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184 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

2-Regular Modules 
  

Nuhad S. AL-Mothafar 
Dept. of Mathematic/ College of Science/ University of Baghdad, Iraq 

Ghaleb A. Humod  
Dept. of Mathematic/ College of Education for Pure Science (Ibn Al-Haitham)/ 

University of Baghdad 
 

             Received in: 28 April 2015   , Accepted in: 7 June 2015 
 
 
Abstract 
        In this paper we introduced the concept of 2-pure submodules as a generalization of pure 
submodules, we study some of its basic properties and by using this concept we define the 
class of 2-regular modules, where an R-module M is called 2-regular module if every 
submodule is 2-pure submodule. Many results about this concept are given. 
 
 

Key Words: 2-pure submodules, 2-regular modules, pure submodules, regular modules. 
 
 
 
 
 

   
  
  

                                                
 
 

 
   
  
   

 
 
 
 
 
 

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 



 

185 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

Introduction 
        Throughout this paper, R denotes a commutative ring with identity and every R-module 
is a unitary. It is well-known that the pure submodules were given by several authors. For 
example [1] and [2]. 
 
Definition (0.1):  [1] 
        Let M be an R-module. A submodule N of M is called pure if the sequence                       
0  EN  EM is exact for every R-module E. 
 
Proposition (0.2):  [1] 
        Let N be a submodule of M. The following statements are equivalent: 
(1) N is a pure submodule of M. 
(2) For each n

ji i
i 1

r m



N, rji  R, mi  M, j = 1,2,…,k, there exists xi  N, i = 1,2,…,n such 

that n n
ji i ji i

i 1 i 1
r m r x

 
 

 for each j. 

(3)  
Proposition (0.3):  [2] 
        Let N be an R-submodule of M. Consider the following statements: 
(1) N is a pure submodule of M. 
(2) N  IM = IN for each ideal I of R. 
(3) N  IM = IN for each finitely generated ideal I of R. 
(4) N  (r)M = (r)N for each principal ideal (r) of R. 
(5) N  rM = rN for each r  R. 
Then (1)  (2)  (3)  (4)  (5). And if M is flat then (1)  (2). 
        Notice that: Anderson was called the submodule N of M pure if it satisfies (2), see [3].  
        Recall that an R-module M is called regular module if every submodule of M is pure              
[2]. M is called a Von Neumman regular module if every cyclic submodule of M is a direct 
summand of M, [4]. 
        This paper is structured in three sections. In section one we give a comprehensive study 
of 2-pure submodules. Some results are analogous to the properties of pure submodules. In 
section two, we study the concept of 2-regular modules. It is clear that every regular module 
is 2-regular, but the converse is not true (see Remarks and Examples (2.2)(1)). Section three is 
concerned with the direct sum of 2-regular modules. It is shown under certain condition, the 
direct sum of 2-regular modules is 2-regular (see corollary 3.3). Also we show that the 2-
regular property of a module is inherited by its submodules (see Corollary 3.7). Other results 
are given in this section. 
 

0- 2-Pure Submodules 
        In this section we introduce the concept of 2-pure submodules. We investigate the basic 
properties of this type of submodules, some of these properties are analogous to the properties 
of pure submodules. 
Definition (1.1):   
        Let M be an R-module. A submodule N of M is called a 2-pure submodule of M if for 
each ideal I of R, I2M  N = I2N. 
Remarks and Examples (1.2):   
(1) It is clear that every pure submodule is a 2-pure, but not the converse. For example: the 

submodule 
{0, 2}

 of the module Z4 as Z-module is 2-pure submodule since if I= 2Z is an 



 

186 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

ideal of Z, then I2Z4
{0, 2}

=4Z4
{0, 2} {0}

. On the other hand I2
{0, 2}

=4
{0, 2} {0}

. 

By the similar simple calculation one can easily to show that I2Z4  
{0, 2}

 = I2
{0, 2}

 for 

every ideal I = nZ of Z where n is any positive integer. Thus 
{0, 2}

 is a 2-pure submodule 

of Z4 but is not pure since if I = 2Z, then IZ4  
{0, 2}

 = 2Z4  
{0, 2}

 = 
{0, 2}

 and I

{0, 2}
 = 2

{0, 2} {0}
. 

(2) In any R-module M, the submodules M and {0} are always 2-pure submodules in M. 
(3) In the module Z as Z-module, the only 2-pure submodules are {0} and Z. To see this, for 

every submodule  nZ of Z, n2 = n21  < n >2Z  nZ, but n2  n2(nZ) = n3Z. 
(4) Every nonzero cyclic submodule of the module Q as Z-module is a non 2-pure 

submodule. 
Proof:   
        Let N be a cyclic submodule of Q as Z-module, generated by an element a

b
 where a and 

b are two nonzero elements in Z. If we take an ideal <n> of Z where  n  is greater than one, 

then <n2> a
b

 = 
2n a

b
  .  

Also, Q = <n2>Q, because for any element c
d
Q we have 2

2
n

n

c c

d d
  <n

2>Q, thus                 

Q = <n2>Q. Therefore <n2>Q a a
b b

   , implies that <n
2>Q 

 

2n
a a

b b
      . 

(5) It is clear every direct summand is 2-pure since every direct summand is pure submodule, 
hence is a 2-pure submodule, but the converse is not true, for example: the submodule 

{0, 3, 6}
 of the module Z9 as Z-module. It is easily to check that I2Z9  

{0, 3, 6}
 = I2

{0, 3, 6}
 for each I of Z. So, 

{0, 3, 6}
 is 2-pure in Z9 but not pure and hence not direct 

summand. Since if we take I = 3Z, then IZ9  
{0, 3, 6}

 = 
{0, 3, 6}

 and I
{0, 3, 6}

 = 
{0}

. 

(6) Let N be a 2-pure submodule of M such that N  K for some submodule K of M, then K 
may not be a 2-pure. For example: consider the module Z as Z-module. Let N = Z and               
K = 2Z. It is clear Z  2Z but 2Z is not 2-pure in Z. 

        The following propositions give some properties of 2-pure submodules. 
 
Proposition (1.3):   
        Let M be an R-module and N be a 2-pure submodule of M. If A is a 2-pure submodule in 
N, then A is a 2-pure submodule in M. 
Proof:   
        Let I be an ideal of R. Since N is a 2-pure submodule in M and A is a 2-pure submodule 
in N, then I2M  N = I2N and I2N  A = I2A. But A  N, implies I2A = I2N  A =                 
(I2M  N)  A = I2M  (N  A) =I2M  A. 
 
 
 



 

187 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

Proposition (1.4):   
        Let M be an R-module and N is a 2-pure submodule of M. If A is a submodule of M 
containing N, then N is a 2-pure submodule in A. 
Proof:   
        Let I be an ideal of R. Since N is a 2-pure submodule in M, hence I2M  N = I2N and 
since N  A  M implies I2A  N = (I2A  I2M)N = I2A  (I2M  N) = I2A  I2N = I2N. 
 
Proposition (1.5):   
        Let M be an R-module and N is a 2-pure submodule of M. If H is a submodule of N, then 
N

H
 is a 2-pure submodule in M

H
. 

Proof:   
        Let I be an ideal of R. Since 

2
2

2

M N I M H N
I ( )

H H H H
(I M H) N

H


  

 
  

2(I M N) (H N)

H

  
         by Modular law 

2

2

I N H

H
N

I ( )
H





 

 
        Recall that a ring R is called an arithmetical ring if every finitely generated ideal of R is 
a multiplication ideal, where an ideal I of R is called a multiplication ideal if every ideal J  I 
there exists an ideal K of R such that J = IK, see [5]. 
        The following proposition gives a characterization of 2-pure submodules of modules 
over some classes of rings. First let us state the following theorem, which can be found in [6]. 
 
Theorem (1.6):   
        Let I = (a1, a2,…, an) be a multiplication ideal in the ring R. Then for each positive 
integer k, (a1, a2,…, an)k = k k k1 2 n( , ,..., )a a a . 

Proof:  see [6]. 
 
Proposition (1.7):   
        Let M be a module over arithmetical ring R. The following statements are equivalent: 
(1) N is a 2-pure submodule of M. 

(2) For each 
n

2

ij i
i 1

r x

 N, rij  R, xi  M, j = 1,2,…,m, there exists ix   N, i = 1,2, …, n such 

that 
n n

2 2

ij i ij i
i 1 i 1

r x r x
 

   for each j. 

Proof:   

(1)  (2) Assume that N is a 2-pure submodule of M, let 
n

2

i ij i
i 1

y r x


  N for any finite sets, 
n

i i 1
{x }   in M, 

m

j j 1
{y }   in N and {rij} in R where i = 1,2,…,n, j = 1,2,…,m. Let I be an ideal of R 



 

188 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

generated by the finite set {r1j,r2j,…,rnj}, then rij I and 2ijr I
2 imply 2ijr xi  I

2M. Thus 
n

2

j ij i
i 1

y r x


   I2M, therefore yj  I2M  N. But I2M  N = I2N, implies yi  I2N. Since R is 

arithmetical ring, hence by theorem (1.6), 2 2 2 21 j 2 j njI (r , r ,..., r ) . Therefore 
n

2

j ij i
i 1

y r x


   for some 

i
x   N. 

(2)  (1)  Let N be any submodule of M. Let yj  I2M  N, 
n

2

j ij i
i 1

y r x


   where ni i 1{x }    M, 
m

j j 1
{y }    N, i = 1,2,…,n, j = 1,2,…,m. Therefore by hypothesis, there exists ix   N such that 

n n
2 2

j ij i ij i
i 1 i 1

y r x r x
 

     I2N implies yj  I2N. Then I2M  N  I2N. The reverse inclusion is 

clear. Thus I2M  N = I2N, and hence N is a 2-pure submodule of M. 
 
 

1- 2-Regular Modules 
        In this section, we introduce and study the class of 2-regular modules. 
 
Definition (2.1):   
        An R-module M is called 2-regular module if every submodule of M is 2-pure. 
 
Remarks and Examples (2.2):   
(1) It is clear that the following implications hold: 

Von Neumman regular  regular  2-regular 
But non of these implications is reversible. For example: the module Z4 as Z-module is              
2-regular since every submodule of Z4 is 2-pure submodule in Z4, but Z4 is not regular 
since the submodule 

{0, 2}
 of Z4 is not pure, see remark and example (1.2)(1). 

(2) The modules Z and Q as Z-modules are not 2-regular modules, see remarks and examples 
(1.2)(3), and (4). 
 

        The following theorem shows that the cyclic 2-pure submodules is enough to make the 
module be 2-regular. 
 
Theorem (2.3):   
        Let M be an R-module. The following statements are equivalent: 
(1) M is 2-regular module. 
(2) Every cyclic submodule of M is 2-pure submodule of M. 
(3) Every finitely generated submodule of M is 2-pure submodule. 
(4) Every submodule of M is a 2-pure submodule of M. 
Proof:   
(1)  (2) it follows by definition (2.1). 
(2)  (1) Assume that every cyclic submodule of M is 2-pure. Let N be a submmodule of M 
and I is an ideal of R. Let x I2M  N implies x  I2M and x  N. Therefore                        
x  I2M  <x> = I2 <x>   I2 N. 
(1)  (3)  It follows by definition (2.1), and the proof of (2)  (1). 
(3)  (2)  It is clear. 
(1)  (4)  It follows by definition (2.1). 
 



 

189 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

2- The Direct Sum of 2-Regular Modules-Basic Results  
        In this section, we study the direct sum and the epimorphic image of 2-regular module; 
various properties of 2-regular modules are discussed and illustrated. 
        We start with the following proposition. 
        The following proposition shows that the factor module of a 2-regular module is also             
2-regular module. 
 
Proposition (3.1):   
        Let M be an R-module. Then M is a 2-regular if and only if M

N
 is 2-regular for every 

submodule N of M. 
Proof:   
()  Let N be a submodule of M and K is any submodule of M containing N. Since M is                   
2-regular then K is 2-pure in M. Thus K

N
 is 2-pure in M

N
 by proposition (1.5), therefore M

N
 

is 2-regular. 
() It is easily by taking N = 0. 
 
        Now, we have several consequences of the proposition (3.1), the first result shows that 
the epimorphic image of 2-regular module is 2-regular. 
 
Corollary (3.2):   
        Let M and M' be R-modules and f: M  M' be an R-epimorphism. If M is 2-regular 
module then M' is 2-regular. 
Proof: Since f:MM' is an R-epimorphism and M is 2-regular. Then M

ker f
 is 2-regular 

module by proposition (3.1).But 
M

M '
ker f

  by the first isomorphism theorem. Therefore M' is       

2-regular. 
 
Corollary (3.3):   
        Let M1 and M2 be R-modules. If M = M1 M2 is 2-regular R-module, then M1 and M2 
are 2-regular R-modules. The converse is true provided ann (M1) + ann (M2) = R. 
The following statements are equivalent: 
 
Proof:   
        For the first assertion, assume that M = M1 M2 is 2-regular R-module. Let                        
i:M  Mi be the natural projective map of M onto Mi for each i = 1, 2. Since I is an             
R-epimorphism then the epimorphic image of M is 2-regular, implies that Mi is 2-regular. 
Conversely, assume M1 and M2 are 2-regular R-modules and M = M1 M2. Let be                        
a submodule of M = M1 M2. Since ann(M1) + ann(M2) = R then by the same way of the 
proof of [7,prop.(4.2),CH.1], N = N1 N2 where N1 is a submodule in M1 and N2 is a 
submodule in M2. Let I be an ideal of R. To show I2M  N = I2N. Since I2M1N1=I2N1 and 
I2M2N2=I2N2 implies that (I2M1N1)(I2M2N2)= I2N1 I2N2. Then (I2M1I2M1)                 
(N1 N2)=I2(N1 N2), therefore M is 2-regular module. 
 
 
        The proof of the following result is similar to that of corollary (3.3). 



 

190 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

Corollary (3.4):   
        Let M1 and M2 be R-modules. If N1 is a 2-pure submodule in M1 and N2 is a 2-pure 
submodule in M2, then N1 N2 is a 2-pure submodule in M1 M2. 
 
Corollary (3.5):   
        Let M1 and M2 be R-modules and M1 M2 is 2-regular R-module, then M1+M2 is                   
2-regular. 
Proof:   
        Define f : M1 M2  M1+M2 by f (m1,m2) = m1 + m2. It is easily to check that f is an 
epimorphism. Since M1 M2 is 2-regular, thus the epimorphic image of M1 M2 is 2-regular 
by corollary (3.2). Therefore M1+M2 is 2-regular. 
 
Corollary (3.6):   
        Let M1 and M2 be 2-regular R-modules such that ann (M1) + ann (M2) = R, then M1+M2 
is a 2-regular   R-module. 
Proof:   
        Since M1 and M2 are 2-regular R-modules then M1 M2 is 2-regular by corollary (3.3) 
implies that M1+M2 is a 2-regular by corollary (3.5). 
 
 
       The following result shows that every submodule of a 2-regular module inherits the                   
2-regular property. 
 
Corollary (3.7):   
        Every submodule of a 2-regular module is a 2-regular module. 
Proof:   
        Let N be a submodule of a 2-regular R-module M. To show that N is 2-regular                     
R-module. Let K be a submodule in N and I is an ideal of R. Thus we have: 
I2N  K = (I2M  N)  K                    since N is 2-pure in M 
              = I2M  (N  K) 
              = I2M  K 
              = I2K                                        since K is 2-pure in M 
Therefore K is 2-pure in N implies N is 2-regular. 
        We end this paper by the following remark. 
 
Remark (3.8):   
        If all proper submodules of an R-module M are 2-regular then M may not be 2-regular, 
for example: the module Z8 as Z-module is not 2-regular. Since 4   is not 2-pure 
submodule of Z8 because 22Z8 4  = 4   but 22 4  = 0  , while every proper 
submodule of Z8 is 2-regular, since 42 Z   and 4  Z2 are 2-regular modules. 
 
 
 
 
 
 
 
 



 

191 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

References 
1. Rotman, J.J., (1979), An Introduction to Homological Algebra, Academic Press, New 

York. 
2. Fieldhous, D.J., (1969), Pure Theories, Math.Ann., 184, 1-18. 
3. Anderson, F.W. and Fuller, K.R., (1992), Rings and Categories of Modules, Springer 

Verlag, New York. 
4. Kasch, F., (1982), Modules and Rings, Academic Press, New York. 
5. Larsen, M.D. and McCarthy, P.J., (1971), Multiplicative Theory of Ideals, Academic 

Press, New York. 
6. Naoum, A.G. and Majid, M.Balboul, (1985), On Finitely Generated Multiplication Ideals 

in Commutative Rings, Beitrage Zur Algebra and Geometric, 19, 75-82. 
7. Abbas, M.S., (1991), On Fully Stable Module, Ph.D.Thesis, University of Baghdad. 

 
 
 
 
 
 
 
 
 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
 
 
 



 

192 | Mathematics 

 ٢٠١٥) عام ٢(العدد ٢٨المجلد                                                                              والتطبيقيةالھيثم للعلوم الصرفة  ابنمجلة  

Ibn Al-Haitham. J. for Pure & Appl. Sci.                                           Vol. 28 (2) 2015 

   2-النمط المقاسات المنتظمة من 
  نھاد سالم عبد الكريم

 جامعة بغداد /كلية العلوم /قسم الرياضيات
 غالب أحمد حمود   

جامعة بغداد / )ابن الھيثم( الصرفةكلية التربية للعلوم /الرياضياتقسم    
  

  ٢٠١٥حزيران  ٧ في:البحث  ، قبل٢٠١٥نيسان  ٢٨ في:أستلم البحث 
 خالصةال

كتعميم لمفھوم المقاسات الجزئية النقية وباستعمال  2 –في بحثنا ھذا نقدم مفھوم المقاسات الجزئية النقية من النمط         
اذا كان  2 –بأنه منتظم من النمط  Rعلى الحلقة  Mإذ يقال ان المقاس  2 –ھذا المفھوم نعرف المقاسات المنتظمة من النمط

  . أعطينا العديد من النتائج حول ھذا المفھوم.2 –كل مقاس جزئي فيه يكون نقيا ً من النمط 
  
  
  

، المقاسات الجزئية النقية، 2 –، المقاسات النتظمة من النمط 2 –المقاسات الجزئية النقية من النمط  المفتاحية:الكلمات 
  المقاسات المنتظمة.