Let R be a commutative ring with non-zero identity. A proper submodule N of an R-module M is weakly primary i

Ashour, Arwa E.

Institutional Repository of the Islamic University of Gaza

Journal of Al Azhar University-Gaza (Natural Sciences),    2011, 13 : 31-40 http://www.alazhar.edu.ps/journal123/natural_Sciences.asp?typeno=1  On Weakly Primary Submodules Arwa Eid Ashour arashour@iugaza.edu.ps     Department of Mathematics,    Islamic University of Gaza, Received 11/10/2010 Accepted 3/01/2011 Abstract: Let  R  be a commutative ring with non-zero identity.  A proper submodule  N of an  R-module  M  is weakly primary if  ,(0 RrNrm ∈∈≠ )Mm∈ implies  Nm∈  or  NMrn⊆  for some positive integer n.  A number of results concerning weakly primary submodules are given.  For example,  we give other characterization of weakly primary submodules.   Also various properties of weakly primary submodules are considered.  For example we show that if  N  is a weakly primary submodule of an  R-module  M  which is not primary,  then IN = 0  for every ideal ):( MNI ⊆.   1. Introduction Weakly primary submodules in a commutative ring with non-zero identity have been introduced by S. E. Atani and F. Farzalipour in [1].  In [2],  the concept of weakly prime submodules was introduced and a number of results concerning weakly prime submodules were given.  The weakly primary submodules and the weakly prime submodules are different concepts.  These two concepts are extensions of the concepts  of weakly primary and weakly prime ideals that have the following definitions: A proper ideal  P  of  R  is said to be weakly prime ideal if  Pab∈≠0  implies  Pa∈  or  ,Pb∈  and it is weakly primary ideal if  Pab∈≠0  implies  Pa∈  or  Pb n ∈  for some positive integer  n.  Several authors have studied in details  weakly primary and weakly prime ideals and prove many important results about these two concepts,  see [1],[3],[4],[5].    In this paper,  we prove several results concerning the weakly primary submodules of a commutative ring.  Some of our results are analogous to the results given in [2].  The corresponding results are Arwa Eid Ashour (32)         Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13  obtained by modification.  We also give various properties of weakly primary submodules. Before we state some results and properties concerning weakly primary submodules,  let us introduce some notations. Throughout this paper all rings will be commutative with non-zero identity.  A proper submodule  N of an  R-module  M  is said to be weakly prime if  ,(0 RrNrm ∈∈≠  )Mm∈  implies  Nm∈  or  NrM ⊆ [2].  If  R  is a ring and  N  is a submodule of an  R-module  M,   the ideal  }:{ NrMRr ⊆∈  will be denoted by  ).:( MN   Then  ):0( M  is the annihilator of  M.   ______________ "2000  Mathematics  Subject  Classification" :  Primary 16D25,  Secondary 16D80,  16N60 It is known that if  N is a primary submodule of an  R-module  M,  then  ):( MN  is a primary ideal of  R  and  ):( MN = NMrRr n ⊆∈ :{  for some positive integer n} is a prime ideal of  R, [6].  In Section 2, we show that this is not in general true in the case of a weakly primary submodule.  We investigate the conditions that make these results satisfied in the case of weakly primary submodules. In [7] some results were proved concerning weakly primary submodules that are not primary.  In Section 3,  we prove that if  N  is a weakly primary submodule of an  R-module  M  that is not primary,  then  IN = 0  for every ideal  ):( MNI ⊆ .   We also give a characterization of weakly primary submodules.  We show that a submodule  N  of an  R-module  M is weakly primary if and only if for any ideal  I  of  R  and any submodule  D of  M with  ,0 NID ⊆≠  either  ):( MNI ⊆  or  .ND ⊆  Finally,  in Section 4,  we show that the direct sum of a finite number of submodules that contain only one weakly primary submodule is weakly primary,  however the direct sum of weakly primary submodules is not in general weakly primary. On Weakly Primary Submodules    Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13           (33)   2. Weakly Primary Submodules Recall that a proper submodule  N  of  a module  M  over a commutative ring  R is  said to be weakly primary if whenever ,(0 RrNrm ∈∈≠  )Mm∈ ,  then  Nm∈  or  NMr n ⊆  for some positive integer n.   Remarks 2.1 1) Clearly,  every primary submodule of a module is weakly primary submodule.  However,  since  0  is always weakly primary ( by the definition),  a weakly primary submodule need not be primary.  Thus the weakly primary submodule concept is a generalization of the concept of primary submodule. 2) We can show directly from the definitions that every weakly prime submodule is weakly primary,  however the converse is not in general true since for example if  =R  Z ,  the set of integers,   M = Z ×  Z  and         L = ( 4 , 0 ) Z + ( 0 , 1) Z ,  then  L is a  weakly primary submodule [8],         however it is not weakly prime submodule since  .)1,2(20 L∈≠  But neither       LM ⊆2  nor L∈)1,2( . 3) It is known that if  N  is a primary submodule of an  R-module  M ,  then   ):( MN  is a primary ideal of  R and ):( MN  is a  prime ideal of  R, [6].  Contrary to what happens for a primary submodules,  if  N  is a weakly primary submodule,  the ideal   ):( MN  is not in general a weakly primary ideal of  R ,  and also ):( MN   is not in general a weakly prime ideal of  R.  For example:  Let  M  denotes the cyclic Z –module  ZZ 6 .  Take }.0{=N  Certainly N is a weakly primary submodule of  M,  but neither  ):( MN = 6Z  is a weakly primary ideal of  R  nor  ):( MN = 6Z  is a  weakly prime ideal of  R.   However we have the following results:   Arwa Eid Ashour (34)         Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13  Proposition 2.2   Let  R  be a commutative ring,  M a faithful cyclic  R-module and N  is a weakly primary submodule of  M.  Then  ):( MN  is a weakly primary ideal of  R. Proof.  Assume  M = Rx and let  ):(0 MNab∈≠  with  ),:( MNa∉   then there exists  Rr∈  such that  ,)( Nrxa ∉  so  .Nax∉   As  ,0 NabM ⊆≠   it follows that  Nabx∈≠0 ( for if  ,0=abx  then  ,0):0():0( ==∈ Mxab  a contradiction).  Since  N  is a weakly primary submodule of  M,  ):( MNbn∈  for some positive integer n.  Thus  ):( MN  is a weakly primary ideal of  R.                    □  Proposition 2.3   Let  R  be a commutative ring,  M a faithful cyclic  R-module and N  is a weakly primary submodule of  M. Then ):( MN  is a weakly prime ideal of  R. Proof.  Assume  M = Rx and let  ∈≠ ab0 ):( MN  with  ∉a ):( MN ,  then ):( MNa∉ .  As in the proof  of the previous proposition,  ):( MNbn∈  for some positive integer n. Thus  ):( MNb∈  and ):( MN  is a weakly prime ideal of  R.        □                                                              Proposition 2.4   Let  R  be a commutative ring,  M a  P-primary  R-module and N  is a weakly primary submodule of  M.  Then  ):( MN  is a weakly primary ideal of  R. Proof.  Let  ):(0 MNab∈≠  with  ),:( MNa∉   then there exists  NMm −∈  such that  .Nam∉   As  ,0 NabM ⊆≠   we have Nabm∈ .  If  ,0=abm  then  ,):0():0( PMmab ==∈  so ):( MNPbn ⊆∈  for some positive integer n.  If  ,0≠abm since  N  is a weakly primary submodule of  M,  ):( MNbn∈  for some positive integer n.  Thus  ):( MN  is a weakly primary ideal of  R.            □  Proposition 2.5   Let  R  be a commutative ring,  M a  P-primary  R-module and N  is a weakly primary submodule of  M.  Then  ):( MN  is a weakly prime ideal of  R. On Weakly Primary Submodules    Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13           (35)  Proof.  Let  ∈≠ ab0 ):( MN  with  ∉a ):( MN ,  then  ).:( MNa∉   Thus there exists  NMm −∈  such that  .Nam∉   As in the proof  of the previous proposition,  ):( MNbn∈  for some positive integer n.  Thus  ):( MNb∈  and  ):( MN  is a weakly prime ideal of  R. □  3.  Characterization of  Weakly Primary Submodules The following two results concerning weakly primary submodules that are not  primary were proved in [7].  Proposition 3.1 Let  R  be a commutative ring,  M an  R-module, and  N  a weakly primary submodule of  M. If  N  is not primary submodule of  M,  then  ):( MN N = 0.   Proposition 3.2 Let  R  be a commutative ring,  M a multiplication  R-module, and  N  a weakly primary submodule of  M.  If  N  is not primary submodule of  M,  then  .02 =N        We give now a generalization of Proposition 3.1 which is the corollary of the next theorem.  Theorem 3.3   Let  N  be a weakly primary submodule of  an  R-module M.  If  ):( MN ,0≠N   then  N  is a primary submodule of  M. Proof.  Let  Nrm∈  for some Rr∈  and  Mm∈ .  If  ,0≠rm   then  N  is weakly primary gives Nm∈  or  ):( MNr n ∈  for some positive integer n.  So assume that  .0=rm   First suppose that  ,0≠rN  then there exists Ns∈  such that  .0≠rs   Thus ,)(0 Nsmrrs ∈+=≠   so N is weakly primary gives  ):( MNr n ∈  for some positive integer n,  or  .)( Nsm ∈+   Hence  ):( MNrn ∈  for some positive integer n,  or  .Nm∈   So we can assume that  .0=rN   Next suppose that  ,0):( ≠MNm  say  0≠mk  where  .):( MNk∈   Then  .)(0 Nmkrmk ∈+=≠   Thus  ):()( MNkr ∈+  or  .Nm∈   So  Arwa Eid Ashour (36)         Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13  ):( MNr∈   or  .Nm∈   Therefore  ):( MNrn ∈  for some positive integer n,  or  .Nm∈   So we can assume  .0):( =MNm   Since  ):( MN ,0≠N   there exists  ):( MNt∈  and  Nd ∈  with  0≠td .  Then  ,))((0 Ndmtrtd ∈++=≠   so  ):()( MNtr ∈+  or  .)( Ndm ∈+  Thus   ):( MNr∈   or  .Nm∈   Therefore ):( MNrn ∈  for some positive integer n,  or  .Nm∈   So  N  is a primary submodule of  M.                          □       Now,  the following result follows immediately from Theorem 3.3.   Corollary 3.4   Let  R  be a commutative ring,  M an  R-module and  N  a weakly primary submodule of  M.  If  N  is not primary submodule of  M,  then for any ideal I  of  R such that  ),:( MNI ⊆  then  IN = 0.  In particular,   ):( MN N = 0.    Remark 3.5   We can give another proof the result stated in  Proposition 3.2 by using Corollary 3.4  as follows:  Since M  is a multiplication module,  .):( MMNN =   Therefore  .):():():( 22 NMNNMNMMNN ⊆==   Since  N is weakly primary,  but not primary,  ):( MN .0=N   Thus  .02 =N         We next give a characterization of weakly primary submodules.  Theorem 3.6   Let  R  be a commutative ring,  M an  R-module, and  N  a proper  submodule of  M.  Then  N is a weakly primary submodule of  M  if and only if for every ideal  I  of  R  and every submodule  D of  M with  ,0 NID ⊆≠  either  ):( MNI ⊆  or  .ND ⊆  Proof. )⇐  Suppose that  Nsm∈≠0  where  Rs∈  and  .Mm∈   Take  RsI = and  .RmD =   Then  ,0 NID ⊆≠   so either  ):( MNI ⊆  or ND ⊆ ;   hence  either  ):( MNs∈  or  Nm∈ .  On Weakly Primary Submodules    Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13           (37)  Thus either  NMsn ⊆  for some positive integer n or Nm∈ .  Therefore  N  is a weakly primary submodule of  M. )⇒   Suppose that  N  is a weakly primary submodule of  M.  If  N is primary,  then the result is clear.  So we can assume that  N is a weakly primary submodule of  M  that is not primary.  Let  NID ⊆≠0  with  .NDx −∈   We show that  ):( MNI ⊆ . Let  .Ir∈   If  ,0 rx≠  then  N  is weakly primary gives  ):( MNr∈ .  So assume that  .0 rx=   First suppose that  0≠rD ,  say  0≠rd  where  .Dd∈   If  Nd∉ ,  then  ):( MNr∈ .  If  Nd ∈ ,  then  ,)( Nrdxdr ∈=+   so ):( MNr∈  or  .)( Nxd ∈+   Thus  ):( MNr∈ .   So we can assume that  0=rD .  Suppose that  0≠Ix ,  say 0≠ax  where  .Ia∈   Then  N  is weakly primary gives  ):( MNa∈ .  As  ,)( Naxxar ∈=+   we get  ):( MNr∈ .   Therefore,  we can assume that   0=Ix .  Since  0,ID ≠   there exist  b I∈  and  1d D∈ such that   1 0.bd ≠   As  ):( MN 0N = ( by Corollary 3.4 )  and  1 10 ( )b d x bd N≠ + = ∈  we can divide the proof into the following cases: Case1.   ( : )b N M∈  and  1( ) .d x N+ ∉         Since   1 10 ( )( ) ,r b d x bd N≠ + + = ∈   we obtain  ):( MNbr ∈+ ,  so        ):( MNr∈ . Case2.  ):( MNb∉ and .)( 1 Nxd ∈+            As .0 1 Nbd ∈≠   We have ,1 Nd ∈   so Nx∈  which is a contradiction.  Therefore in all the cases, we have ):( MNr∈ .  Thus ):( MNI ⊆ and the proof is complete.  □  Proposition 3.7  Let  M  be a multiplication R-module and  N1 ,  N2 ,  …,  Nk  be submodules of  M.   Let  N  be a primary submodule of  M  with  ):( MN M = 0.  If  1,kiiN N=⊆∏   then   1 10.kki ii iN N= == =∏ I     Arwa Eid Ashour (38)         Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13  Proof.   We have  Ni = Ii M  for some ideals  Ii ( i = 1, 2, …, k ) of  R.   Then   N1  N2 … Nk = I1 I2 … Ik  M ⊆  N,  and so  I1I2…Ik ):():( MNMN ⊆⊆ .   Since  ):( MN  is a prime ideal of  R,  because  N  is a primary submodule of  M,  see [6],  then ):( MNI j ⊆  for some  j = 1, 2, …, k.   Therefore,   Nj = Ij M  ):( MN⊆ M = 0.  Thus  Nj= 0 for some  j =1, 2, …, k.  Since  ∏= ==⊆⊆kikijii NNN1 1,0I   then  1 10.kki ii iN N= == =∏ I         □   Since every primary submodule is weakly primary,  but the converse is not true,  see Remarks 2.1,   then we can generalize the previous result as follows.  Proposition 3.8  Let  M  be a multiplication  R-module that is either   i) a P-primary R-module,  or  ii)  a faithful cyclic  R-module. Let   N1 ,  N2 ,  …,  Nk  be submodules of  M  and  N  be a weakly primary submodule of  M  with  ):( MN M = 0.  If  1,kiiN N=⊆∏   then ∏==kiiN1.0      Proof.    Note first that by Propositions 2.3 and 2.5,  ):( MN  is a weakly prime ideal of  R.  Now,  as in the proof of Proposition 3.7,  ∏=⊆kii NN1,  and so  ):(...21 MNIII k ⊆ .   If  ,0...21 ≠kIII   then the proof is similar to the proof of Proposition 3.7.   So let  ,0...21 =kIII   then  ∏====kiki MMIIIN121 .00...               □ 4.   Direct Sum of  Weakly Primary Submodules       We show in this section that the direct sum of a finite number of submodules that contain only one weakly primary submodule is weakly primary,  however the direct sum of weakly primary On Weakly Primary Submodules    Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13           (39)  submodules is not in general weakly primary.  We start by the following result.  Proposition 4.1   Let  1M  and  2M  be unitary R-modules over a  ring R.  Let  ,21 MMM ⊕=   and  .21 MMN ⊕⊆   Then the following are satisfied: (1)   2MQN ⊕=   is a weakly primary submodule of  M  if and only if  Q is a weakly          primary submodule of  1M . (2)  QMN ⊕= 1  is a weakly primary submodule of  M  if and only if  Q is a weakly          primary submodule of  2M . Proof.   We will prove (1) and  the proof of (2) will be similar. )⇒   Let  2MQN ⊕=  be a weakly primary submodule of  M.  Let   Qrq∈≠0 ,  Qq∉ .   Then  2)0,( MQq ⊕∉ ,  while  2)0,(0 MQqr ⊕∈≠ .   Since 2MQN ⊕=  is a weakly primary submodule of  M ,   there exists a positive integer  n such that  .)( 221 MQMMrn ⊕⊆⊕  Hence QMr n ⊆1   for some positive integer  n.   Thus Q is a weakly primary submodule of  1M .  ⇐ )   Suppose that  Q is a weakly primary submodule of  1M .  Let  2),(0 MQkqr ⊕∈≠  with  Rr∈  and  ).(),( 2MQMkq ⊕−∈   Then  .0 Qrq∈≠    Since Q is a weakly primary submodule of  1M ,  and Qq∉ ,  then  QMr n ⊆1   for some positive integer  n.   Thus 2MQMrn ⊕⊆  for some positive integer  n.  Therefore  2MQ⊕   is a weakly primary submodule of  M.                                         □        Now,  we can easily prove the following result.  Proposition 4.2    Let  1M ,  2M , … ,   nM   be unitary R-modules over a  ring R.  Let  nMMMM ⊕⊕⊕= ...21 .    iQ   is a weakly primary submodule of  iM   for some },,...2,1{ ni∈   if and only if  )( ,1 jnijji MQ ≠=⊕  is a weakly primary submodule of  .M   Arwa Eid Ashour (40)         Journal of Al Azhar University-Gaza (Natural Sciences), 2011, 13      Finally,  note that the direct sum of weakly primary submodules is not necessarily  weakly primary,  as in the following example: Let  =R  Z ,  the set of integers,  and  M = Z ⊕Z ,  then   )3()4( ⊕=N  is not a weakly primary submodule of  M,   since  ,)1,4(30 N∈≠   but neither  NMn ⊆3  for any positive integer  n,   nor .)1,4( N∈    However,  (4) and (3) are weakly primary submodules of  Z.  References: [1] Atani,  S. E. and Farzalipour,  F. :  On weakly primary ideals,  Georgian Mathematical Journal,  12(2005), No.3, 423-429. [2] Atani,  S. E. and Farzalipour,  F. :  On weakly prime submodules,  Tamkang Journal of Mathematics,  38(2005), No.3, 247-252. [3] Anderson,  D. D. and Smith, E. :  weakly prime ideals.  Houston Journal of Mathematics,  29(2003),  831-840. [4] Dauns,  J. : Prime modules,  Journal für die reine Angewandte Mathematik,  2(1978),  156-181. [5] Lu, C. P. : Spectra of modules, Comm. In Algebra,  23(1995),  3741-3752. [6] El-Atrash,  M. S. and Ashour,  A. E. : On primary compactly packed modules, The Islamic University Journal,  13(2005),  No.2,  117-128. [7] Atani,  S. E. :  The product of multiplication submodules,  Honam Mathematical Journal,  27(2005), 1-8. [8] Ashour,  A. E. : On primary compactly packed modules,  Ph.D. Thesis,  The joint program of Ain Shams University( Cairo, Egypt) and El-Aqsa University( Gaza, Palestine ),  2005. 

On weakly primary submodules

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