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214 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

On e-Small Submodules 
 

Inaam M.A. Hadi 
Sameeah H. Aidi   

Dept. of Mathematics/ College of Education for Pure Science( Ibn Al-Haitham) 
University of Baghdad 

Received in :3/June/2015,Accepted in :20/September/2015 
 
 
 

Abstract 
        Let M be an R-module, where R is a commutative ring with unity. A submodule N of M 
is called e-small (denoted by N 

e
  M) if N + K = M, where K 

e
  M implies K = M. We give 

many properties related with this type of submodules. 
 
 
 
Keywords: Small submodule, -small submodule, e-small submodule, and e-coclosed 
submodule. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 



 

215 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

1- Introduction 
        Throughout this work, R is a commutative ring with unity and M is an R-module.                   

A proper submodule N of M is called small (N ≪ M), if N + K = M where K  M implies             
K = M, [1]. A submodule N of M is called -small if N + K = M with 

M

K
is singular implies,       

K = M, [2]. A submodule N of M is called essential (N
e
 M) if N  W  (0) for any non zero 

submodule W of M, [3]. An R-module M is called singular (non singular) if Z(M) = M              
(Z(M) =(0)), where Z(M) = {x  M: 

R
ann (x)  R}. Zhou and Zhang in [4] introduce a new 

type of small submodule namely e-small submodule and give some basic properties of this 
kind of submodules. 
        In this paper, we continuo the work of Zhou [4] and give many other properties of e-
small submodule and study the behavior of e-small submodules in certain class of module. 
 
 
2- Preliminary 
 
Definition (2.1): [4] 
        Let N be a submodule of a module M. N is said to be e-small in M (denoted by N

e
M), 

if N + L =M with  L 
e
  M implies L = M. 

 
Remark (2.2):  
        Obviously, every small (-small) submodule of an R-module M is e-small [4], but the 
converses are not true in general, for example: 
        In the Z-module Z12, the submodule N = 12

e
2 Z    but N   Z12, also N 

δ
  Z12. 

Also, in the Z-module Z6, N = 6
e

3 Z   , but N 
δ
  Z6 and N   Z6, [4]. 

 

Proposition (2.3): [4,Proposition 2.3] 
        Let N be a submodule of M. The following statements are equivalent: 
(1) N 

e
  M 

(2) If X + N = M, then X  M with 
M

X
a semisimple module. Where X  M, means there 

exists W  M such that W  X =M. 
Corollary (2.4): [4] 
        If M is a projective module, then every e-small submodule N of M is -small. 
 
        The next proposition explains how close the notion of e-small submodules to small 
submodules. 
 
Proposition (2.5): [4,Proposition 2.5] 
         Let M be an R-module 
(1) Assume N, K, L are submodules of M with K  N: 

(a) If N 
e
  M, then K 

e
  M and 

e

N M

K K
 . 

(b) N + L 
e
  M if and only if N 

e
  M and L 

e
  M. 



 

216 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

(2) If K 
e
  M and f : M  N is a homeomorphism, then f (K) 

e
  N. In particular, if                

K
e
M  N, then K 

e
  N. 

(3) Assume that K1  M1 < M, K2  M2  M and M = M1  M2, then K1  K2 
e
  M1  M2 

if and only if K1 
e
  M1 and K2 

e
  M2. 

 
 
3- Main Results 
 
Proposition (3.1): 
        Let M be an R-module, let m  M. Then Rm 

e
  M if and only if there exists an 

essential maximal submodule N with m  N. 
Proof: 
()  Suppose there exists an essential maximal submodule N such that m  N. Hence               
M = Rm + N  and so Rm 

e
  M. 

()  Suppose Rm 
e
  M. Hence there exists an essential submodule W (W  M) such that          

M = Rm + W. Let C = {N 
e
  M: Rm + N = M}. Then B  . By Zorn's lemma, there exists a 

maximal element N in C such that Rm + N = M. We claim that N is a maximal submodule. 
Suppose N is not maximal, so there exists a submodule K of M with N  K  M. But N 

e
  M, 

so K 
e
  M and M = Rm + N  Rm + K, thus Rm + K = M and hence K  C. But this 

contradicts the maximality of N. Therefore N is a maximal submodule, and N
e
 M with mN. 

 
Proposition (3.2): 
        Let M be an R-module, let K  N  M be submodules of M. If K 

e
  M and N  M, then 

K 
e
  N. 

Proof: 
        Since N  M, then M = N  W for some W  M. To prove K 

e
  N. Assume N =K + U 

for some U 
e
  N. Then M = (K + U)  W = K + (U  W). We claim that (U  W) 

e
  M.           

To see this : let m  M and m  0. As M = N  W, m = n + w for some n  N, w  W.             
If n  0, then there exists r  R\{0} such that 0  rn  U, hence rm = rn + rw  0 (because if 
rm = 0, then rn = – rw  N  W = (0) and hence rn = 0 which is a contradiction). Thus                 
0  rm  (U  W). 
Now if n = 0, then m = w and so 0  1m = 1w  W  (U  W). Therefore (U  W) 

e
  M. 

Since K 
e
  M, we get U  W = M. Now, assume x  N  M, so that x = u1 + w1 for some           

u1  U, w1  W. It follows that x – u1 = w  (N  W) = (0), hence x = u1  U. Thus N = U 
and K 

e
  N. 

 
 



 

217 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

        Recall that a submodule N of an R-module M is called coclosed whenever K  N, 
N M

K K
  implies N = K [5], [6]. 

 
        Hasan in [7], gave the following definition: 
 
 
Definition (3.3): 
        Let N be a submodule of an R-module M. N is called e-coclosed if whenever K  N, 

e

N M

K K
 , then N = K. 

 
Remarks (3.4): 
(1) It is known that every direct summand is coclosed. However a direct summand may not 

be e-coclosed for example: 

Let M be the Z-module Z6, let N = 2 M  . 
e

N M

0 (0) 
 , but N  0  . 

(2) It is clear that every e-coclosed submodule is coclosed, but the converse is true by the 
same example in (1), N is coclosed and it is not e-coclosed. 

 
Proposition (3.5): [7,Lemma 4.2.8] 
        Let A be a submodule of an R-module M. If A is e-coclosed, then for each X  A,                   
X 

e
  M implies X 

e
  A. 

Proof: 

        To prove X 
e
  A. Assume A = X + Y for some Y 

e
  A. We claim that 

e

A M

Y Y
 . To see 

this, let 
M A C

Y Y Y
   for some 

e

C M

Y Y
 . Then M = A + C, so M = X + Y+ C implies M=X + C. 

Since 
e

C M

Y Y
 , we have 

e
C M . Hence C = M since X 

e
  M. This implies 

M C

Y Y
  and 

e

A M

Y Y
 . But A is e-coclosed in M, so that Y = A. Thus X 

e
  A. 

 
Proposition (3.6): 
        Let M be a non singular R-module. A proper submodule N of M is e-small if and only if 
it is -small. 
Proof: 
()  it is clear by Remark (2.2). 

()  Let N < M. Assume N + K = M with 
M

K
 is singular. Since M is nonsingular, then by           

[8, Proposition 1.21,p.32], K 
e
  M. But N 

e
  M, so K = M. Thus N 

δ
  M. 

 
Proposition (3.7): 
        Let M be an indecomposable R-module. A proper submodule N of M is small if and only 
if it is e-small. 
Proof: 
()  It is clear by Remark (2.2). 



 

218 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

()  Let N < M. Assume N + K = M with K  M. Since N 
e
  M, then by Proposition (2.3),      

K  M  and 
M

X
 is semisimple. But M is indecomposable and K  (0), so K = M.             

Thus N   M. 
 
        Now we get the following corollaries. 
 
Corollary (3.8): 
        Let M be a an indecomposable R-module and let N < M. The following statements are 
equivalent: 
(1) N   M. 
(2) N 

δ
  M. 

(3) N 
e
  M. 

 
 
        Since every uniform module is indecomposable we have the following result which 
follows directly by Corollary (3.8). 
 
Corollary (3.9): 
        Let M be uniform R-module and let N < M. Then the following statements are 
equivalent: 
(1) N   M. 
(2) N 

δ
  M. 

(3) N 
e
  M. 

 
 
        Recall that for an R-module M, if M has maximal submodule. Then                        

e
Rad  M={N

e
 MN is maximal in M}, and if M has no maximal submodule,                       

e
Rad  M = M, [4]. 

 
 
        The following is a characterization of 

e
Rad M. 

 
Theorem (3.10): [4,Theorem 2.10] 
       Let M be an R-module. Then 

ee
Rad(M) {N M N M}    . 

 
Corollary (3.11): [4,Corollary 2.11] 
        Let M and N be R-modules. 
(1) If f : M  N is an R-homeomorphism, then 

e e
(Rad(M)) Rad(N)f  . 

(2) If every proper essential submodule of M is contained in a maximal submodule of M, 
then 

e
Rad(M)  is the largest e-small submodule of M. 

 
 



 

219 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

        Recall that an R-module M is called multiplication if for each N  M, there exists an 
ideal I of R such that N = IM. Equivalently M is multiplication if for each N  M,                      
N = (N:M)M, where (N

R
: M) = {r  R: rM  N}, [9]. 

 
Corollary (3.12):  
        Let M be a finitely generated or multiplication R-module. Then 

e
Rad(M)  is the largest           

e-small submodule of M. 
Proof:  
        Since M is finitely generated or multiplication, then every proper submodule is contained 
in maximal submodule. Hence the result is followed by Corollary (3.11). 
 
 
 
Proposition (3.13):  
        Let M be an module, let m  M then Rm 

e
M  if and only if m  

e
Rad(M) . 

Proof:  
        Suppose Rm 

e
M , then Rm 

e
Rad M , hence m  

e
Rad(M) . Conversely, let                    

m  
e

Rad(M) . Assume 
e

Rad (M) M . Suppose Rm 
e

M , then by Proposition(3.1), there 

exists an essential maximal submodule N in M and m  N. Hence m  
e

Rad(M)  which is a 

contradiction. Thus Rm 
e

M . 

If 
e

Rad (M) M , then M has no essential maximal submodule. Hence for each m  M,             

Rm 
e

M  (by Proposition (3.1)). 

 
Proposition (3.14):  
        An arbitrary sum of e-small submodules of a module M is an e-small submodule of M if 
and only if 

ee
Rad(M) M . 

Proof:  
()  Since 

e
Rad(M) = the sum of all e-small submodules (by Theorem (3.10)), 

ee
Rad(M) M . 

()  Suppose 
ee

Rad(M) M . Let {K} be a family of e-small submodules of M. 

α
eeα

K Rad(M) M


  . Therefore α
eα

K M

   by Proposition (2.5 (a)). 

 
Proposition (3.15):  
        Let M be an R-module. Then 

e
Rad(M) M  if and only if all finitely generated 

submodules are e-small submodules of M. 
Proof:  
()  Suppose 

e
Rad(M) M  and Let N be a finitely generated submodule of M. Hence 

1 nN Rx ... Rx    where x1, …, xn  M = 
e

Rad(M) , then by Proposition (3.13), i
e

Rx M  

and by Proposition (2.5(1.b)) 
e

N M . 



 

220 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

()  Let m  M. Then <m> = Rm is finitely generated, so by hypothesis, 
e

Rm M  and hence 

<m> 
e

Rad(M) . Thus 
e

m Rad(M) . 

 
Next (3.16): 
        It is known that for a module M, if Rad(M) M  then M/Rad M has no nonzero small 
submodulele. However this statement can not be generalized for 

e
Rad(M) ,as the following 

example shows. 
 
Example (3.17):  

        Consider the Z-module Z24 24 24
ee

Rad(Z ) 2 Z   . But 24 2
Z

Z
2 

  and 2 2
e

Z Z . 

 
Proposition (3.18):  
        Let M be a faithful finitely generated multiplication R-module, let N < M. Then the 
following statements are equivalent 

e
N M  if and only if (N:M) 

e
R . 

 
Proof:  
()  Assume (N:M) + K = R with K 

e
R . Then (N:M)M + KM = M, thus N + KM = M. But 

K 
e

R , so by [9,theorem 2.13], KM 
e
  M and since N 

e
  M, we get KM = M. Therefore      

K = R by [9,theorem 3.1]. 
()  Assume N + K = M with K 

e
  M. Since M is multiplication N = (N:M)M, K = (K:M)M, 

and 
eR

(K : M) R  by [1,theorem 2.13]. Thus (N:M)M + K:M)M = M and since M is a finitely 

generated faithful multiplication R-module, then (N:M) + (K:M) = R. As (N:M) 
e

R  and 

e
(K : M) R , we have (K:M) = R. It follows that K = M, and N 

e
  M. 

 
Corollary (3.19):  
       Let M be a faithful finitely generated multiplication R-module, let N < M. The following 
statements are equivalent: 
(1) N 

e
  M. 

(2) (N:M) 
e
  R. 

(3) N = IM for some I 
e
  R. 

 
 
References 
1. Fleury, P., (1974), Hollow Modules and Local Endomorphism Rings, Pac.J.Math., 53, 

379-385. 
2. Zhou, Y.Q., (2000), Generalizations of Perfect Semiperfect and Semiregular Rings, 

Algebra College, 7, 305-318. 
3. Kasch, F., (1982), Modules and Rings, Academic Press, Inc-London. 
4. Zhou, D.X. and Zhang, X.R., (2011), Small-Essential Submodules and Morita Duality, 

Southeast Asian Bull. Math., 35, 1051-1062. 



 

221 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

5. Golan, J.S., (1971), Quasi-Semiperfect Modules, Quart, J.Math.Oxford, 2, 22173-182. 
6. Lomp,C., (1991), On Dual Goldie Dimension, Diplama Thesis, Univ. of Dussel Dorf. 
7. Marhoon, H.K., (2014) Some Generalizations of Monoform Modules, Ms.C Thesis, 

University of Baghdad. 
8. Goodearl, K.R., (1976), Ring Theory Nonsingular Rings and Modules, Marcel Dekkl. 
9. Elbast, Z.A. and Smith, P.F., (1988), Multiplication Modules, Commuinication in 

Algebra, 10, 4. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  



 

222 | Mathematics 

 2015) عام 3العدد ( 28مجلة إبن الهيثم للعلوم الصرفة و التطبيقية                                                                            المجلد  

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 28 (3) 2015 

  
   e-الجزئية الصغيرة من النمط  حول المقاسات 

  
  

  أنعام محمد علي هادي
  سميعة حسون عيدي

  

جامعة بغداد /)ابن الهيثم ( كلية التربية للعلوم الصرفة /قسم الرياضيات   
  

2015/أيلول/20قبل البحث في: ،2015/حزيران/3استلم البحث في:  
 

 
 

  خالصةال
(يرمز له  e –يسمى مقاسا ً جزئيا ً من النمط  Mفي  Nحلقة ابدالية ذات محايد. المقاس  R، إذ Rمقاسا ً على  Mليكن 

 Mبالرمز
e
N   اذا كان (N + K = M إذ ،M 

e
K   تؤدي الىK = M اعطينا العديد من الخواص المتعلقة لهذا .

  النمط من المقاسات الجزئية.
  
  
  

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