n−Primary Submodules


 

δ-Hollow Modules 
 

Inaam  M. A. Hadi   
     Sameeah Hasoon Aidi*  

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

  
 Received   in : 8  May 2014   ,  Accepted    in:  22  June 2014 
 

Abstract 
        Let R be a commutative ring with unity and M be a non zero unitary left R-module. M is 
called a hollow module if every proper submodule N of M is small (N ≪ M), i.e. N + W ≠ M 
for every proper submodule W in M. A δ-hollow module is a generalization of hollow 
module, where an R-module M is called δ-hollow module if every proper submodule N of M 

is δ-small (N
δ
 M), i.e. N + W ≠ M for every proper submodule W in M with 

M
W

 is singular. 

In this work we study this class of modules and give several fundamental properties related 
with this concept. 
 
Key Words: Small submodule, δ-small submodule, hollow module, δ-hollow module, 

singular module, nonsingular module. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
* This paper is a part of the thesis submitted by the second author. 

241 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

Introduction 
        Throughout this article all rings are commutative rings with identity, and all modules are 
unitary left R-module. A proper submodule L of a module M is called small (denoted by           
L ≪ M), if for every proper submodule K of M, L + K ≠ M. A module M is called hollow if 
every proper submodule of M is called small, [1]. As a generalization of the concept small 
submodule, Zhou in [2] introduce the concept δ-small submodule, where a submodule N of an 
R-module M is called δ-small (denoted by N

δ
 M) if whenever N + K = M and M/K is 

singular module, then K = M. In fact an R-module M is called singular (non singular) if     
Z(M) = {m ∈ M: 

R
ann (m) is an essential ideal of R} = M ((0)), [3], and a submodule N of an 

R-module M is called essential in M (denoted by N
e
≤ M  or  N ∗→  M) if N ∩ W ≠ (0) for 

any non zero W ≤ M, [4]. The concept of δ-hollow module appeared in [5], where an R-
module M is called δ-hollow, if every proper submodule of M is a δ-small in M. Hence 
hollow module is δ-hollow, but the converse is not true. 
        The aim of this work is to give a comprehensive study of the class of δ-hollow modules. 
It is of interest to know how far the old theories of hollow module extend to the new situation. 
 
1- Preliminary 
        In this section, we give some definitions and propositions which are useful in our work. 
 
Definition 1.1: 
        A non zero module M is called a hollow module if every proper submodule N of M is a 
small submodule of M (N ≪ M) that is N + W ≠ M for every W < M, [1]. 
 
Definition 1.2: 
        Let M be an R-module. A submodule A of a module M is called a δ-small submodule of 
M (denoted by A

δ
 M) if  M ≠ A + B for any proper c-singular B of M, (where B is c-singular 

if 
M
B

 is singular module), see [2].. 

 
        An R-module M is called δ-hollow if every proper submodule is δ-small in M, [5]. 
 
        An R-module M is called semisimple if every submodule of M is a direct summand of M             
[3], [4]. 
 
Proposition 1.3: [2] 
        Let M be an R-module and A be a submodule of M. Then the following are equivalent 
(1) A 

δ
  M. 

(2) If M = A + B, then M = Y ⊕ B, for projective semisimple submodule Y of A. 

(3) If M = A + B with M
B

 Goldie torsion, then M = B, where R-module M is called Goldie 

torsion if Z2[M] = M, and Z2(M) is defined by 2
Z (M) M
Z(M) Z(M)

= , (see [3]). 

 
Proposition 1.4: [2] 
(1) Let A and B be submodules of an R-module M such that A ≤ B. If A 

δ
  B then A 

δ
  M. 

242 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

(2) Let A and B be submodules of an R-module M such that A ≤ B. If B 
δ
  M then A 

δ
  M. 

(3) Let A and B be submodules of an R-module M such that A ≤ B, then B 
δ
  M  if and only 

if A 
δ
  M and 

δ

B M
A A
 . 

(4) Let M and N be an R-modules and let  f : M → N be a homeomorphism. If A is a 
submodule of M such that A 

δ
  M, then f (A) 

δ
  N. 

(5) Let A and B be submodules of an R-module M. Then A + B 
δ
  M if and only if A 

δ
  M 

and B 
δ
  M. 

(6) Let M = M1 ⊕ M2 be an R-module, let A1 ≤ M1 and A2 ≤ M2. Then A1 ⊕ A2 
δ
  M1 ⊕ M2  

if and only if  A1 
δ
  M1  and  A2 

δ
  M2. 

 
Proposition 1.5: [6] 
        Let A and B be submodules of an R-module M such that A ≤ B. If B is a direct summand 
of M and A 

δ
  M then A 

δ
  B. 

 
 
        Recall that an R-module M is called indecomposable if the only direct summand of M 
are (0), M, [4]. 
        An R-module M is called a prime module if 

R
ann M = 

R
ann N, for each non zero 

submodule of M, [7]. 
        A non zero R-module is called uniform module if every non zero submodule of M is 
essential in M (N 

e
≤  M), [4]. 

 
Proposition 1.6: 
        Let M be an R-module, then: 
(1) Let A be a proper submodule of an indecomposable R-module M. Then A 

δ
  M if and 

only if A ≪ M, [6, proposition 1.2.13]. 
(2) Let A be a submodule of singular R-module M. Then A 

δ
  M if and only if A ≪ M,           

[6, proposition 1.2.14]. 
(3) Let M be torsion module over an integral domain R and A be a submodule of M. Then                

A 
δ
  M if and only if A ≪ M, [6, corollary 1.2.16], where an R-module over integral 

domain R is called torsion if  
T(M) = {m ∈ M : ∃ r ∈ R/{0}, rm = 0} = M, [4]. 

(4) Let M be a prime R-module with Z(M) ≠ 0 and A be a proper submodule of M. Then              
A

δ
 M if and only if A ≪ M, [6, proposition 1.2.17]. 

(5) Let M be a uniform R-module and A be a submodule of M. Then A
δ
 M if and only if            

A ≪ M, [6, proposition 1.2.18]. 
(6) Let M be an R-module. Then every non singular semisimple submodule A of M is              

δ-small in M, [6, proposition 1.2.3]. 
 

243 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

2- Basic Properties of δ-Hollow Modules 
        In this section, we give the basic properties about δ-hollow modules. We see that under 
certain conditions hollow modules and δ-hollow modules are equivalent. Also we noticed that 
some properties of hollow modules can be generalized to δ-hollow modules. 
 
Remarks and Examples 2.1: 
 
(1) It is clear that Z6 is non singular semisimple, so every submodule of Z6 is non singular 

semisimple, hence every submodule is small by proposition (1.6 (6)). Thus Z6 is a            
δ-hollow module. But Z6 is not hollow. Also notice that Z6 is decomposable. 
 

(2) It is clear that every hollow module is a δ-hollow module. 
Hence each of the Z-module Z4, Z8 and pZ ∞  are δ-hollow. 

 

(3) Z12 as Z-module is not a δ-hollow module since 3 4< > ⊕ < >  = Z12 and 12 4
Z

Z
4< >

  and 

Z4 is a singular Z-module. However 4< >  ≠ Z12. 
 
 
        By using proposition (1.6) hollow modules and δ-hollow modules are coincident under 
certain class of modules. 
 
Theorem 2.2: 
        Let M be an R-module. Then: 
(1) If M is an indecomposable module, then M is a hollow module if and only if M is a δ-

hollow module. 
(2) If M is a singular module, then M is a hollow module if and only if M is a δ-hollow 

module. 
(3) If M is a prime module with Z(M) ≠ 0, then M is a hollow module if and only if M is a δ-

hollow module. 
(4) If M is a uniform R-module then M is a hollow module if and only if M is a δ-hollow 

module. 
(5) If M is a torsion module over a commutative integral domain R then M is a hollow 

module if and only if M is a δ-hollow module. 
 
Proposition 2.3: 
        Epimorphic image of δ-hollow module is δ-hollow. 
Proof: 
        Let M be a δ-hollow module, let M' be a module and let f : M → M' be an 

epimorphism. Suppose N' is a proper submodule of M' with N' + K' = M' and  
M '
K '

 is singular. 

This implies  f – 1(N) ≨ M because if  f – 1(N) = M then  f f – 1(N') = f (M) = M' and N' = M' 
which is a contradiction. Thus f – 1(N')

δ
 M. Also N'+K'= M' implies that f – 1(N')+ f – 

1(K')=M. 

To check 1
M
(K)f −

 is a singular R-module. We show that  1 1
M M

Z( )
(K) (K)f f− −

= . 

244 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

Let  m + f – 1(K') ∈ 1
M
(K)f −

. We must prove 
R

ann (m + f – 1(K'))
e
≤ R. Since  f (m) + K'∈

M '
K '

 

(which is singular). Thus f (m) + K' ∈Z(
M '
K '

).  

So 
R

ann ( f (m) + K') 
e
≤  R. Let J be any ideal of R, J ≠ 0 so 

R
ann  ( f (m)+ K') ∩ J ≠ 0. Thus 

there exists  j ∈ J, j ≠ 0 and j ( f (m) + K') = M
K '

0 , then j f (m) + K' = K'. Thus    j  f (m) ∈ K',  

which implies f (jm) ∈ K', hence  jm ∈ f – 1(K'). Thus  j(m + f – 1(K')) = f – 1(K') = 
1
M
( K ')

0
f −

, that 

is   j ∈ 
R

ann (m + f – 1(K')) ∩ J, and hence  
R

ann (m + f – 1(K')) 
e
≤  R.  

Thus  m + f – 1(K') ∈ Z( 1
M
(K )f − ′

), that is 1
M
(K )f − ′

 is singular. Since f – 1(N') 
δ
  M and 

1

M
(K )f − ′

 is singular, we get f – 1(K') = M (since M is δ-hollow). It follows that                            

f ( f – 1(K')) = f (M) = M', hence K' = M'. Thus M' is a δ-hollow module. 
 
 
Corollary 2.4: 

        Let M be an R-module. If M is a δ-hollow module then 
M
N

 is a δ-hollow module for 

every proper submodule N of M. 
Proof: 

        Let N be a proper submodule of a δ-hollow M. Let π: M → 
M
N

 be the natural 

epimorphism, then 
M
N

 is a δ-hollow module by proposition (2.3). 

 
Corollary 2.5: 
        A direct summand of a δ-hollow module is a δ-hollow module. 
Proof: 
        Let M be a δ-hollow R-module and N be a direct summand of M. Hence M = N ⊕ K for 

some submodule K of M. By second isomorphism theorem 
M
K

 ≃ N. But M
K

 is δ-hollow by 

corollary (2.4). Thus N is δ-hollow. 
 
Proposition 2.6: 

        Let M be an R-module and K 
δ
  M. If 

M
K

 is a δ-hollow module then M is a δ-hollow 

module. 
Proof: 

        Let N < M with M = N + L, where L is a submodule of M and 
M
L

 is singular R-module 

then 
M N L N K L K
K K K K

+ + +
= = + . But 

M (L K) M
/

K K L K
+

+
  by third fundamental theorem. 

We shall prove 
M

L K+
 is singular. 

245 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

Let  
M

m
L K

∈
+

, so m = m + (L + K). But 
R

ann (m + L) ≤ 
R

ann (m + (L + K)), since if                   

r ∈ 
R

ann (m + L), then rm + L = M
L

0  = L implies that  rm ∈ L ≤ L + K. Therefore                    

rm + (L + K) = L + K = M
L K

0
+

, which implies that  r ∈ 
R

ann (m + (L + K)). But 
M
L

 singular 

implies 
R

ann (m + L) 
e
≤  R, hence 

R
ann (m + (L + K)) 

e
≤  R. Thus  m + (L + K) ∈ 

M
Z( )

L K+
, so 

M
L K+

 is singular. Hence 
L K M

K K
+

=  since 
M
K

 is δ-hollow. It follows that L + K = M. But      

K 
δ
  M, implies  L = M. Therefore  M is δ-hollow. 

 
3- δ-Hollow Modules and Other Related Modules 
        In this section, we give some relationships between δ-hollow modules and other related 
modules. 
        Let M be a module, then: 
M is called amply supplemented module if for any two submodules U and V of M with                   
U + V = M, V contains a supplement of U in M, where a submodule A of M is called a 
supplement of B (B ≤ M) if M = A + B and A ∩ B ≪ A. Equivalently A is a supplement of B 
if A + B = M and B is a minimal element in the set of submodules L ≤ M with B + L = M, [8]. 
        Recall that every hollow module is amply supplemented, see [11,proposition (1.3.5)]. 
        We shall give analogus statement for δ-hollow, but first recall that an R-module is called 
δ-amply supplemented if for any two submodules U and V of M with U + V = M, V contains 
a δ-supplemented of U in M, where a submodule N of M is called δ-supplement of a 
submodule W of M if N + W = M, N ∩ W 

δ
  N, [9], [10]. 

Proposition 3.1: 
        Every δ-hollow module is a δ-amply supplemented. 
Proof: 

        Let U proper submodule of M and U + M = M. Since U + M = M and 
M

(0)
M

=  singular 

and U ∩ M = U. But U 
δ
  M, since M is δ-hollow. 

        Recall that a submodule N of a module M is called δ-coclosed in M (briefly N 
scc
≤  M) if 

N
K

is singular and 
δ

N M
K K
  implies N = K for any submodule K of M contained in  N, [12]. 

 
Proposition 3.2: 
        Let M be a module and L be a non zero submodule of M which is δ-hollow, then either L 
is δ-small submodule of M  or  a δ-coclosed submodule of M, but not both. 
Proof: 

        Suppose L is not δ-coclosed submodule of M, so there exists K < L such that 
δ

L M
K K
  

and 
L
K

 is singular . But L is δ-hollow and K < L, hence K 
δ
  L and 

δ

L M
K K
 . Hence                     

246 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

L 
δ
  M. If  L  δ-coclosed submodule of M, and suppose that  L 

δ
  M  then  

δ

L M
M

(0) (0)
= , 

and hence  L = 0, which is a contradiction. 
 
Proposition 3.3: 
        Every non zero δ-coclosed submodule of a δ-hollow module is δ-hollow. 
Proof: 
        Let M be a δ-hollow module and let N be a non zero submodule of M such that N 

δcc
≤  M . 

To show that N is δ-hollow. 
Let L < N ≤ M then L < M  and so  L 

δ
  M. But L < N and N is δ-coclosed implies that            

L 
δ
  N by [12, corollary (2.6)]. Thus  N is a δ-hollow module. 

Proposition 3.4: 
        Let M be a δ-hollow module and let N be a direct summand of M. Then N is δ-hollow. 
Proof: 
        Let A be a proper submodule of N. Since M is δ-hollow, A 

δ
  M and by                   

proposition (1.5) A 
δ
  N. Therefore N is δ-hollow. 

Proposition 3.5: 

        Let M be a singular R-module, let N 
δ
  M. If 

M
N

 is a finitely generated R-module, then 

M is finitely generated. 
Proof: 

        As 
M
N

 is finitely generated, 
M
N

 = R(x1 + N) + … + R(xn + N) for some  x1, …, xn ∈ M. 

We claim that M = Rx1 + … + Rxn. Let m ∈ M then  m + N = r1(x1 + N) + … + rn(xn + N), so 
that m – r1x1 – … – rnxn ∈ N. This implies  m = r1x1 + … + rnxn + n  for some n ∈ N. Thus      
M = <x1 , …, xn> + N. But  M / <x1 , …, xn> is singular (since M is singular) and N 

δ
  M by 

hypothesis M = <x1 , …, xn>. 
Corollary 3.6: 
        Let M be a singular R-module and N be a proper submodule of module M. If M is a δ-

hollow module and 
M
N

 is finitely generated then M is finitely generated. 

Proof: 
        It is clear by proposition (3.5). 
Corollary 3.7: 
        Let M be an R-module with every factor of M is singular and let N < M. If M is a δ-

hollow and 
M
N

 finitely generated, then M is finitely generated. 

Proof: 
        It is clear by proposition (3.5). 
 
Note: 
        Let M be an R-module. If every non zero factor of M is indecomposable, then by 
[13,41.4(1)] M is hollow module, which implies that M is δ-hollow. But the converse is not 

247 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

true, for example Z6 as Z6-module is δ-hollow but does not imply that every non zero factor of 
Z6 is indecomposable, since 6 6

Z
Z

(0)
  is not indecomposable. 

        Recall that, an R-module. M is called δ-lifting, if for every submodule N of M, there 
exist submodules K, K' ≤ M such that M = K ⊕ K' with K ≤ N and  N ∩ K' 

δ
  M, [8]. 

        It is clear that every lifting is δ-lifting. 
Proposition 3.8: 
        Every indecomposable and δ-lifting module is δ-hollow. 
Proof: 
        Let M be indecomposable and δ-lifting module and N be a proper submodule of M. 
Since M is δ-lifting, then M= K ⊕ K' where K ≤ N and N ∩ K' 

δ
  K'. But M is 

indecomposable, so K' = 0 and M = K. Then M ≤ N < M which is a contradiction. Hence        
K' = M  and so  N ∩ K' = N ∩ M = N. Thus  N 

δ
  M. It follows that M is δ-hollow. 

Proposition 3.9: 
        Every δ-hollow module is δ-lifting. 
Proof: 
        Let N be a proper submodule of δ-hollow module M, then M = (0) ⊕ M and {0} ≤ N 
where N ∩ M = N 

δ
  M. Thus M is δ-lifting. 

Proposition 3.10: 
        Let M be an R-module. Then the following statements are equivalent: 
(1) M is indecomposable and δ-lifting. 
(2) M is δ-hollow and indecomposable. 
(3) M is hollow. 
Proof: 
(1) ⇒ (2)  Let N < M. Since M is δ-lifting then M = K ⊕ K' with K ≤ N and N ∩ K' 

δ
  M. As 

M is indecomposable, then K' = 0  or  K = 0. 
If K' = 0, then K = M, which implies that M ≤ N. That is a contradiction. So K = 0, 
hence K' = M and N ∩ K'=N ∩ M=N 

δ
  M. Thus M is δ-hollow. 

(2) ⇒ (3)  It is clear by proposition (2.2(1)). 
 
(3) ⇒ (1)  If M is hollow, then M is indecomposable by [11,proposition 1.3.9]. But M is 

hollow, hence M is lifting by [11, proposition 1.3.16], which implies that M is    
δ-lifting. 

 
        The following is needed for the next result. 
Definition 3.11: [2] 
        A pair (P, f ) is a δ-projective cover of an R-module M, if P is a projective module and              
f :P→M is an epimorphism and ker f 

δ
 P. 

 
Proposition 3.12: 
        Let (P, f ) be δ-projective cover of M. Then M is δ-hollow if and only if P is δ-hollow. 
Proof: 

248 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

(⇒) Since M is δ-hollow module and since f :P→M is an epimorphism, then 
P

M
ker f

  by 

the first fundamental theorem and hence 
P

ker f
 is  δ-hollow. But 

P
ker f

 is δ-hollow and 

ker f 
δ
 P. So by proposition (2.6), P is δ-hollow. 

(⇐) Let N be a proper submodule of M, then f – 1(N) is a proper submodule of P. Since P is   
δ-hollow, then f – 1(N) 

δ
  P, and hence f f – 1(N)

δ
 M by proposition (1.3(4)). But                      

f f – 1(N) = N, so N 
δ
  M. Thus M is δ-hollow. 

 
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 

Collog, 7, 305-318. 
3. Goodearl ,K.R. (1976), Ring Theory, Nonsingular Rings and Modules, Marcel Dekkel. 
4. Kasch, F. (1982) Modules and Rings, Academic Press, Inc-London. 
5. Nematollah, M.J. (2009) On δ-Supplemented Submodules, Tarbiat Modlen Univ., 20th 

Seminar on Algebra 2-3 ordibehesht, 1388 (Apr 22-23), 155-158. 
6. Hassan, S.S. (2011) Some Generalizations of δ-lifting Modules, M.Sc. Thesis, University 

of Baghdad. 
7. Desale,G.and Nicholoson,W.K.(1981)Endoprimitive Ring, J.of Algebra, 70:                

548-560. 
8. Mohamed,S.H. and Muller,B.J. (1990)Continuous and Discrete Modules, Cambridge 

Univ. Press, Cambridge. 
9. Wang, Y. (2007)δ-Small Submodules and δ-Supplemented Modules, International J. 

Math. And Mathematical Sciences, 1-8. 
10. Kosan ,M.T. (2007) δ-lifting and δ-supplemented Modules, Algebra Colloguim, H (1), 53-

60. 
11. Ali, P.M.H. (2005) Hollow Modules and Semihollow Modules, M.Sc. Thesis, Univ. of 

Baghdad. 
12. Lomp, C. and Büyükasik, E. (2009) when δ-Semiperfect Rings are Semiperfect, 

Turk.J.Math., 33, pp.1-8. 
13. Wisbauer, R. (1991) Foundations of Modules and Rings Theory, Gordon and Breach 

Science Publisher Reading. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

249 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 



 

 δالمقاسات المجوفھ من النمط  
 

 إنعام محمد علي ھادي
 سمیعھ حسون عیدي

جامعة بغداد / (ابن الھیثم)كلیة التربیة للعلوم الصرفة /قسم الریاضیات   
 

 2014حزیران 22،قبل البحث في 2014ایار  8استلم البحث
 
 
 

 خالصةال
مقاسا ً مجوفا ً اذا كان  M. یُدعى Rمقاسا ً غیر صفري أیسر أحادي على  Mحلقة إبدالیة ذات محاید ولیكن  Rلتكن         

مقاس جزئي  Wلكل   N + W ≠ M، ھذا یعني   (N ≪ M)مقاسا ً جزئیا ً صغیرا ً  Mفي  Nكل مقاس جزئي فعلي 
اذا  S –من النمط  مقاسا ً مجوفا ً Mیُدعى ، إذ ،تعمیماً  للمقاس المجوفδ –. ندرس المقاس المجوف من النمط Mفعلي في 

(δ M -مقاس جزئي صغیر من النمط  Mفي  Nكان كل مقاس جزئي فعلي 
δ
  (N ھذا یعني انN + W ≠ M  لكل

بحیث  Mمن  Wمقاس جزئي فعلي 
M
W

في ھذا العمل الصنف من المقاسات ونعطي العدید من ندرس مقاس منفرد.  

 الخواص االساسیة المتعلقة بھذا المفھوم.
 

، δ، مقاس مجوف ، مقاس مجوف من النمط δمقاس جزئي صغیر ، مقاس جزئي صغیرة من النمط  الكلمات المفتاحیة :
 مقاس منفرد ، مقاس غیر منفرد.

 

250 | Mathematics 

@a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 27 (2) 2014 


	1- Preliminary
	2- Basic Properties of (-Hollow Modules
	3- (-Hollow Modules and Other Related Modules
	لتكن R حلقة إبدالية ذات محايد وليكن M مقاسا ً غير صفري أيسر أحادي على R. يُدعى M مقاسا ً مجوفا ً اذا كان كل مقاس جزئي فعلي N في M مقاسا ً جزئيا ً صغيرا ً (N ≪ M)  ، هذا يعني N + W ( M  لكل W مقاس جزئي فعلي في M. ندرس المقاس المجوف من النمط – (...
	الكلمات المفتاحية : مقاس جزئي صغير ، مقاس جزئي صغيرة من النمط (، مقاس مجوف ، مقاس مجوف من النمط (، مقاس منفرد ، مقاس غير منفرد.