). A. W. Chatters and S. M. Kheuri [4] defined the concept of c-retractable
module, where an R-module M is called c-retractable if HomR(M,C) 6= {0} for any
nonzero complement submodules C of M . This notion is a generalization of both the re-
tractable modules and the extending modules. They have shown that if M is a nonsingular
c-retractable module such that SS is extending, then M is extending. But the converse is
not true in general. On the other hand it is shown in [22] that if M is a retractable wd-
Rickart module, then every indecomposable submodule of M is a simple direct summand.
Motivated by the definition of the modules mentioned above and the results on retractable
and c-retractable modules, we investigate the c-retractibility. Our aim in this paper is to
give some new results on c-retractable modules. In general, c-retractable modules need
not be projective and vice versa. Connections between projectivity and c-retractibility
are investigated. Conditions are found under which, a c-retractable module is extending,
quasi-continuous, quasi-injective and retractable. With the help of c-retractability, we
investigated when the notions of K-nonsingularity and Baer modules are equivalent. Also,
we characterize semisimple artinian rings in termes of c-retractable modules.
Our paper is structured as follows:
In the second section, we are going to give preliminary definitions which we will use
throughout this paper.
In the third section, we are going to show among others, the following results:
(1) Every projective module over a right SI-ring is c-retractable.
(2) Let M be a wd-Rickart module in which local summands are summand. Then M is
uniform-extending and c-retractable if and only if M is extending.
(3) Let M be a dual Baer c-retractable R-module. Then the following hold:
(i) M is a direct sum of uniform submodules.
(ii) M = Z2(M) ⊕ (⊕i∈IMi) with all Mi nonsingular uniform quasi-Baer and End(Mi)
semi-local quasi-Baer.
(iii) M is ADS if and only if M is quasi-continuous.
(iv) M is auto-invariant if and only if M is quasi-injective.
(5) The following conditions are equivalent for a ring R:



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 160

(a) R is semisimple artinian.
(b) Every c-retractable R-module is a C4-module.
(c) Every c-retractable R-module is pseudo-projective.
(6) Let M be a locally noetherian c-retractable R-module. Then M is homo-related to a
direct sum ⊕i∈IUi of uniform submodules of M .

For an R-module M , S = EndR(M) denotes the endomorphism ring of M . For φ ∈ S,
Imφ stands for image of φ. The notations N ≤M , N ≤e M and N ≤⊕ M mean that N
is a submodule of M , an essential submodule and a direct summand of M , respectively.

2. Preliminaries

In this section, we are going to give preliminary definitions which we will use
throughout this paper.

Definition 1. Let S be a submodule of an R-module M . A submodule C of M is said to
be complement to S in M if C is maximal with respect to the property that C ∩ S = {0}.

Definition 2. A submodule C of an R-module is a complement in M (C ⊆c M , for short)
if there exists a submodule S of M such that C is complement to S in M .

Definition 3. 1. An R-module M is called extending module if every complement sub-
module of M is a direct summand.
2. An R-module M is called continuous if it is extending and satisfies the following con-
dition: (C2) Every submodule of M that is isomorphic to a direct summand M is itself a
direct summand of M .
3 An R-module M is called quasi-continuous if it is extending and satisfies the following
condition: (C3) If N and K are direct summands of M with N ∩K = 0, then N ⊕K is
a direct summand of M .

Definition 4. Let M be an R-module, put Z(M) = {m ∈ M : annR(m) ≤e R}. M
is called nonsingular if Z(M) = {0}, and singular if Z(M) = M . The Goldie torsion
submodule Z2(M) of M is defined by Z(M/Z(M)) = Z2(M)/Z(M). M is Z2-torsion if,
Z2(M) = M .

Definition 5. A module M has finite uniform dimension n (written Udim(M) = n) if
there is an essential submodule V ≤e M that is a direct sum of n uniform submodules.

3. Main results

Definition 6. An R-module is called c-retractable if HomR(M,C) 6= 0 for each 0 6= C ⊆c
M .

Remark 1. Cleary, every retractable module is c-retractable. The converse is not true in
general. For example: Q as a Z-module is c-retractable while it is not retractable..



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 161

Example 1. Every extending module is c-retractable.

Remark 2. If R = Z[x], then R is c-retractable by ([4], Example 2.4). Clearly, R⊕R is
a c-retractable R-module. However, R⊕R is not extending. (see [4], Example 2.4).

Remark 3. ([4], Example 3.2)
Let R be the ring of all 2 by 2 upper triangular matrices which have arbitrary real numbers
on the diagonal and an arbitrary complex number in the (1, 2)-position and let eij be the
element of R with 1 in the (i, j)-position and 0 elsewhere. Set P = e11R and let K denote
the field of real numbers. Hence, P is a nonsingular projective R-module which is not a
c-retractable R-module while the R-module M = R ⊕ P is c-retractable. This shows that
a direct summand (hence a submodule or a factor module) of a c-retractable module need
not be c-retractable.

Proposition 1. Let M be a c-retractable R-module. Then M/N is
c-retractable for any fully invariant complement submodule N ≤M .

Proof.
Let K/N ⊆c M/N where N ≤ K ≤M and N is a fully invariant complement submodule
of M . Then, K ⊆c M by Proposition 6.28 in [11]. Thus, there exists a nonzero homomor-
phism f : M −→ K. Now, f(N) ⊆ N by hypothesis, and so f : M/N −→ K/N defined
by f(m+N) = f(m) +N for all m ∈M is a nonzero homomorphism.

Proposition 2. Let M be a c-retractable R-module such that HomR(M/C,C) contains
a monomorphism for any C ⊆c M . Then M/C is c-retractable.

Proof.
LetN/C ⊆c M/C. By the c-retractable condition onM , there is a nonzero homomorphism
g : M −→ N . From this and by our assumption, HomR(M/C,N/C) 6= 0.

Proposition 3. Let M be a c-retractable R-module. If M = L⊕N such that HomR(L,N) =
0, then N is a c-retractable R-module.

Proof.

Note that EndR(M) =

[
EndR(L) HomR(N,L)

0 EndR(N)

]
. Hence, EndR(M)

[
L
0

]
⊆

[
L
0

]
.

It follows that (L⊕0) is a fully invariant complement submodule of M . Now, an application
of Proposition 1 shows that N is c-retractable.

Proposition 4. Let M be a c-retractable R-module and 0 6= C ⊆c M .
If HomR(M/C,C) = 0, then C is c-retractable.

Proof.
Let 0 6= K ⊆c C. Thus, there exists 0 6= f ∈ S such that Imf ⊆ K.
If f(C) = 0, then the rule m+ n −→ m+Kerf yelds a nonzero homomorphism
M/C −→M/Kerf ∼= Imf which is in contradiction with our assumption
HomR(M/C,C) = 0. Thus, f(C) 6= 0, hence f |C is a nonzero endomorphism of
C with image in K.



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 162

Proposition 5. If an arbitrary direct sum of copies of M is c-retracatable, then M is
c-retractable.

Proof.
This follows from ([17], Proposition 2.10).

Remark 4. A projective module need not be c-retracatble and vice-versa. In fact a simple
is c-retractable but not be projective. Moreover, by Remark 3, there is a projective module
which is not c-retractable. In the following, we show that certains classes of projective
modules are c-retractable.

Following [24], we call an R-module SI if every singular module is M -injective.
Recall that a ring R is called right SI, if every singular R-module is injective.

Lemma 1. ([24], Proposition 2.2)
Every homomorphic image of a SI-module is a SI-module.

Lemma 2. ([24], Proposition 2.7)
The following conditions are equivalent for a ring R.
(1) R is a right SI-ring.
(2) Every R-module is a SI-module.

Theorem 1. Let R be any ring. Then every projective SI R-module is retractable and
hence c-retractable.

Proof.
Let M be a nonzero projective SI R-module. Let 0 6= m ∈M . For a given submodule A
of mR, there exists a submodule C of mR such that C ⊕ A ≤e mR. Thus, mR/(C ⊕ A)
is singular. Since M is a SI-module, M/(C ⊕ A) is a SI-module by Lemma 1. Hence,
mR/(C⊕A) is M/(C⊕A)-injective and hence a direct summand of M/(C⊕A). It follows
that M has a submodule B such that M/B is isomorphic to mR/(C ⊕ A). Hence there
exists a nonzero homomorphism f : M −→ mR/(C ⊕ A). By the projective condition
on M , f can be lifted to a nonzero of homomorpism g : M −→ mR. Therefore, M is
c-retractable.

Corollary 1. Let R be a right SI-ring. Then every projective R-module is c-retractable.

Theorem 2. Let R be a right perfect ring. Then the following statements are equivalent
for a hereditary R-module:
(1) M is c-retractable.
(2) HomR(M,C) contains an epimorphism for any 0 6= C ⊆c M .
(3) M is extending.

Proof.
(1)⇒ (2) follows a similar argument to the one used in ([14], Theorem 2.2).
(2) ⇒ (3). Let 0 6= C ⊆c M . By (2), there exists an epimorhism f : M −→ C. Then
IC : C −→ C can be lifted to a nozero homomorphism g : C −→M , and hence C ≤⊕ M .



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 163

Therefore, M is extending.
(3)⇒ (1) It is easy to see.

Recall that a family {Ni}I of independent submodules of a module M is said to be a
local summand, if for any finite subset A ⊂ I, ⊕ANα is a direct summand of N .
An R-module M is called uniform-extending if every uniform submodule is essential in a
direct summand of M .

Recall that a module M is called wd-Rickart if the image any endomorphism of M
contains a nonzero direct summand.

Lemma 3. If M is an R-module such that every nonzero complement submodule contains
a nonzero direct summand, then M is c-retractable.

Proof.
This is clear.

Lemma 4. Let M be a wd-Rickart R-module. Then M is c-retractable if and only if every
nonzero complement submodule of M contains a nonzero direct summand.

Proof.
The suffiency follows from Lemma 3. Conversely, assume that M is any wd-Rickart c-
retractable module. Let 0 6= C ⊆c M . Since M is c-retractable, there is a nonzero
endomorphism ϕ of M such that Imϕ ⊆ C. Thus, the wd-Rickart property of M implies
that C contains a nonzero direct summand.

Lemma 5. If M is any wd-Rickart c-retractable R-module, then every indecomposable
complement submodule of M is uniform.

Proof.
Let M be any wd-Rickart c-retractable module. Let C be an indecompsable complement
submodule of M . Let D any nonzero complement submodule of C. Since D ⊆c M , we
infer from Lemma 4 that D contains a nonzero direct summand E of M . As E ≤ C ≤M
and E ≤⊕ M , E ≤⊕ C. Since C is indecomposable, C = E = D. It follows that D is a
direct summand of C, and hence C is an extending module. Since C is indecomposable,
C is uniform.

Theorem 3. Let M be a wd-Rickart R-module for which local summands are summand.
Then M is uniform-extending and c-retractable if and only if M is extending.

Proof.
Suppose that M is a uniform-extending c-retractable module. Since local summands of
M are summand, M is a direct sum of indecomposable modules (see [14], Theorem 2.17).
Thus by Lemma 5, M is direct sum of uniform modules. Therefore, by ([6], 8.5), M is
extending. The converse implication is clear.



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 164

Corollary 2. Let M be a wd-Rickart quasi-discrete R-module. Then M is uniform-
extending and c-retractable if and only if M is extending.

Proof.
This follows from Theorem 3 and the fact that any local summand of a quasi-discrete
module is a summand (see [6], Corollary 4.13).

Remark 5. By Lemma 5, an indecomposable wd-Rickart c-retractable module is uniform.

Recall that a module M is called simple radical, if M 6= 0 such that Rad(M) = M
and M has no proper nonzero submodules N with Rad(N) = N . Hence a simple radical
c-retractable module is uniform.

Let M be an R-module and N ≤ M . Put D(N) = {ϕ ∈ S : Imϕ ⊆ N}. M is called
dual Baer if for every N ≤M , there is e2 = e ∈ S such that D(N) = eS.
Recall that an R-module M is said to be ADS if for every decomposition M = S ⊕ T
and every complement T ′ of S, we have M = S ⊕ T ′.Recall that an R-module M is called
quasi-Baer if, for all fully invariant submodules N ≤M , LS(N) = Se, with e2 = e ∈ S.

Proposition 6. Let M be a dual Baer c-retractable R-module. Then the following state-
ments hold:
(1) M is a direct sum of uniform submodules.
(2) M = Z2(M) ⊕ (⊕i∈IMi) with all Mi nonsingular uniform quasi-Baer and End(Mi)
semi-local quasi-Baer.
(1) If R is a right self-injective ring, then M = Z2(M) ⊕M ′ where M ′ is nonsingular
semisimple.

Proof.
(1) Suppose M is dual-Baer c-retractable. By Corollary 2.6(i) in [10], M is a direct sum of
indecomposable submodules. By ([24], Theorem 3.1), M is wd-Rickart. Thus, according
to Lemma 5, M is a direct sum of uniform submodules.
(2) Suppose M has the stated condition. Then by (1), M is a direct sum of uniform
modules. Hence by ([2], Corollary 2.3, Theorems 3.2 and 3.9), M = Z2(M) ⊕M ′ where
M ′ is quasi-Baer. Since M is dual Baer, we infer from Corollaries 2.5 and 2.6 in [2] that
M ′ = ⊕i∈IMi with all Mi indecomposable. Thus, M = Z2(M) ⊕ (⊕i∈IMi) where each
Mi is indecompsable. Consequently, each Mi is nonsingular uniform by Lemma 5. On the
other hand since M ′ is quasi-Baer, it follows from ([18], Theorem 3.17) that each Mi is
quasi-Baer for each i ∈ I. The last part follows from ([10], Corollary 2.5 and Proposition
2.17) and ([18], Theorem 4.1).
(3) By (2), M = Z2(M)⊕ (⊕i∈IMi) with all Mi nonsingular uniform. Let M ′ = ⊕i∈IMi.
Thus, since R is right self-injective, all Mi are simple, proving the result.

Theorem 4. Let M be a dual Baer R-module. Then the following statements are equiv-
alent:
(1) M is ADS and c-retractable.
(2) M is continuous.
(3) M is quasi-continuous.



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 165

Proof.
(1) ⇒ (2) Suppose M is ADS and c-retractable. Since M is dual Baer, we infer from
Proposition 6(1) that M = ⊕i∈IMi is a direct sum of uniform modules. Thus, every
Mi is quasi-continuous for every i ∈ I. On the other hand since M is ADS, we infer
from Lemma 3.1 in [1] that ⊕i 6=j∈IMj is Mi-injective for every i ∈ I. Therefore M is
quasi-continuous by ([14], Theorem 2.13). Now, let ϕ be an essential monomorphism of
M . Then Imϕ ≤e M . Since M is dual Baer, Imϕ ≤⊕ M . Hence, Imϕ = M . Therefore,
according to ([14], Lemma 3.14), M is continuous.
(3)⇒ (1) This implication is clear.

Corollary 3. Let M be a dual Baer c-retractable R-module such that every nonsingular
summand is ADS. Then M = Z2(M)⊕M ′ where M ′ is nonsingular quasi-continuous.

Proof.
By Proposition 6(2), M = Z2(M) ⊕ (⊕i∈IMi) with all Mi nonsingular uniform. Let
M ′ = ⊕i∈IMi. Thus, by our assumption, M ′ is ADS. Therefore, applying the same
techniques as in the proof of Theorem 4, one can show easily that M ′ is quasi-continuous.

Proposition 7. Let M be a d-Rickart R-module with S is left T -nilpotent. Then the
following statements are equivalent:
(1) M is ADS and c-retractable.
(2) M is quasi-continuous.

Proof.
(1) ⇒ (2) Since M is d-Rickart and S is left T -nilpotent, it follows from Proposition
3.4.11 in [12] that M = ⊕niMi with all Mi indecomposable. Since d-Rickart modules are
wd-Rickart, we infer from Lemma 5 that M = ⊕niMi with all Mi uniform. On the other
hand since M is ADS, we infer from Lemma 3.1 in [1] that ⊕i 6=jMj is Mi-injective for
every 1 ≤ i ≤ n. Thus M is quasi-continuous by ([14], Lemma 2.14).
(2)⇒ (1) This implication is clear.

Let M be an R-module. The left annihilator of N ≤ M in S = EndR(M) is denoted
by LS(N) = {φ ∈ S : φN = {0}}.

Let M be a module. A submodule N of M is said to be an automorphism-invariant
submodule if ϕN ⊆ N for automorphism ϕ of M . M is called auto-invariant if it is an
automorphism-invariant submodule of its injective hull.

Proposition 8. Let M be a dual Baer R-module. Then M is auto-invariant and c-
retractable if and only if M is quasi-injective.

Proof.
Suppose M is auto-invariant and c-retractable. Since M is dual Baer, we infer from Propo-
sition 6(1) that M = ⊕i∈IMi is a direct sum of extending modules. Thus, by Corollary
15 in [13], M is quasi-injective. The converse implication is clear.



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 166

Recall that an R-module M is called C4 if, whenever A and B are submodules of M
with M = A ⊕ B and f : A −→ B is an homomorphism with Kerf ≤⊕ A, we have
Imf ≤⊕ B.

Proposition 9. If every 2-generated R-module is a C4-module, then every dual Baer
c-retractable R-module is semisimple.

Proof.
Let M be any dual Baer c-retractable R-module. Thus, as in the proof of Theorem
4, M = ⊕i∈IMi where each Mi is uniform. Now, we have to show that each Mi is
semisimple. For any 0 6= m ∈ E(Mi), let 0 6= N ≤ mR and take 0 6= n ∈ N . By our
assumption, mR⊕ nR is a C4-module. Consider the inclusion map i : nR −→ mR. Thus
i(nR) = nR ≤⊕ mR. Since mR is indecomposable, nR = mR, and hence N = mR. Thus,
every cyclic submodule of mR is a direct summand. It follows that mR is semisimple.
Hence, E(Mi) is semisimple. Consequently, Mi is semisimple. Therefore, M is semisimple.

Theorem 5. The following conditions are equivalentes for a ring R:
(1) R is semisimple artinian.
(2) Every c-retractable R-module is a C4-module.
(3) Every c-retractable R-module is pseudo-projective.

Proof.
(1)⇒ (2) is clear.
(2)⇒ (1) Let I be a right ideal of R. Clearly, I⊕R is c-retractable, and hence a C4-module
by (2). Consider the inclusion map i : I −→ R. Therefore, i(I) = I ≤⊕ R. Hence, RR is
semisimple. Thus, R is semisimple artinian.
(1)⇒ (3) is clear.
(3) ⇒ (1) Let S be a simple R-module. Then there is a free R-module F and an epi-
morphism f : F −→ S. Hence, S ⊕ F is c-retractable by ([19], Proposition 1.4). By our
assumption, S ⊕ F is pseudo-projective. Now, Consider the exact sequence

0−→Kerf g−→ M
f−→ 0. So, by the proof of ([15], Proposition 3.9), this sequence splits.

Consequently, S ≤⊕ F , and hence S is projective. Therefore, R is semisimple.

Remark 6. Theorem 5 shows that the condition ”right V -ring” in ([15], Proposition 3.9)
is superfluous.

Recall that an R-module is called Baer if, for all N ≤M , LS(N) = Se, with e2 = e ∈ S.
A module M is called K-nonsingular if, ∀ϕ ∈ End(M), Kerϕ ≤e M implies ϕ = 0.

Proposition 10. Let M be a K-nonsingular c-retractable R-module. Then S is right
nonsingular.

Proof.
See proof of ([16], Proposition 3.6).

Proposition 11. Let M be a c-retractable R-module such that SS is extending. Then M
is K-nonsingular if and only if M is Baer.



A. D. Diallo, P. C. Diop , M. Barry / Eur. J. Pure Appl. Math, 13 (1) (2020), 158-169 167

Proof.
Suppose M is K-nonsingular. By Proposition 10, S is right nonsingular. Let N be a
submodule of M . Thus, LS(N) is a complement right ideal in S. Because SS is extend-
ing, then LS(N) = S(1 − e) for some e = e2 ∈ S, and hence M is Baer. The converse
implication follows from ([18], Lemma 2.15).

Recall that a module is locally noetherian if any of its finitely generated submodules
is noetherian. An R-module M is said to be homo-related to an R-module L if there are
α : M −→ L and β : L −→M such that βα 6= 0.

Theorem 6. Let M be a locally noetherian c-retractable R-module. Then M is homo-
related to a direct sum ⊕i∈IUi of uniform submodules of M .

Proof.
Suppose M is a c-retractable locally noetherian module. Hence, every submodule of M
contains a uniform submodule. Thus, by Zorn’s Lemma, M conains a maximal local
direct summand N = ⊕i∈IUi where each Ui is uniform. Also by the locally noetherian
condition on M again, R/r(m) ∼= mR is noetherian for any element m in M . Hence, R
satisfies ACC on right ideals of the form r(m) where m ∈ M . Thus, according to ([6],
8.1), N is a complement submodule of M . Since M is c-retractable, there exists a nonzero
homomorphism f : M −→ N . It follows that M is homo-related to N .

Corollary 4. Let R be a right noetherian ring. Then every c-retractable R-module is
homo-related to a direct sum ⊕i∈IUi of uniform submodules of M .

Theorem 7. Let M be a nonsingular c-retractable R-module such that every Udim(mR) <
∞ for every element m ∈ M . Then M is homo-related to a direct sum ⊕i∈IUi of inde-
composable nonsingular submodules of M .

Proof.
Suppose M has the stated condition. By Zorn’s Lemma, M conains a maximal local direct
summand N = ⊕i∈IUi where each Ui is indecomposable nonsingular. Let m ∈ M . Then
R/r(m) is a nonsingular R-module which has finite uniform dimension. By ([6], Section
5.10), R has ACC on right ideals of the form r(m) where m ∈ M . Thus, according to
([6], 8.1), N is a complement submodule of M . Since M is c-retractable, there exists a
nonzero homomorphism f : M −→ N . It follows that M is homo-related to N .

Proposition 12. Let M be a c-retractable R-module with Udim(M) ≥ 2. Then M is
retractable.

Proof.
Suppose M has the stated condition. Let 0 6= N ≤M . Since Udim(N) <∞, N contains
a uniform submodule U . After replacing U by an essential closure, we may assume that U
is a complement submodule of M . By our assumption, there is a nonzero homomorphism
M −→ U . Therefore, M is retractable.



REFERENCES 168

Corollary 5. Let M be an R-module with Udim(M) ≥ 2. Then M is wd-Rickart c-
retractable if and only if M is semisimple.

Proof.
Suppose M is wd-Rickart c-retractable. Since Udim(M) ≥ 2, M is a finite direct sum of
indecomposable submodules. By Proposition 12, M is retractable. Therefore, according
to ([23], Proposition 2.17), M is semisimple. The converse implication is clear.

Proposition 13. The following statements are equivalent for an R-module M with udim(M) =
n ≥ 2.
(1) M is c-retractable.
(2) HomR(M,U) 6= 0 for every uniform submodule U of M .
(3) HomR(M,U) 6= 0 for every cyclic uniform submodule U of M .
(4) M is retractble.

Proof.
(1)⇒ (2) follows from Proposition 12.
(2)⇒ (3) Clear.
(3) ⇒ (4) Let 0 6= N ≤ M . Let 0 6= m ∈ N . By hypothesis, mR has finite uniform
dimension and hence mR contains a uniform submodule U . Let 0 6= u ∈ U . By (3),
HomR(M,uR) 6= 0. Hence, M is retractable.
(4)⇒ (1) is clear.

Remark 7. By Proposition 12, every c-retractable module M with udim(M) = n ≥ 2 is
retractable. Note that the condition udim(M) = n ≥ 2 can not be dropped. In fact, Q as
a Z-module is c-retractable uniform but it is not retractable.

References

[1] A. Alahmadi, S. K Jain and A. Leroy (2012), ADS modules, J. Algebra 352:215-22.

[2] S. Asgari and A. Haghany, Generalizations of T -extending Modules relative to fully
invariant submodules, J. Korean Math. Soc. 49(2012), No. 3, pp. 503-514.

[3] Sh. Asgari and A. Haghany and A. R. Rezaei, (2014). Modules whose t-closed sub-
module have a summand as a complement, Comm. Algebra, 42:5299-5318.

[4] A. W. Chatters and S. M. Kheuri, Endomorphism rings of modules over nonsingular
CS rings, J. London Math. Soc. 21(1980), 434-444.

[5] N. Ding, Y. Ibrahim, M. Yousif and Y. Zhou (2017), C4-Modules, Comm. Algebra
45(4):1727-1740.

[6] N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer (1994), Extending modules,
Pitman Research Notes in Mathematics 313. Harlow: Longman.



REFERENCES 169

[7] A. Ghorbani and M. R. Vedadi, Epi-retractable modules and some application, Bull.
Iranian Math. Soc. 35 (2009), no. 1, 155-166.

[8] K. R. Goodearl, Ring theory, nonsingular rings and modules. Marcel Dekker, Inc New
York and Basel (1976).

[9] Y. Ibrahim, X. H. Nguyen, M. Yousif and Y. Zhou, Rings whose cyclics are C3-
module, J. Algebra. Appl. 15 (8) (2016)(18 pages).

[10] D. Keskin Tutunu and R. Tribak (2010), On dual Baer modules 52:261-269.

[11] T. Y. Lam, Lectures on modules and rings, G.T.M.(189), Springer-Verlag, Berlin-
Heidelber, New York, 1999.

[12] G. Lee, Theory of Rickart modules, Ph. D. Thesis, M.S., Graduate, School of the
Ohio State University (2010).

[13] T. K Lee and Y. Zhou (2013), Modules which are invariant under automorphisms
injective hulls, J. Algebra Appl. 12(2):9pp.

[14] S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, LMS Lecture
Note Series, 147. Cambridge University Press, Cambridge, 1990.

[15] B. M. Pandeya, A. K Chaturvedi and A. J. Gupta, Applications of epi-retractable
modules, Bull. Iranian Math. Soc. 35 No. 2(2012), pp 469-477.

[16] S. T. Rizvi and C. S. Roman, On K-nonsinglar modules and applications. Comm.
Algebra, 35 (2007): 2960-2980.

[17] S. T. Rizvi and C. S. Roman, On Direct Sum of Baer Modules, J. Algebra, 59 (2009):
632-696.

[18] S. T. Rizvi and C. S. Roman, Baer and quasi-Baer modules. Comm. Algebra, 32
(2004): 103-123.

[19] P. F. Smith, Modules with many homomorphisms, J. of Pure and Appl. Algebra, 197:
305-321 (2005).

[20] P. F. Smith and A. Tercan (1993). Generalizations of CS-modules. Comm. Algebra
21:1809-1847.

[21] P. F. Smith and A. Tercan (1993). Direct summands of modules which satisfy (C11).
Algebra Colloq. 11:231-237.

[22] R. Tribak (2015), On Weakly Dual Rickart Module and Dual Baer Modules, Comm.
Algebra 43:8, 3190-3206.

[23] M. R. Vedadi, Essentially retractable modules, Journal of Science, Islamic Republic
of Iran 18(4), 2007, 355-360.

[24] M. F. Yousif, SI-Modules, Math. J. Okayama Univ. 28, 133-146(1986).