This research considers three basic concepts concerning primary submodules, which are important, at least to some mathematicians. The first concept is the concept of primary radical of a submodule, the second concept is the concept of minimal primary submodules, and the third one is the concept of primary compactly packed modules. In this paper, we generalize these three concepts over noncommutative rings and generalize many results concerning them, in the rings that are not necessary commutative

Ashour, Arwa E.

English

Institutional Repository of the Islamic University of Gaza

IUG Journal of Natural and Engineering Studies 
Vol.20, No.1, pp 63-71 2012, ISSN 1726-6807, http://www.iugaza.edu.ps/ar/periodical/   
 
On  Primary  Compactly Packed  Modules  Over  
Noncommutative Rings 
 
Arwa  Eid  Ashour 
Assistant Professor,   Department of Mathematics, 
Islamic University of Gaza, 
P.O. Box 108,  Gaza Strip,  Palestine, 
Tel: +970-8-2060420, 
E-mail: arashour@iugaza.edu.ps 
 
Abstract: This research considers three basic concepts concerning primary 
submodules,  which are important,  at least to some mathematicians. 
     The first concept is the concept of primary radical of a submodule, the 
second concept is the concept of minimal primary submodules,  and the third 
one is the concept of primary compactly packed modules. 
     In this paper,  we generalize these three concepts over noncommutative 
rings and generalize many results concerning them,   in the rings that are not 
necessary commutative.  
 
مة الرص المعرفة على الحلقات غير المقاسات االبتدائية المحزو
 االبدالية
المقاسات الجزئية  االبتدائيـة،  و التـي تهتم هذه الدراسة بثالثة مفاهيم رئيسية تتعلق ب : ملخ ص 
  .تعتبر ذات أهمية كبرى،  على األقل لبعض الرياضيين
 هو مفهوم المقاسات المفهوم الثاني  ، لجزئيةلجذر االبتدائي للمقاسات ا المفهوم األول هو مفهوم ا   
  .المقاسات االبتدائية المحزومة الرصب فهو يتعلقالمفهوم الثالث أما .  الجزئية  االبتدائية األصغرية
نعمم العديد مـن  على الحلقات غير االبدالية و نقوم في هذا البحث بتعميم المفاهيم الثالثة السابقة   
  . النتائج المتعلقة بها
1.  Introduction 
Primary modules have long been known in commutative rings for their 
center stage role in some theories in commutative algebra.   As long as for 
their importance in generalizing many concepts associated prime 
submodules such as the concepts of prime radical of a submodule,  and 
compactly packed modules. 
        More recently,  primary submodules have been introduced in 
noncommutative rings in the hope of extending and generalizing the 
concepts associated primary submodules over noncommutative rings. 
Let  R  be an arbitrary ring.  A nonzero submodule  N  of an  R-module  M  
is primary 
 if    for every nonzero submodule                
Arwa  Eid  Ashour 
 64
     , where  (  )   (  )  rann N rann N= ,N N⊆  
for some positive integer  n }.  This definition     0=Nr n ,    ( ) {rann N r r R= ∈   
of a primary submodule over noncommutative rings was introduced recently 
in [1].  In fact, this definition is a generalization of the definition of prime 
submodules in an arbitrary ring and an extension of the definition of 
primary submodules over commutative rings,  see[1 ].   The definition of  
primary submodules over noncommutative rings was a motive for us to 
generalize many concepts and results associated to prime submodules over 
noncommutative rings,  and primary submodules over commutative rings.  
        In the beginning of this paper ,  in Section 2,  we introduce the 
definition of primary radical of a submodule as follows:   Let  N  be a 
submodule of an  R-module  M.  If there exist primary submodules 
containing  N ,   then the intersection of all primary submodules containing  
N  is called the primary radical of  N  and is denoted by  prad( N ).   If there 
is no primary submodule containing  N ,  then  prad( N )=M.  In particular  
prad( M )=M. 
Also in this section,  we study some properties of primary radical of a 
submodule. 
       In[2],  and [3],  Chin Pi Lu proved some results on minimal prime 
submodules in commutative rings.   We generalize the concept of minimal 
prime submodules to the concept of minimal primary submodules over 
commutative rings,  see[4],[5].   
             In  Section three we define the minimal primary submodules over 
non commutative rings.  Thus we define a primary submodule Q of an  R-
module  M  to be a minimal primary over a submodule  N  if  QN ⊆   and 
we show that there is no smaller primary submodule with this property. We 
also prove some results concerning minimal primary submodules over 
noncommutative rings.   
Key  Words:  Primary submodules over  noncommutative rings, primary 
radical of a submodule over  noncommutative rings,  minimal primary 
submodules over  noncommutative rings,  primary compactly packed 
modules over  noncommutative rings.   
"2000  Mathematics  Subject  Classification" :  Primary 16D25,  Secondary 
16D80,  16N60. 
 The compactness property of prime ideals was studied by Ries, C.M. 
and Viswanthan, T.M. in 1970,  see[6].  Then in 1995,   Yong Hwan Cho.,  
defined the comprimely  packed rings,   see[7].  The previous studies, in [6] 
and [7],  were generalized to modules in 2002,  on the rings that are 
commutative, and the concept of compactly packed modules was 
On  Primary  Compactly Packed  Modules  over   
 65
introduced,  see[8].   Then,  we generalized these results on primary 
modules,  and defined the primary compactly packed modules over 
commutative rings in [4],[5],[9].   In  Section 4,   we study more properties 
of primary radical of a submodule,  and introduce the concept of primary 
compactly packed modules over any arbitrary ring.   We define a proper 
submodule  N  of an  R-module  M  to be primary compactly packed if for 
each family   λαα ∈}{P  of  primary submodules of   M  with   
U
λα
α
∈
⊆ PN ,    βPN ⊆   for  some  λβ ∈ .    Moreover,   the module M  
is called primary compactly packed if every submodule of  M  is primary 
compactly packed.   Also,  we give equivalent definitions of primary 
compactly packed modules and study some various properties of primary 
compactly packed modules. 
 Finally,    we remark that throughout this paper,   we will work 
exclusively with left unitary modules and all rings are assumed to be rings 
with identity. 
2. Primary Radical of Submodules 
We start this section by the definition of primary radical of a submodule,  as 
follows. 
Definition 2.1    Let  N  be a submodule of an  R-module  M.  If there exist 
primary submodules containing  N ,   then the intersection of all primary 
submodules containing  N  is called the primary radical of  N  and is denoted 
by  prad( N ).   If there is no primary submodule containing  N ,  then  prad( 
N )=M.  In particular  prad( M )=M.    
We say that a submodule  N  is a primary radical submodule if  prad( N )=N.  
Examples 2.2 
1)  It is clear that every primary submodule is primary radical submodule. 
2)  Let R=Z,  the set of integers.  Since every ideal of  R  is a submodule of  
R,  primary ideals of  R are primary submodules of  R.  So for  
,...21 21 kkpppn
ααα=   where             
spi `  are prime numbers,  and since   ,)()(
1
I
k
i
i
ipn
=
= α    then in  Z,  every ideal 
is primary radical submodule of   R. 
           The following result can be easily noticed. 
Proposition 2.3  Let  N  and  L be submodules of an  R-module  M.  Then 
1)   ).(NpradN ⊆   
2)    If    ,LN ⊆   then  ).()( LpradNprad ⊆  
Arwa  Eid  Ashour 
 66
3)    );())(( NpradNpradprad =   that is,   the primary radical of  N  is a 
primary  
        radical submodule.  
4)    ).()()( LpradNpradLNprad ∩⊆∩  
5)    )).()(()( LpradNpradpradLNprad +=+   
 
Theorem 2.4  For any  R-module  M,  if  M  satisfies the ACC for primary 
radical submodules,  then every primary radical submodule of  M   is the 
primary radical of a finitely generated submodule. 
Proof.   Assume that there exists a primary radical submodule  N which is 
not primary radical of a finitely generated submodule.  Let  Ne ∈1  and let  
N1 =  prad ( e1 R ).   Then  .1 NN ⊆   So there exists  .12 NNe −∈    Let  
).( 212 ReRepradN +=    Then  .21 NNN ⊂⊂ So that there exists   
.23 NNe −∈    Continuing in this process,  we will have an ascending chain 
of primary radical submodules  ...,321 ⊂⊂⊂ NNN  
which is a contradiction. 
 The following Theorem follows from Example 2.2 (1). 
Theorem 2.5  If every primary radical submodule is the primary radical of a 
finitely generated submodule,   then every primary submodule is the 
primary radical of a finitely generated submodule. 
Proposition 2.6  Let  N and  L  be submodules of an R-module  M  such that 
whenever   ,QLN ⊆∩   we have  QN ⊆   or   QL ⊆  for any primary 
submodule  Q  of  M.   Then  ).()()( LpradNpradLNprad ∩=∩   
Proof.  By part 4 of  Proposition 2.3,  ).()()( LpradNpradLNprad ∩⊆∩    
Now if  ,)( MLNprad =∩   then clearly  MLpradNprad == )()(   and so  
).()()( LpradNpradLNprad ∩=∩   If ,)( MLNprad ≠∩   then there 
exists a primary submodule  Q such that  .)( QLN ⊆∩   By hypothesis,  
QN ⊆   or   QL ⊆ ,   so that  QNprad ⊆)(   or  .)( QLprad ⊆    Since this 
is true for all primary submodule  Q  containing   ,LN ∩   then 
)())()(( LNpradLpradNprad ∩⊆∩   and therefore  
).()()( LpradNpradLNprad ∩=∩  
 
 We generalize Proposition 2.6 as follows. 
Proposition 2.7  Let  kNNN ,...,, 21   be submodules of an  R-module  M  
such that whenever  ,...21 QNNN k ⊆∩∩∩    we have QNi ⊆   for some  i 
On  Primary  Compactly Packed  Modules  over   
 67
= 1,2,…, k,   for any primary submodule  Q  of  M.   Then  
II
k
i
i
k
i
i NpradNprad
11
)()(
==
=  . 
3.  Minimal Primary submodules 
       We define the minimal primary submodules over noncommutative rings 
as follows. 
Definition 3.1  A primary submodule  Q  of an  R-module  M  is called a 
minimal primary submodule over a submodule  N  if QN ⊆  and there is no 
smaller primary submodule with this property.   Thus a primary submodule  
Q  is a minimal primary submodule of an  R-module  M  if it does not 
strictly contain any other primary submodule. 
Lemma 3.2  Let  IiiQ ∈}{  be a nonempty family of primary submodules of 
an  R-module  M .   Then either I
Ii
iQ
∈
= }0{   or  I
Ii
iQ
∈
 is a primary 
submodule of the  R-module  M . 
Proof.  It is easy to show that I
Ii
iQ
∈
 is a submodule of  M .   Now,  suppose 
that  I
Ii
iQ
∈
≠ }0{ .   Let  N  be a nonzero submodule of  I
Ii
iQ
∈
.   Then  N  is a 
nonzero submodule of  IiQi ∈∀, .    Since  iQ   is primary  Ii ∈∀ ,   then  
rann(N )=rann( iQ ) ,  Ii ∈∀ .    Now,    )(Nrannr ∈    if and only if    
)( iQrannr ∈     if and only if  Ii ∈∀  there exists a positive integer in  such 
that   0=i
n Qr i    if and only if      ,0)( =
∈
I
Ii
i
s Qr     where   ∑
∈
=
Ii
ins    if and 
only if      )(I
Ii
iQrannr
∈
∈ .     Thus  rann(N )=rann(I
Ii
iQ
∈
) ,  and  I
Ii
iQ
∈
 is 
primary. 
Theorem 3.3  If an  R-module  M  satisfies the  ACC  on submodules,  and   
A≠0  is a submodule of  M   that is contained in a primary submodule  Q of  
M ,   then  Q  contains a minimal primary submodule over  A. 
Proof.   Denote by  Ω ,  the set of all primary submodules which contain A,   
and are contained in  Q.    Then  Ω∈Q  ,    and therefore  Ω  is nonempty.   
If  Q   and  Q  belong to Ω ,   then we write  QQ ≤  if  QQ ⊆ .    This gives 
a partial order on  Ω .   We shall prove that Ω  is an inductive system.   Let 
Σ  be a nonempty totally ordered subset of  Ω .   Let  Q  be the intersection 
of all members of  Σ .   By the previous  Lemma,  Q  is a primary 
submodule of  M ,   or  Q =0.   Since  ,0 QQA ⊆⊆≠     then  Q  is primary,  
Arwa  Eid  Ashour 
 68
and  .Ω∈Q    Also since  BQ ⊆   for every  Σ∈B ,   we have  QB ≤   for 
every  Σ∈B .   Thus  Q   is an upper bound for  Σ .   Therefore,  Ω  is an 
inductive system.   By Zorn's  Lemma,   Ω   contains a maximal element  
∗Q .   Since   Ω∈∗Q ,   it is primary submodule with   QQA ⊆⊆ ∗ .   
Suppose now that  1Q  is a primary submodule satisfying  QQA ⊆⊆ 1 .    
Then   Ω∈1Q   and  1QQ ≤
∗ .    Consequently,  since  ∗Q  is maximal in Ω ,    
1QQ =
∗ .    This shows that  ∗Q  is a minimal primary submodule of  A and 
completes the proof. 
4. Primary Compactly Packed Modules 
   Now,   we can generalize the concept of primary compactly packed 
modules that was introduced in [ 4 ] over the rings that are not necessary 
commutative as follows. 
Definition 4.1  A proper  submodule  N  of a unitary  R-module  M  is 
primary compactly packed if for each family   λαα ∈}{P    of primary 
submodules of   M  with   U
λα
α
∈
⊆ PN ,    βPN ⊆   for  some  λβ ∈ .    
Moreover,   the module M  is called primary compactly packed if every 
submodule of  M  is primary compactly packed. 
Theorem 4.2    Let  M  be an R-module.   Then the following statements are 
equivalent. 
a) M  is primary compactly packed. 
b) For each proper submodule N  of  M ,   there exists    Na ∈  such that  
).()( RapradNprad =  
c)  For each proper submodule N  of  M ,   if  λαα ∈}{N  is a family of 
submodules of  M  and  U
λα
α
∈
⊆ NN ,   then  )( βNpradN ⊆  for   
      some  λβ ∈ .     
d) For each proper submodule N  of  M ,   ,   if  λαα ∈}{N  is a family of 
primary radical submodules of  M  and  U
λα
α
∈
⊆ NN ,   then  
βNN ⊆  for  some  λβ ∈ .     
Proof. ( a → b ) :  Let  N  be a proper submodule of  M ,   it is clear that   
)()( NpradRaprad ⊆   for each  Na ∈ .    Suppose that  
)()( RapradNprad ⊄  for each  Na ∈ .    Then for  each Na ∈ ,  there 
On  Primary  Compactly Packed  Modules  over   
 69
exists a primary submodule  aP   for which  aPRa ⊆  and  PaN ⊄ .   
However,   U
Na
RaN
∈
= U
Na
aP
∈
⊆ ;    that is,   M  is not primary 
compactly packed,   which is a contradiction. 
( b→ c ):  Let  N  be a proper submodule of  M,   and let  λαα ∈}{N  be a 
family of submodules of  M  such that  U
λα
α
∈
⊆ NN .    By  (b)   ,   there 
exists  Na ∈  such that  ).()( RapradNprad =    Then  U
λα
α
∈
∈ Na   and 
hence   βNa ∈  for  some  λβ ∈ ,   so that  βNRa ⊆  for  some  
λβ ∈ ,   and hence  ,   )()()( βNpradRapradNpradN ⊆=⊆   
for  some  λβ ∈ . 
( c →d ) :   Let  N  be a proper submodule of  M  and let  λαα ∈}{N  be a 
family of primary radical submodules of  M  such that  U
λα
α
∈
⊆ NN .    
Then by  (c),   there exists λβ ∈  such that   )( βNpradN ⊆ .   Since  
βN  is primary radical submodule of  M ,   then  βNN ⊆ . 
( d →a ) Let  N  be a proper submodule of  M  and suppose that λαα ∈}{N  
is a family of submodules of  M  that satisfies U
λα
α
∈
⊆ NN .    Since    
αN  is primary submodule of  M  for  each  λα ∈ ,     αN =  prad( αN )   
for  each  λα ∈ .    Thus  U
λα
α
∈
⊆ NN = U
λα
α
∈
)(Nprad .    By  (d),   
there exists λβ ∈ such that  )( βNpradN ⊆ = βN .    Thus  M  is 
primary compactly packed. 
Theorem 4.3     If  M  is primary compactly packed which has at least one 
maximal submodule,   then  M  satisfies the ACC on primary radical 
submodules. 
Proof.   Let  ...21 ⊆⊆ NN   be an ascending chain of primary radical 
submodules of  M  and let   U
i
iNL = .   If  L =  M  and  H is a maximal 
Arwa  Eid  Ashour 
 70
submodule of  M,   then  U
i
iNH ⊂ .    Since M  is primary compactly 
packed,   by Theorem 4.2,   jNH ⊆  for some  j.   Therefore,     by 
maximality of  H,  jNH =   for some  j.    Since  U
i
injj NNN ⊆⊆ + ,   
,...2,1=n , and  jN  is maximal,   either  njj NN +=   for every ,...2,1=n ,   
thus  ,MNN
i
ij == U   which is impossible,    or  ,MNN
i
inj ==+ U    
which is also impossible.   Thus  L  is a proper submodule of  M.   Since  M  
is primary compactly packed,   by Theorem 4.2,  jNL ⊆   for some  j  and 
hence  ...,... 2121 ===⊆⊆⊆ ++ jjj NNNNN    therefore the ACC is 
satisfied on primary radical submodules. 
Theorem 4.4  Let  MM →Φ :    be an  R-module isomorphism.   If  M  is 
primary compactly packed,  then  M   is primary compactly packed. 
Proof.   Let  M  be primary compactly packed,  and suppose that  
U
λα
α
∈
⊆ KN   where N  is a proper submodule of M  and  αK is a primary 
submodule of  M  for  each  λα ∈ .   Since Φ  is an  R-module 
isomorphism,  then  =Φ⊆Φ
∈
−− )()( 11 U
γα α
KN    ))(( 1U
γα α∈
−Φ K    
Since αK is a primary submodule of  M  for  each  λα ∈ ,    by [ 1 ] ,    
)(1 αK
−Φ  is a primary submodule of  M   for  each  λα ∈ .    But  M  is 
primary compactly packed.   Thus      
there exists λβ ∈ such that  )()( 11 βKN
−− Φ⊆Φ .    Therefore   
βKN ⊆   for some λβ ∈ ,  and hence  N  is primary compactly packed.   
Thus  M   is primary compactly packed. 
     The following definition was introduced in the rings that are 
commutative,  see[9]. 
We give a generalization to it in noncommutative rings. 
Definition 4.5   A module  M  of an arbitrary ring  R  is called a Bezout 
module if every finitely generated submodule of  M  is cyclic.   
Theorem 4.6   Let  M  be a Bezout module.   If  M  satisfies the ACC on 
primary radical submodules,   then M  is primary compactly packed. 
On  Primary  Compactly Packed  Modules  over   
 71
Proof.  Let  N  be a proper submodule of  M.  By  Proposition 2.3,   prad (N) 
is a primary radical submodule;  hence,  by Theorem 2.4,   there exists 
afinitely generated submodule  L  of  M  such that  prad (N)=prad (L)  and 
hence  L  is a cyclic submodule of  M,  because  M  is Bezout.   It follows 
from Theorem 4.2 that  M  is primary compactly packed.  
                                                  
REFERENCES 
 
[ 1 ] Ashour, A. E.: Primary  Ideals and Primary  Modules  over  
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[ 6 ] Ries, C.M.  and  Viswanthan, T.M. :  A Compactness Property for 
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On Primary Compactly Packed Modules over non commutative rings

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On Primary Compactly Packed Modules over non commutative rings

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