EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 3, 2017, 521-528
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Weakly G-Supplemented modules

Celil Nebiyev1,∗, Hasan Hüseyin Ökten2

1 Department of Mathematics, Ondokuz Mayıs University, Turkey
2 Technical Sciences Vocational School, Amasya University, Turkey

Abstract. In this work, we define weakly g-supplemented modules and cofinitely weak g-supplemented
modules. We investigate some properties of these modules. We show that the finite sum of
weakly g-supplemented modules is weakly g-supplemented, an arbitrary sum of cofinitely weak g-
supplemented modules is cofinitely weak g-supplemented. We also define g-semilocal modules and
give some equivalencies for weakly g-supplemented, cofinitely weak g-supplemented and g-semilocal
modules.

2010 Mathematics Subject Classifications: 16D60, 16D80

Key Words and Phrases: g-Small submodules, generalized radical, g-Supplemented modules,
weakly supplemented modules

1. Introduction

Throughout this paper all rings have an identity and all modules are unital left modules.
Let R be a ring and M be an R-module. We denote a submodule N of M by N ≤M .

If M/N is finitely generated for N ≤M , then N is called a cofinite submodule of M . Let
M be an R-module and T ≤ M . If K = 0 for every K ≤ M with T ∩K = 0, then T is
called an essential submodule of M and it is denoted by T E M . K is called a generalized
small (briefly, g-small) submodule of M if for every T E M with M = K +T implies that
T = M , this is written by K �g M (in [5], it is called an e-small submodule of M and
denoted by K �e M). If T is both essential and maximal submodule of M , then T is
called a generalized maximal submodule of M . The intersection of all generalized maximal
submodules of M is called the generalized radical of M and it is denoted by RadgM (in
[5], it is denoted by RadeM). If M have no generalized maximal submodules, then the
generalized radical of M is defined by RadgM = M . Let U and V be submodules of M .
If M = U +V and V is minimal with respect to this property, or equivalently, M = U +V
and U ∩V � V , then V is called a supplement of U in M . If M = U +V and M = U +T
with T E V implies that T = V , or equivalently, M = U + V and U ∩ V �g V , then
V is called a g-supplement of U in M . If every submodule of M has a supplement in

∗Corresponding author.

Email addresses: cnebiyev@omu.edu.tr (C. Nebiyev), hokten@gmail.com (H. Ökten)

http://www.ejpam.com 521 c© 2017 EJPAM All rights reserved.



C. Nebiyev, H. Ökten / Eur. J. Pure Appl. Math, 10 (3) (2017), 521-528 522

M , then M is called a supplemented module. M is called a g-supplemented module, if
every submodule of M has a g-supplement in M . Let U, V ≤ M . If M = U + V and
U ∩ V � M , then V is called a weak supplement of U in M . If every submodule of M
has a weak supplement in M , then M is called a weakly supplemented module. If every
cofinite submodule of M has a weak supplement in M , then M is called a cofinitely weak
supplemented module.

There are some important properties of g-small submodules in [1, 3, 4] and [5].

Lemma 1 ([4, 5]). Let M be an R-module and K, N ≤M . The following conditions are
hold.

(i) If K ≤ N and N �g M , then K �g M .

(ii) If K �g N , then K is an g-small submodule in submodules of M which contain N .

(iii) If f : M → N is an R-module homomorphism and K �g M , then f (K)�g N .

(iv) If K �g L and N �g T for L, T ≤M , then K + N �g L + T .

Corollary 1. Let M be an R-module and K ≤ N ≤M . If N �g M , then N/K �g M/K.

Corollary 2. Let M be an R-module, K �g M and L ≤M . Then (K + L) /L�g M/L.

Lemma 2 ([1]). Let M be an R-module. Then RadgM =
∑

L�gM
L.

Lemma 3 ([1]). Let M be an R-module. If M has at least one proper essential sub-
module and every proper essential submodule of M is contained in a generalized maximal
submodule, then RadgM �g M .

Lemma 4 ([1]). If M is a finitely generated R-module and M has at least one proper es-
sential submodule, then every proper essential submodule of M is contained in a generalized
maximal submodule of M .

2. Weakly G-Supplemented Modules

Definition 1. Let M be an R-module and U, V ≤M . If M = U + V and U ∩ V �g M ,
then V is called a weak g-supplement of U in M . If every submodule of M has a weak
g-supplement in M , then M is called a weakly g-supplemented module.

Clearly, we see that g-supplemented modules are weakly g-supplemented. We also see
that every weakly supplemented module is weakly g-supplemented.

Lemma 5. Let M be an R-module, M1 ≤M , U ≤M and M1 be a weakly g-supplemented
module. If M1 + U has a weak g-supplement in M , then U has also a weak g-supplement
in M .



C. Nebiyev, H. Ökten / Eur. J. Pure Appl. Math, 10 (3) (2017), 521-528 523

Proof. Let X be a weak g-supplement of M1 + U in M . Then M1 + U + X = M and
(M1 + U) ∩ X �g M . Since M1 is weakly g-supplemented, (U + X) ∩M1 has a weak
g-supplement Y in M1, i.e. M1 ∩ (U + X) + Y = M1 and M1 ∩ (U + X) ∩ Y �g M1.
Following this, we have M = M1 ∩ (U + X) + Y + U + X = U + X + Y and

U ∩ (X + Y ) ≤ X ∩ (U + Y ) + Y ∩ (U + X) ≤ X ∩ (M1 + U) + Y ∩M1 ∩ (U + X)�g M.

Hence X + Y is a weak g-supplement of U in M .

Corollary 3. Let M be an R-module U ≤ M and Mi ≤ M for i = 1, 2, . . . , n. If
U +M1+M2+ . . .+Mn has a weak g-supplement in M and Mi is a weakly g-supplemented
module for every i = 1, 2, . . . , n, then U has a weak g-supplement in M .

Proof. Clear from Lemma 5.

Lemma 6. Let M = M1 + M2. If M1 and M2 are weakly g-supplemented modules, then
M is a weakly g-supplemented module.

Proof. Clear from Lemma 5.

Corollary 4. A finite sum of weakly g-supplemented modules is weakly g-supplemented.

Proof. Clear from Lemma 6.

Lemma 7. Let M be an R-module, X ≤ U ≤ M and V be a weak g-supplement of U in
M . Then (V + X) /X is a weak g-supplement of U/X in M/X.

Proof. Since V is a weak g-supplement of U in M , we have M = U+V and U∩V �g M .
Thus (U ∩ V + X) /X �g M/X by Lemma 1. Since M = U + V , it is easy to see that
M
X = U+V

X = U
X + V+X

X and U
X ∩

V+X
X = U∩V+X

X �g
M
X . Therefore (V + X) /X is a weak

g-supplement of U/X in M/X.

Theorem 8. If M is a weakly g-supplemented module, then every factor module of M is
weakly g-supplemented.

Proof. Clear from Lemma 7.

Corollary 5. If M is a weakly g-supplemented module, then the homomorphic image of
M is weakly g-supplemented.

Definition 2. Let M be an R-module. If M/RadgM is semisimple, then M is called a
g-semilocal module.

Clearly, we see that every semilocal module is g-semilocal.

Lemma 9. For an R-module M , the following statements are equivalent.



C. Nebiyev, H. Ökten / Eur. J. Pure Appl. Math, 10 (3) (2017), 521-528 524

(i) M is g-semilocal.

(ii) For every U ≤ M there exists a submodule V ≤ M such that U + V = M and
U ∩ V ≤ RadgM .

(iii) There exists a decomposition M = M1 ⊕M2 such that M1 is semisimple, RadgM E
M2 and M2/RadgM is semisimple.

Proof. Clear from [2, Proposition 2.1].

Lemma 10. Any homomorphic image of a g-semilocal module is g-semilocal.

Proof. Let M and N be R-modules, f : M −→ N be an R-module epimorphism and
M be g-semilocal. Since M is g-semilocal, M/RadgM is semisimple. Let

ϕ : M/RadgM −→ N/RadgN, x + RadgM −→ ϕ (x + RadgM) = f (x) + RadgN

be a map. It easy to check that ϕ is an R-module epimorphism, since f (RadgM) ≤
RadgN . Since every homomorphic image of a semisimple module is semisimple, N/RadgN
is semisimple. Hence N is g-semilocal.

Lemma 11. Let M = M1 + M2. If M1 and M2 are g-semilocal, then M is g-semilocal.

Proof. Since M1 and M2 are g-semilocal, M1/RadgM1 and M2/RadgM2 are semisim-
ple. Then M1

RadgM1
⊕ M2

RadgM2
is semisimple. Let

f :
M1

RadgM1
⊕ M2

RadgM2
−→ M

RadgM
,

(x1 + RadgM1, x2 + RadgM2) −→ f (x1 + RadgM1, x2 + RadgM2) = x1 + x2 + RadgM

be a map. It is easy to check that f is an R-module epimorphism. Since every homomor-
phic image of a semisimple module is semisimple, M/RadgM is semisimple. Hence M is
g-semilocal.

Corollary 6. Let M = M1 +M2 + . . .+Mn. If Mi is g-semilocal for every i = 1, 2, . . . , n,
then M is g-semilocal.

Proof. Clear from Lemma 11.

Lemma 12. If M is a weakly g-supplemented module, then M is g-semilocal.

Proof. Let U/RadgM be any submodule of M/RadgM . Since M is weakly g-supplemented,
there exists a submodule V of M such that M = U +V and U ∩V �g M . Since U ∩V �g

M , then by Lemma 2, U ∩ V ≤ RadgM . Then by M
RadgM

= U+V
RadgM

= U
RadgM

+
V+RadgM
RadgM

and

U

RadgM
∩V + RadgM

RadgM
=

U ∩ V + RadgM

RadgM
=

RadgM

RadgM
= 0,

M

RadgM
=

U

RadgM
⊕V + RadgM

RadgM
.

Hence M is g-semilocal.



C. Nebiyev, H. Ökten / Eur. J. Pure Appl. Math, 10 (3) (2017), 521-528 525

Lemma 13. Assume M be an R-module and RadgM �g M . If M is g-semilocal, then
M is weakly g-supplemented.

Proof. Let U be any submodule of M . Since M is g-semilocal, (U + RadgM) /RadgM
is a direct summand of M/RadgM . Then there exists a submodule V of M such that

RadgM ≤ V and M
RadgM

=
U+RadgM
RadgM

⊕ V
RadgM

. By M
RadgM

=
U+RadgM
RadgM

⊕ V
RadgM

= U+V
RadgM

,
M = U + V . Since

U ∩ V + RadgM

RadgM
=

U + RadgM

RadgM
∩ V

RadgM
= 0, U ∩ V ≤ RadgM �g M.

Hence V is a weak g-supplement of U in M .

Corollary 7. Assume M be an R-module with RadgM �g M . Then M is weakly g-
supplemented if and only if M is g-semilocal.

Proof. Clear from Lemma 12 and Lemma 13.

Lemma 14. Let M be a finitely generated R-module. Then RadgM �g M .

Proof. If M has at least one proper essential submodule, since M is finitely generated,
by Lemma 4, every proper essential submodule of M is contained in a generalized maximal
submodule of M . Then by Lemma 3, RadgM �g M . If M have no proper essential
submodules, then RadgM = M �g M also holds.

Lemma 15. Let M be a finitely generated R-module. Then M is weakly g-supplemented
if and only if M is g-semilocal.

Proof. By Lemma 14 and Corollary 7, this is clear.

Corollary 8. RR is weakly g-supplemented if and only if RR is g-semilocal.

Proof. By Lemma 15, this is clear.

Proposition 1. Let M be a weakly g-supplemented R-module. Then for every U, V ≤M
with M = U + V , there exists a weak g-supplement K of U in M with K ≤ V .

Proof. Assume U, V ≤M with M = U +V . Since M is weakly g-supplemented, U ∩V
has a weak g-supplement T in M . In this case, M = U∩V +T and U∩V ∩T �g M . Since
M = U +V = U ∩V +T , M = U +V ∩T . Let K = V ∩T . Then M = U +V ∩T = U +K
and U ∩ K = U ∩ V ∩ T �g M . Hence K is a weak g-supplement of U in M with
K ≤ V .

Example 1. Let p and q be prime numbers and consider the ring

R = Zp,q = {a
b
| a, b ∈ Z, b 6= 0, p 6 |b and q 6 |b}.

By [2, Remark 3.3], RR is weakly supplemented but not supplemented. Since every nonzero
submodule of RR is essential in RR, RR is weakly g-supplemented but not g-supplemented.



C. Nebiyev, H. Ökten / Eur. J. Pure Appl. Math, 10 (3) (2017), 521-528 526

3. Cofinitely Weak G-Supplemented Modules

Definition 3. Let M be an R-module. If every cofinite submodule of M has a weak
g-supplement in M , then M is called a cofinitely weak g-supplemented module.

Clearly we see that every weakly g-supplemented module is cofinitely weak g- supple-
mented.

Lemma 16. Assume M be a finitely generated R-module. If M is cofinitely weak g-
supplemented, then M is weakly g-supplemented.

Proof. Clear, since every submodule of M is cofinite.

Lemma 17. Let M be a cofinitely weak g-supplemented module. Then every factor module
of M is cofinitely weak g-supplemented.

Proof. Let M/X be any factor module of M and U/X be a cofinite submodule of

M/X. Since M
U
∼= M/X

U/X , U is a cofinite submodule of M . Since M is cofinitely weak

g-supplemented, U has a weak g-supplement V in M . Then by Lemma 7, (V + X) /X is a
weak g-supplement of U/X in M/X. Hence M/X is cofinitely weak g-supplemented.

Corollary 9. Any homomorphic image of a cofinitely weak g-supplemented module is
cofinitely weak g-supplemented.

Proof. Clear from Lemma 17.

Lemma 18. Let M be an R-module, M1 ≤ M , U be a cofinite submodule of M and M1

be a cofinitely weak g-supplemented module. If M1 + U has a weak g-supplement in M ,
then so does U .

Proof. Let X be a weak g-supplement of M1 + U in M . Then M1 + U + X = M and
(M1 + U) ∩ X �g M . Since U is a cofinite submodule of M , U + X is also a cofinite
submodule of M . Then by M1

M1∩(U+X)
∼= M1+U+X

U+X = M
U+X , M1 ∩ (U + X) is a cofinite

submodule of M1. Since M1 is cofinitely weak g-supplemented, M1 ∩ (U + X) has a weak
g-supplement Y in M1, i.e. M1 ∩ (U + X) + Y = M1 and M1 ∩ (U + X) ∩ Y �g M1.
Following this, we have M = M1 ∩ (U + X) + Y + U + X = U + X + Y and

U ∩ (X + Y ) ≤ X ∩ (U + Y ) + Y ∩ (U + X) ≤ X ∩ (M1 + U) + Y ∩M1 ∩ (U + X)�g M.

Hence X + Y is a weak g-supplement of U in M .

Corollary 10. Let M be an R-module, U be a cofinite submodule of M and Mi ≤M for
i = 1, 2, . . . , n. If U + M1 + M2 + . . . + Mn has a weak g-supplement in M and Mi

is a cofinitely weak g-supplemented module for every i = 1, 2, . . . , n, then U has a weak
g-supplement in M .



C. Nebiyev, H. Ökten / Eur. J. Pure Appl. Math, 10 (3) (2017), 521-528 527

Proof. Clear from Lemma 18.

Lemma 19. Any sum of cofinitely weak g-supplemented modules is cofinitely weak g-
supplemented.

Proof. Let {Mi}i∈I be a family of cofinitely weak g-supplemented submodules of an
R-module M and M =

∑
i∈I Mi. Let U be any cofinite submodule of M . Since U is

cofinite submodule of M , there exists a finite subset {i1, i2, . . . , in} of I such that
M = U+Mi1 +Mi2 +. . .+Min . Since U+Mi1 +Mi2 +. . .+Min has a weak g-supplement 0
in M and Mik is cofinitely weak g-supplemented for k = 1, 2, . . . , n, then by Corollary 10,
U has a weak g-supplement in M .

Proposition 2. Let R be a ring. The following statements are equivalent.

(1) RR is g-semilocal.

(2) RR is weakly g-supplemented.

(3) Every finitely generated R-module is g-semilocal.

(4) Every finitely generated R-module is weakly g-supplemented.

(5) R(I) is cofinitely weak g-supplemented for every index set I.

(6) Every R-module is cofinitely weak g-supplemented.

Proof. (1)⇔ (2) Clear from Corollary 8.
(1)⇒ (3) Assume M be a finitely generated R-module and let M = 〈m2,m2, . . . ,mn〉.

Then M = Rm1 +Rm2 + . . .+Rmn. Since RR is g-semilocal and Rmi (i = 1, 2, . . . , n) is
an homomorphic image of RR, by Lemma 10, Rmi is g-semilocal. Then by Corollary 6,
M is g-semilocal.

(3)⇔ (4) Obtained from Lemma 15.
(4) ⇒ (5) By hypothesis, RR is weakly g-supplemented. Hence RR is cofinitely weak

g-supplemented. Because of this, by Lemma 19, R(I) is cofinitely weak g-supplemented
for every index set I.

(5)⇒ (6) Clear from Corollary 9, since every R-module is RR-generated.
(6) ⇒ (2) By hypothesis, RR is cofinitely weak g-supplemented. Since RR is finitely

generated, by Lemma 16, RR is weakly g-supplemented.

Proposition 3. Let M be weakly g-supplemented R-module and U be a cofinite submodule
of M . Then for every V ≤M with M = U + V and U ∩ V is a cofinite submodule of M ,
there exists a weak g-supplement K of U with K ≤ V .

Proof. Similar to proof of Proposition 1.



REFERENCES 528

References

[1] B. Koşar, C. Nebiyev, and N. Sökmez. G-supplemented modules. Ukrainian Mathe-
matical Journal, 67(6):861–864, 2015.

[2] C. Lomp. On semilocal modules and rings. Communications in Algebra, 27(4):1921–
1935, 1999.

[3] T. C. Quynh and P. H. Tin. Some properties of e-supplemented and e-lifting modules.
Vietnam Journal of Mathematics, 41(3):303–312, 2013.

[4] N. Sökmez, B. Koşar, and C. Nebiyev. Genelleştirilmiş küçük alt modüller, 2010.
Presented at the XXIII. Ulusal Matematik Sempozyumu, Erciyes Üniversitesi, Kayseri-
Türkiye.

[5] D. X. Zhou and X. R. Zhang. Small-essential submodules and morita duality. Southeast
Asian Bulletin of Mathematics, 35(6):1051–1062, 2011.