232 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 

ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  

http://asrjetsjournal.org/  

 

Weak ss-Lifting Modules 

Burcu Nişancı Türkmen* 

Amasya University, Faculty of Arts and Sciences, Department of Mathematics, Amasya, 05100, Turkey 

Email: burcunisancie@hotmail.com 

 

Abstract 

Let M be a module. M is called weak ss-lifting if it is ss-supplemented and its ss-supplement submodules are 

direct summand. In this paper, we provide the basic properties of weak ss-lifting modules. In particular, we 

show that every direct summand of a weak ss-lifting module is weak ss-lifting. Moreover, we prove that a ring R 

is semiperfect with semisimple radical if and only if every projective left R-module is weak ss-lifting. 

Keywords: weak (ss-) lifting module; ss-supplemented module; semiperfect ring, local commutative principle 

ideal ring. 

1. Introduction  

A submodule N of M will show that N   M. Rad(M) and Soc(M) will indicate radical and socle of M, 

respectively. A submodule K of M is called a supplement of N in M if M = N + K and N  K<<K. The module M 

is called supplemented if every submodule of M has a supplement in M. A submodule K of M has ample 

supplements in M if every submodule T of M such that M = K + T contains a supplement of K in M. The module 

M is called amply supplemented if every submodule of M has ample supplements in M [7]. For a module M, two 

submodules N and K of M are called mutual supplements if, M = N + K, N  K<< N and N   K<<K [1]. In [9], 

Zhou and Zhang generalized the concept of socle of a module M to that of Socs(M) by considering the class of 

all simple submodules of M that are small in M in place of the class of all simple submodules of M, that is,  

Socs(M) =∑                        It is clear that Socs(M)   Rad(M) and Socs(M)   Soc(M). 

A module M is called lifting, if for any submodule N of M, there exists a direct summand K of M such that K 

 N and N/K<<M/K, equivalently M is lifting if and only if M is amply supplemented and every supplement 

submodule of M is direct summand.  

------------------------------------------------------------------------ 

* Corresponding author. 



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We say N lies above K in M if N/K <<M/K. Detailed studies about lifting modules can be found at the [1]. As a 

proper generalization of lifting modules, Nebiyev and Pancar defined this way strongly  -supplemented 

modules in [5]. A supplemented module M is called strongly  -supplemented if every supplement submodule 

of M is a direct summand in M. Then this definition is available in [1] as weak lifting modules and there exists 

basic properties of these modules. A submodule K of M is called a ss-supplement of N in M if M = N + K and N 

  K   Socs(K). The module M is called ss-supplemented if every submodule of M has a ss- supplement in M. A 

submodule K of M has ample ss-supplements in M if every submodule T of M such that M = K + T contains a ss-

supplement of K in M. The module M is called amply ss-supplemented if every submodule of M has ample ss-

supplements in M. The class of ss-supplemented modules was first studied by Kaynar et.al. in [4]. In this article, 

motivated by weak lifting modules, we define weak ss-lifting modules and provide the basic properties of weak 

ss-lifting modules. In particular, we show that every direct summand of a weak ss-lifting module is weak ss-

lifting in Proposition 2.5. In Theorem 2.8, we characterize the rings whose projective modules are weak ss-

lifting. Throughout this paper, R will always denote an associative ring with identity element and modules will 

be left unital. Rad(R) will denote the Jacobson radical of the ring R. We will use the notation N<< M to stress 

that N is small submodule of M. We refer to [1,4,7] for any undefined notion arising in the tex. 

2. Results 

 In this section, we investigate some properties of weak ss-lifting modules. We mainly study the relation 

between the notion of weak ss-lifting modules and some other notions. Definition 2.1. A ss-supplemented 

module M is called weak ss-lifting if every ss-supplement submodule of M is a direct summand. 

It is clear that the following implications hold: 

weak ss-lifting   weak lifting 

Lemma 2.2. Let M be a weak lifting module and Rad(M)   Soc(M). Then M is weak ss-lifting. Proof. Clearly, M 

is supplemented. By [4, Lemma 19], M is ss-supplemented because Rad(M)   Soc(M). Let V be a ss-

supplement in M. Therefore, we can write M = U + V and U  V<<V for some submodule U   M. Since M is 

weak lifting, V is a direct summand of M. It means that M is weak ss-lifting.□ Now we give an example which 

shows that a weak lifting module need not be weak ss-lifting. 

Example 2.3. Let F be a field, R = F[[X]] and K be the quotient field of the commutative domain R. Put  

M =R K. It is clear that M is a weak lifting module but M is not weak ss-lifting module since Soc(M) = 0. 

It is clear that every weak ss-lifting module is ss-supplemented. Now we show that the converse is true 

accepting special condition in the following Lemma. Recall from [7, 41.13] that a module M is called  -

projective if, for every two submodules N, K of M and identity homomorphism . IM : M   M with M = N + K, 

there exists p   End(M) with Im(p)   N and Im(IM - p) K.  

Lemma 2.4. Let M be a ss-supplemented and  -projective module. Then M is a weak ss-lifting module. 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 74, No  1, pp 232-236 

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Proof. Since M is ss-supplemented and  -projective, M is amply ss-supplemented by [4, Proposition 37]. Let K 

be a ss-supplement of some submodule N   M. Since M is amply ss-supplemented, there exists a submodule N' 

of N such that M = N + K = N' + K and N'   K   Socs(N'). It follows that N' and K are mutual supplements in M. 

By [7, 41.14(2)], we obtain that K   N' = 0. Therefore M = N'   K. Thus M is weak ss-lifting.□  

Proposition 2.5. Let M be a weak ss-lifting module. Then every direct summand of M is weak ss-lifting. 

Proof. Given M = U   V. By [4, Proposition 26], we deduce that U is ss-supplemented as a factor module of M. 

Let K be a ss-supplement of N in U. It is clear that K is a ss-supplement of N   V in M. Since M is weak ss-

lifting, we can write M = K   W for some submodule W   M. Then Applying modular law, we get that U = U 

 M = U   (K   W) = K   (U   W). Therefore K is a direct summand of U. Hence U is a weak ss-lifting 

module.□ Theorem 2.6. Let Mi be projective module for every 1 < i < n. Then M =     
 Mi is weak ss-lifting if 

and only if every Mi is weak ss-lifting. Proof. ( ) Since every Mi is a direct summand of M for 1 < i < n, the 

proof follows from Proposition 2.5. ( ) Since every Mi is ss-supplemented for 1 < i < n, M is ss-supplemented 

by [4, Corollary 24]. It follows from [7] that M is projective. Then M is  -projective by [1, 4.13]. So M is weak 

ss-lifting by Lemma 2.4. Proposition 2.7. Let M be a  -projective module with Rad(M)   Soc(M). Then the 

following statements are equivalent. 

(1) M is ss-supplemented; 

(2) M is weak ss-lifting; 

(3) M is supplemented; 

(4) M is weak lifting. 

Proof. (1)   (2) Since M is  - projective, the proof follows from Lemma 2.4. 

(2) (1) Clear. 

(1)   (3)   (4) Clear by [4, Lemma 39].□ 

Recall that an R-module M is semiperfect if every factor module of M has a projective cover. If the ring R as a 

left R-module is semiperfect then the ring R is semiperfect. 

Theorem 2.8. For any ring R, the following statements are equivalent. 

(1) R is semiperfect and Rad(R)   Soc(RR); 

(2) RR is ss-supplemented; 

(3) RR is weak ss-lifting; 

(4) every projective left R-module is weak ss-lifting; 

(5) every projective left R-module is ss-supplemented. 

Proof. By Lemma 2.2 and [4, Theorem 41].□ 



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Theorem 2.9. For a ss-supplemented module M, the following statements are equivalent. 

(1) M is weak ss-lifting; 

(2) every ss-supplement submodule of M lies above a direct summand; 

(3) (a) every non-zero ss-supplement submodule of M contains a non-zero direct summand of M; 

(b) every ss-supplement submodule of M contains a maximal direct summand of M. 

Proof. (1)  (2) Clear. 

(2)  (1) Let U be a submodule of M and V be a ss-supplement of U in M. By hypothesis, there exist 

submodules K   M and L   M such that M = K   L, K   V and V   L < L. Then V = V   M = V   (K   L) = K 

  (V   L), V   L << M and M = U + V = U + K + (V   L) = U + K. Since V is a ss-supplement of U, we have V 

= K. Thus M = V   L. Hence M is weak ss-lifting. 

(1)   (3) Clear. 

(3)   (1) Let V be a ss-supplement of U in M. Suppose that W   V and M = W   S. Then V = W   (V   

S) and V   S is a ss-supplement of U + W in M. If V   S = 0, then by 3-(a) there exists a non-zero direct 

summand T of M such that T   V  S. Then we have W   T is a direct summand of M and W   T   V. This 

contradicts the choice of W. Thus V   S = 0 and V = W. So V is a direct summand of M. Therefore M is weak ss-

lifting. Proposition 2.10. Let R be a commutative noetherian local ring with maximal ideal m. The following 

statements are equivalent for a projective R-module M and Rad(M)   Soc(M ). 

(1) M is weak lifting; 

(2) M is weak ss-lifting; 

(3)   M           where Jk are ideals of R 

(a) there exists l > 1 such that the set { k   K | m
l
  Jk } is finite, 

(b) the ideals { Jk | k   K } are linearly ordered by inclusion, and 

(c) if Ji   Jh then mJh   Ji. 

Proof. Follows from Lemma 2.2 and [3, Proposition 3.7]. Corollary 2.11. Let R be a commutative noetherian 

local ring with maximal ideal m. The following statements are equivalent for a projective R-module M and 

Rad(M)   Soc(M). 

(1) M is lifting; 

(2) M is weak lifting; 

(3) M is weak ss-lifting; 

(4)    M =    Rak and for every pair (k, n)  K x K, Rak   Ran is weak ss-lifting. □ 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2020) Volume 74, No  1, pp 232-236 

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In the remaining indication of this article we denote Bm(k,k + 1) the direct sum of arbitrarily many copies of 

R/   and R/     where m is a maximal ideal of R and k is a non-negative integer. 

Theorem 2.12. Let R be a local commutative principal ideal ring with maximal ideal m. If M is a projective R-

module and Rad(M)   Soc(M), then the following statements are equivalent: 

(1) M is weak lifting; 

(2) M is weak ss-lifting; 

(3)  M   Bm(k, k + 1) or M   R
(a)

 for some non-negative integers a and k. 

Proof. Clear by Proposition 2.10.□ 

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