2009) 3( 22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة               المجلد

 حول المقاسات الجزئیة األولیة الضعیفة

 

 أنعام محمد علي هادي

ابن الهیثم،جامعة بغداد-قسم الریاضیات ،كلیةالتربیة   

  

  الخالصة

یكـون  Mفـي  Nُنعـرف ان مقاسـاً  جزئیـاً  فعلیـاً  . Rمقاساً  أیسـر علـى  Mحلقة ابدالیة ذا محاید ولیكن  Rلتكن         

r  (N:Mأو    x  Nیـؤدي الـى   r x  N ≠ 0 و   ، x  Mو   ، r  Rأولیـاً  ضـعیفاً  اذا كـان لكـل  فـي .  (

ا كـان ، یسـمى أولیـاً  ضـعیفاً  اذRفـي  Pان مثالیـاً  فعلیـاً  اذ الحقیقة ان هذا المفهوم هو تعمیم لمفهوم مثالي أولي ضعیف، 

  . a  P أو  b  Pیؤدي الى ان     a b  P ≠ 0و  ، a, b  Rلكل 

  .خواص مختلفة عن المقاسات الجزئیة األولیة الضعیفة قد أعطیت

  

  

  

  

  

 
 
 
 
 
 
 
 
 
 
 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I        VO L.22 (3)  2009 

 
On Weakly Prime Submodules 

 
 
 
 

I. M.A.Hadi  
Departme nt of Mathematics,Ibn-Al-Haitham College of Education  
Unive rsity of Baghdad  
 

Abstract 
        Let R be a commutative ring with unity  and let M  be a left R-module. We define a 
p rop er submodule N of M  to be a weakly  p rime if whenever  r  R,  x  M , 0  r x  N 
imp lies  x  N  or  r  (N:M ). In fact this concept is a generalization of the concept weakly   
p rime ideal, where a p rop er ideal P of R is called a weakly p rime, if for all a, b  R, 0  a b  
P implies a  P or b  P. Various p rop erties of weakly  p rime submodules are considered. 

    
 1.Introduction 
        Throughout this p ap er, R be a commutative r ing with identity  and M  be a unity  R-
module. A p rop er submodule N of M  is said to be Prime if whenever        r  R , x  M  , rx  
N implies either x  N or r  (N:M ), where                   (N:M ) = {r  R: r M   N}, see (1). 
        Semip rime submodules was given by  Dauns in (2), as a gener alization of p rime 
submodules, where a p rop er submodule N of M  is semip rime if r

k
x  N, for r  R, x  M , k 

 Z+ (set of p ositive integers) imp lies rx  M . Also Eman A.A. in (3) studied these notions. 
        In 1999, quasi-prime submodules was introduced and st udied by  M untaha (see (4)), as  
another generalization of p rime submodules, where a p rop er submodule N of M  is quasi-
p rime if r1r2m  N, for r1, r2  R, m  M  imp lies r1m  N or r2m  N; equivalently , N is 
quasi-prime if the ideal (N:M ) is p rime for all m  M . 
        In 2004, M .Behoodi and H.Koohi in (5) gave the notion of weakly p rime submodules,  
where a p rop er submodule N of M  weakly  p rime if (N:K) is a  p rime id eal, for all submodu les  
K of M . Also this notion was st udied by  A.Azizi in (6), 2006. 
        By  Th.2.14 in (4), we obtain that the two concep ts weakly p rime submodules and quasi-
p rime submodules are equivalent. 
        In this p ap er, we give another generalization of prime submodules namely  weakly  p rime 
submodules, however this concept is different from the concept of quasi-p rime submodule 
(see Remarks 2.1.(5)).  
        In fact, D.D.Anderson and E.Smith in (7) gave the fo llowin g: A p rop er ideal I of R is  
said to be a weakly  p rime if 0  ab  I, for a, b  R, then a  I or      b  I. We d efine a 
p rop er submodule N of M  is weakly  p rime if whenever          r  R , x  M  , 0  rx  N 
imp lies x  N or r  (N:M ). Moreover S.E.Atani and F.Farzalip our in (8) introduced the 
notion of weakly p rimary  submodules, where a prop er submodule N of M  is a weakly p rimary 
if whenever r  R , x  M , 0  rx  N imp lies x  N or r

n
  (N:M ) for some n  Z+. Also 

they gave that : a prop er ideal of R is a weakly p rimary if it is a weakly  p rime submodule of 
the R-module R, (see (8)). 
        In this p aper we study  weakly  p rime submodules and give many  basic p rop erties related 
to this concept. 
 

 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I        VO L.22 (3)  2009 

2.Basic Properties 

     As we mentioned in the introduction, we introduce the following : 
De fini tion 2.0 :  A p rop er submodule N of an R-module M is weakly  p rime if whenever rR, 
x M , 0 rxN implies  xN or  r(N:M ). 
 In this section, we will give basic p rop erties of weakly  p rime submodules. Some of these are 
extension of the results about weakly p rime ideals, which are given in (7). 
        Let us start with t he following: 
 
 Remarks 2.1:  
(1) It is clear that every  p rime submodule is weakly  p rime. However, since (0) the zero 

submodule of any module) is alway s weakly  p rime (by  definition), a weakly  p rime 
submodule may  not be p rime; for examp le: the zero submodule of the Z-module Z4 is 
weakly  p rime, but it is not p rime. 
M oreover it is easy  to check that in the class of torsion free modules, the concepts of 
p rime submodule and weakly  p rime submodule are equivalent. 

(2) Every  weakly  p rime ideal P of a ring R is a weakly  p rime submodule of the R-module 
R. 

(3) Every  weakly  p rime submodule is weakly  p rimary, but the converse is false as the 
following examp le shows. 

The submodule N = )4(  of the Z-module Z8 is weakly  p rimary but it is not weakly  
p rime. 

(4) It is easy  to check that :if P is a weakly  p rimary submodule of an R-module M  and 
(P:M ) is a semiprime ideal, then P is weakly p rime. 

(5)  (a) Weakly  p rime submodule need not be quasi-p rime as t he following examp le shows: 
The zero submodule of the Z-module Z12 is weakly  p rime, but it is not quasi-prime 

since (0 : 3)  = 4Z which is not  a prime ideal of Z. 
 (b) Quasi-p rime submodule need not be weakly  p rime submodule, as the following 
examp le shows 
If M  is the Z-module ZZ, and N = 2Z(0), then N is a quasi-p rime submodule of M  
(see (4), Rem.2.1.2(1)). But N is not weakly  p rime submodule, since (0,0)  2 (3,0)  
N, (3,0)  N and 2  (N:M ) = (0). 

(6) If P is a p rop er submodule of an R-module M . Then P is a weakly  p rime  R-submodule 
of M  iff P is a weakly  p rime R / I-submodule of M , where I is an ideal of R with  I  
ann M . 

        The following result gives characterizations of weakly  p rime submodules. 
 The orem 2.2 :    Let M  be an R-module. The following asserations are equivalent: 
1. P is a weakly  p rime submodule of M . 
2. (P:x) = (P:M )  (0:x) for any x  M , x  P. 
3. (P:x) = (P:M ) or (P:x) = (0:x) for any x  M , x  P. 
4. If (0)  (a)N  P, then either N  P or (a)  (P:M ), where a  R, N is a submodule of 

M . 
Proof.  (1)  (2)   Let r  (P:x) and x  P. T hen  r x  P. Supp ose  r x ≠ 0. Hence  r  (P:M ) 
because P is weakly  p rime and x  P. If r x = 0, then               r  (0:x). Thus (P:x)  (P:M )  
(0:x). Now if r  (P:M )  (0:x), then either             r  (P:M ) or r  (0:x). Hence, when r  
(0:x), r x = 0  P and so r  (P:x). If                 r  (P:M ) then  r M   P, and this imp lies  r x 
 P. Hence  r  (P:x). Thus       (P:M )  (0:x)  (P:x) and therefore (P:M )  (0:x) = (P:x). 
 (2)  (3)   It is well-known t hat t he union of two ideals A, B of R is an ideal if A  B or B  
A. By  condition, the ideals (P:M ) is the union of the ideals         (P:M ), (0:x), so either (P:M ) 
 (0:x) or (0:x)  (P:M ). Thus either (P:x) = (0:x) or (P:x) = (P:M ). 
 

 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I        VO L.22 (3)  2009 
 

 (3)  (4)   If 0  (a) N  P. Sup p ose that N   P and (a)   (P:M ). N   P imp lies that 
there exists x  N and x  P, hence ax  aN  P; that is a  (P:x). By  condition (3), either 
(P:x) = (P:M ) or (P:x) = (0:x). If (P:x) = (P:M ), we get        a  (P:M ) which is a contradiction. 
Thus (P:x) = (0:x) and so ax = 0. 
On t he other hand, 0  (a) N  P implies that there exists y  N such that             0  ay  P 
and so a  (P:y). M oreover we can see that y  P, for if we assume that y  P, then by  
condition 3, either (P:y) = (P:M ) or (P:y) = (0:y).                    If (P:y) = (P:M ), then a  (P:M ) 
which is a contradiction. If (P:y) = (0,y), we get ay = 0 which is a contradiction. M oreover 0  
ay = ay + ax = a(y + x)  P; that is a  (P:y + x). But y + x  P because x  P, y  P, hence 
by  condition 3 , either (P: y + x) = (P:M ) or (P: y + x) = (0: y + x). If (P: y + x) = (P:M ) then          
a  (P:M ) which is a contradiction. If (P: y + x) = (0: y + x), then a(y + x) = 0 and hence ay + 
ax = ay + 0 = 0 which is a contradiction. Therefore our assump tion is false and so either N  
P or (a)  (P:M ). 
 (4)  (1)   Let r  R, x  M , such that 0  r x  P. T hen 0  (r) (x)  P. By  condition (4), (x) 
 P or (r)  (P:M ) and hence either x  P or r  (P:M ); that is, P is weakly p rime. 
 Remark 2.3 
        It is known t hat if P is a p rime submodule of an R-module M , then (P:M ) is a prime ideal 
of R. However the "weak" analogs of t his st atement is not true in general, for examp le: 

The zero submodule of the Z-module Z4, is weakly  p rime, but (0

:  Z4) = 4 Z is not a weakly  

p rime ideal of Z. 
        We give the following: 
Proposi tion 2.4:    If P is a weakly p rime submodule of a faithful R-module M , then (P

R
: M ) 

is a weakly p rime ideal of R. 
Proof.  Let a, b  R. If 0  a b  (P:M ); then a b M   P. Since M  is faithful,  a b M   (0), 
hence 0  (a) (b M )  P and so by  Th.2.2 either (a)  (P:M ) or          b M   P; that is, either a 
 (P:M ) or b  (P:M ). Thus (P:M ) is a weakly p rime ideal of R. 
     The converse of prop .2.4 is not true as t he following examp le shows: 

        Let M  be the Z-module Z  Z, let P = (0)  2Z. Then (P

:  M ) = (0) which is a weakly  

p rime ideal in Z, however P is not a weakly  p rime submodule of M  because (0,0)  2(0,1)  

P, but  (0,1)  P and 2  (P

:  M ) = (0). 

        Also, we have the following:- 
  Proposi tion 2.5 :     Let P be a weakly p rime submodule of an R-module M . Then (P

R
:  M ) is 

a weakly p rime ideal of R , where  annRR / . 

Proof.  By  remark 2.1 (6), P is a weakly  p rime R -submodule of M . But M  is a faithful R -

module, so by  p rop .2.6, (P
R
:  M ) is a weakly p rime ideal of R . 

       Recall that an R-module M  is called multip lication module if for each submodule N of M , 
N = I M  for some ideal I of R, equivalently N = (N:M )M  (see (9)). 
        In the class of finitely  generated faithful multiplication modules, we have the following:   
The orem 2.6:     Let M  be a faithful finitely  generated multiplication R-module, let N be a 
p rop er submodule of M . Then the following st atements are equivalent 

1. N is a weakly  p rime submodule of M . 
2. (N

R
: M ) is a weakly p rime ideal of R. 

3. N = I M  for some weakly p rime ideal I of R. 
Proof.  (1)  (2)   It holds by  p rop . 2.4 
 

 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I        VO L.22 (3)  2009 
 
(2)  (1)   Let r  R, x  M , such that 0  r x  N. (x) is a submodule of M , hence (x) = J M   
for some ideal J of R. Thus 0  r J M   N = (N:M ) M . But M  is a faithful finitely generated 

multiplication R-module, so by  (10, Th.3.1)                r J  (N
R
: M ). M oreover r J  (0) and 

since (N
R
: M ) is weakly  p rime ideal, either  r  (N

R
: M ) or J  (N

R
: M ) (see Th.3 in (7)). 

Hence either r  (N
R
: M ) or (x) = J M   (N

R
: M ) M  = N, that is r  (N

R
: M )  or  x  N. Thus N 

is weakly  p rime. 

 (2)  (3)   Since (N
R
: M ) is weakly  p rime and N = (N

R
: M ) M , so condition (3) hold. 

 (3)  (2)   By  (3), N = I M  for some weakly  p rime ideal I of R. But M  is a multiplication 

module, so N = (N
R
: M ) M . Hence I M  = (N:M ) M  and so by       (10, Th.3.1)  I = (N

R
: M ). 

    Proposi tion 2.7 :     Let P be a weakly  p rime submodule of an R-module M . If P is not 
p rime, then (P:M ) P = (0). 
Proof.  Sup p ose (P:M ) P  0. We will show that P is p rime. Let r x  P. If         r x  0, then 
either x  P  or  r  (P:M ), since P is weakly p rime. Now assume   r x = 0. First sup p ose  rP  
(0), so t here exists  y  P such that 0  r y  P. Hence  0  r y = r (x + y)  P  which imp lies 
that either  x + y  P or  r  (P:M ). Hence either  x  P  or  r  (P:M ). Now we can assume 
that  r P = 0 and     (P:M ) x = 0. Since (P:M ) P  (0), there exists  s  (P:M ), y  P such that            
0  s y  P. T hus 
(r + s) (x + y) = r x + s x + r y + s y 
                      = 0 + 0 + 0 + s y 
                      = s y 
That is 0  (r + s) (x + y)  P. Then P is weakly  p rime gives  x + y  P  or           r + s  
(P:M ). Hence  x  P  or  r  (P:M ). 
         Now we have the following: 
  Proposi tion 2.8:  Let M  and M be R-modules and let  f : M   M be an R-epimorp hism. If 
N is a weakly  p rime submodule of M  such that  ker f  N, then f (N) is a weakly  p rime 
submodule of M. 
Proof.  Let r  R, y  M, such that 0  r y  f (N). T hen there exists  x  N  such that 0  r y 
= f (x), and since f  is an ep imorp hism, y = f (x1) for some      x1 M . Thus  f (r x1 – x) = 0 and 
so r x1 – x  ker f  N. It follows that   0  r x1  N and since N is weakly  p rime either x1  
N  or  r (N

R
: M ). Thus  y = f (x1)  f (N)  or  r  ( f (N): M); that is f (N) is weakly p rime. 

        As a p articular case of p rop .(2.8), we have the following: if N, W are submodules of an 
R-module M  such that N  W and N is weakly  p rime, then   N / W is a weakly  p rime R-
submodule of M  / N. 
        The following result discussos t he localization of weakly  p rime submodules. 
 Proposi tion 2.9 :     Let P be a weakly  p rime R-submodule and S be a multiplicative subset 

of R with (P
R
:  M )  S = . Then PS is weakly  p rime            RS-submodule of M S. 

Proof.  Let 
b

a
 RS and 

c

x
 M S such that 0s  

b

a

c

x
  PS. Hence 0s  

cb

xa
  PS and so there 

exists  y  P and  d  S such that 
d

y

cb

xa
  , and this imp lies that t here exists  t  S  such that  

t a d x = t b c y. On t he other hand 
1

0


cb

xa
= (0s) which imp lies that  f a x  0  for all  f  S. 

Hence 0  t a d x  P. But  P is a weakly  p rime R-submodule of M , so either  t d x  P  or  a   
 
 
 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I        VO L.22 (3)  2009 

 (P:M ) and hence either 
cdt

xdt
  PS  or  

b

a
  (P:M )S. Because (P

R
:  M )S  (PS

SR
: M S), we have 

either  
c

x
 PS  or  

b

a
  (PS

SR
: M S). 

        As a generalization of Cohen theorem, the following was given in ((3),Prop .4.15,ch.1). 
        Let M  be a finitely  generated R-module, then M  is Noetherian iff every  p rime submodule 
is finitely  generated. 
        Since every  p rime submodule is weakly p rime (by Rem.2.1.(11)), we have the following: 
 Proposi tion2.10 :    Let M  be a finitely generated R-module. Then M  is Noetherian if every  
weakly  p rime submodule is finitely generated. 
        Not eice that t he condition M  is finitely  generated that cannt be dropp ed from Prop .2.10, 
as the following examp le shows: 

        The Z-module Z p
 is not finitely  generated, also it is not Noetherian. However if G is 

a nonzero submodule, then 
1

G   
i

z
p

 for some i  Z+, and 0  P(
1


i +1

z
p

)  G. 

But P  (G: Z p
) = 0 and 

1


i +1
z

p
  G; that is G is not weakly  p rime. Thus (0) is the 

only weakly  p rime submodule of Z p
 and it is obviously finitely  generated. 

 The orem 2.11:    Let M 1, M 2 be R-modules and let N be a prop er submodule of M 1. Then W 
= N  M 2 is a weakly p rime submodule of  M  = M 1 M 2  N is a weakly  p rime submodule 
of M 1 and for  r  R, x  M 1 with r x = 0, but x  N, r  (N:M 1) imp lies  r  ann M 2. 
Proof. () Let r  R, x  M 1 such that 0  r x  N. Then (0,0) ≠ r (x,0)  W, but W is 

weakly  p rime, so either (x,0)  W  or  r  (W
R
: M ). Thus either x  N or  r  (N

R
: M 1), so t hat 

N is weakly  p rime. Now, if r  R, x  M 1 such that        r x = 0, x  N and r  (N:M 1). 
Assume that r  ann M 2, so t here exists m  M 2 such that r m2  0. Thus r (x,m2) = (r x,r m2) 
= (0, r m2)  (0,0) and hence      (0,0)  r (x,m2)  N  M 2 = W. Since W is weakly p rime, so 

either (x,m2)  N  M 2  or  r  (N  M 2
R
: M 1  M 2). Thus either x  N  or   r  (N

R
: M 1) 

which is a contradiction with hy p othesis. 
 ()  Let r  R, (x,y)  M . Assume (0,0)  r (x,y)  N  M 2, so if  r x  0, then either  x  N  
or  r  (N:M 1), since N is weakly  p rime. Thus either                (x,y)  N  M 2  or  r  (N  
M 2:M 1  M 2). If  r x = 0. Sup p ose x  N and        r  (N1:M 1), then by hy p othesis  r  ann 

M 2  and so  r (x,y) = (0,0)  which is a contradiction. Thus either  x  N  or  r  (N1
R
: M 1)  and 

hence either   (x,y)  N  M 2  or  r  (N1  M 2
R
: M 1  M 2). 

       It is known t hat if Q is a p rimary submodule then ):(  Q  is a p rime ideal, see (11, 

p rop . 2.11, p .41). Sometimes Q is called P-p rimary, see(11, p.42). 
        Now we have the following result: 
 Corollary 2.12:    Let Q be P-p rimary submodules of an R-module M 1 with  Q = (0). If 
N is a weakly  p rime submodule of M 1 and M 2 is an R-module such that P  annR M 2, then N 
 M 2 is a weakly p rime submodule in   M 1  M 2.  
Proof.  Let r  R, x  M 1 with  r x = 0. If  x  N1 (so  x  0) and  r  (N:M 1). We will p rove 
that  r  ann M 2 and hence the result is obtained by p revious t heorem. Sup p ose that  r  ann 
M 2. Hence  r  P. 
        On t he other hand,  r x = 0 =  Q, but  Q  is a P-p rimary submodule by  (11,p rop .1.1, 
p .15), so either  x   Q = 0  or  r  P, which is a contradiction. Thus  r  ann M 2. 
 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I        VO L.22 (3)  2009 
 
  Remark 2.13:    Let M 1, M 2 be R-modules. If (0) is a p rime submodule of M 1, then (0)  M2 
is a weakly p rime submodule of M  = M 1  M 2. 
Proof.  Let r  R, (x,y)  M . If (0,0)  r (x,y)  (0)  M 2, then  r x = 0  and    r y  M 2. Since 
(0) is p rime in M 1, either  x = 0  or  r  (0:M 1). Hence either         (x,y) = (0,y)  (0)  M 2  or  
r  ((0) + M 2:M 1  M 2); that is (0)  M 2 is weakly  p rime in M . 
        Thus we can give the following examp le: 
N = (0)  Z4 is a weakly p rime submodule of the Z-module Z  Z4. 
        Next we have the following: 
Proposi tion 2.14:    Let M 1, M 2 be R-modules. If N = U  W be a weakly  p rime submodule 
in M  = M 1  M 2, then U, W are weakly p rime submodules in M 1, M 2 resp ectively. 
Proof.  The proof is a straight forword, so it is omitt ed. 
       Remark 2.15:    The converse of p rop osition 2.14 is not true in general as the following 
examp le shows. 
  Example:    (0) is a weakly p rime submodule of the Z-module Z, (2Z) is a prime submodule 
of the Z-module Z so it is weakly  p rime. But N = (0)  2Z is not weakly  p rime in the Z-
module Z  Z. 
        For t he next results we will assume that R = R1  R2 where each Ri is a commutative ring 
with identity , M i be an Ri-module, where i = 1,2. and            M  = M 1  M 2 be the R-module 
with action (r1,r2) (m1,m2) = (r1 m1, r2 m2) where ri  Ri, mi  M i, i = 1,2. 
 Proposi tion 2.16 :    If P is a prop er R1-submodule of M 1, then the following st atements are 
equivalent 
1. P is a p rime R1-submodule of M 1. 
2. P  M 2 is a prime R-submodule of M  = M 1  M 2. 
3. P  M 2 is a weakly p rime R-submodule of M  = M 1  M 2. 
Proof.  (1)  (2)  Let (r1,r2)  R, (x,y)  M  such that (r1,r2) (x,y)  P  M 2. Then  r1 x  P 
and since P is p rime, either x  P  or  r1  (P

1

:
R

M 1). If      x  P, t hen (x,y)  P  M 2. If r1  

(P
1

:
R

M 1), then (r1,r2)  (P  M 2
R
: M ). Thus     P  M 2 is a prime R-submodule of M . 

 (2)  (3)  It holds by  remark 2.1 (1). 
 (3)  (1)  Let r  R1, x  M 1 such that r x  P. Then for each y  M 2, y  0, (0,0)  (r,1) 
(x,y)  P  M 2. But P  M 2 is a weakly p rime R-submodule of M , so either (x,y)  P  M 2  or  

(r,1)  (P  M 2
R
: M ). Thus either x  P  or                r  (P

1

:
R

M 1); that is P is a p rime R1-

submodule of M 1. 
        Similarly  we have 
 Proposi tion 2.17:    If P is a p rop er R2-submodule of M 2, then the following st atements are 
equivalent 

1. P is a p rime R2-submodule of M 2. 
2. M 1 P is a p rime R-submodule of M  = M 1  M 2. 
3. M 1 P is a weakly  p rime R-submodule of M  = M 1  M 2. 
Proposi tion 2.18:     Let M 1, M 2 be R1, R2-modules resp ectively. If     P =P1  P2 is a weakly  
p rime R-submodule of M  = M 1  M 2, then either P = 0 or P is a p rime submodule of M . 
Proof.  Assume P  0, so either P1  0  or  P2  0. Sup p ose that P2  0, hence there exists y  
P2, y  0. Let r  (P1

1

:
R

M 1) and let x  M 1, then                       (0,0)  (r,1) (x,y) = (r x,y)  P1 

 P2 = P. Since P is weakly  p rime in M , either (x,y)  P  or  (r,1)  (P1  P2: M 1  M 2). Hence 
if (x,y)  P, t hen x  P1 and so M 1 = P1 which implies P = M 1  P2. If (r,1)  (P1  P2: M 1  
M 2), then M 2 = P2 which implies P = P1  M 2. Hence by  p rop ositions 2.16, 2.17, P is a p rime 
R-submodule of M . 
 

 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I        VO L.22 (3)  2009 

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