In this paper we generalize the concept of primary finitely compactly packed modules over noncommutative rings. This concept generalizes the concepts of primary compactly packed modules over noncommutative rings, and primary finitely compactly packed modules over commutative rings. We first find the relation between primary finitely compactly packed modules and primary compactly packed modules over noncommutative rings. We also prove several results on the primary finitely compactly packed modules over noncommutative rings. In addition, we introduce the definition of Dauns-primary submodules over noncommutative rings, and investigate the relation between this concept and the concepts of Dauns-prime submodules, and primary submodules over noncommutative rings

Ashour, Arwa E.

English

Institutional Repository of the Islamic University of Gaza

IUG Journal of Natural and Engineering Studies Vol.20, No.1, pp 55-61 2012, ISSN 1726-6807, http://www.iugaza.edu.ps/ar/periodical/    Primary Rimary Finitely Compactly Packed Modules And Dauns-Primary Submodules 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: In this paper we generalize the concept of primary finitely  compactly  packed modules over noncommutative rings.  This concept generalizes the concepts of primary compactly packed modules over noncommutative rings,  and primary finitely  compactly  packed modules over commutative rings. We first find the relation between  primary finitely compactly packed modules and primary compactly packed modules over noncommutative rings. We also prove several results on the primary finitely compactly packed modules  over noncommutative rings.  In addition,  we introduce the definition of Dauns-primary submodules over noncommutative rings,  and investigate the relation between this concept and the concepts of Dauns-prime submodules, and primary submodules over noncommutative rings. Key Words: Primary submodules over  noncommutative rings,  primary finitely compactly packed modules over  noncommutative rings ,  primary compactly packed modules over  noncommutative rings,   Dauns-primary submodules over  noncommutative rings,  Dauns-prime submodules over  noncommutative rings.  ﺔﻴﺌﺯﺠﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺍ ﻭ ﺹﺭﻟﺍ ﺔﻤﻭﺯﺤﻤﻟﺍ ﺔﻴﻬﺘﻨﻤﻟﺍ ﺔﻴﺌﺍﺩﺘﺒﻻﺍ ﺕﺎﺴﺎﻘﻤﻟﺍﺕﺎﻘﻠﺤﻟﺍ ﻰﻠﻋ ﺔﻓﺭﻌﻤﻟﺍ ﺱﻨﻭﺩ ﻁﻤﻨﻟﺍ ﻥﻤ ﺔﻴﺌﺍﺩﺘﺒﻻﺍﺔﻴﻟﺍﺩﺒﻻﺍ ﺭﻴﻏ  ﺹﺨﻠﻤ:  ﺹﺭﻟﺍ ﺔﻤﻭﺯﺤﻤﻟﺍ ﺔﻴﻬﺘﻨﻤﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺍ ﻡﻭﻬﻔﻤ ﺙﺤﺒﻟﺍ ﺍﺫﻫ ﻲﻓ ﻡﻤﻌﻨ ﺭـﻴﻏ ﺕﺎـﻘﻠﺤﻟﺍ ﻰـﻠﻋ ﺔﻴﻟﺍﺩﺒﻻﺍ  . ﻭﻫ ﻡﻭﻬﻔﻤﻟﺍ ﺍﺫﻫ ﻥﺃ ﺙﻴﺤ ﻤﻭﻬﻔﻤﻟ ﻡﻴﻤﻌﺘﻲ ﺕﺎﺴﺎﻘﻤﻟﺍ  ﺍ  ﺭﻴﻏ ﺕﺎﻘﻠﺤﻟﺍ ﻰﻠﻋ ﺹﺭﻟﺍ ﺔﻤﻭﺯﺤﻤﻟﺔﻴﻟﺍﺩﺒﻻﺍ  ﺔﻴﻟﺍﺩﺒﻻﺍ ﺕﺎﻘﻠﺤﻟﺍ ﻰﻠﻋ ﺹﺭﻟﺍ ﺔﻤﻭﺯﺤﻤﻟﺍ ﺔﻴﻬﺘﻨﻤﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺍ ﻭ  . ﻥﻴﺒ ﺔﻗﻼﻌﻟﺍ ﺩﺠﻨ ﺔﻴﺍﺩﺒﻟﺍ ﻲﻓ ﺍ ﺔﻤﻭﺯﺤﻤﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺍ ﻭ ﺹﺭﻟﺍ ﺔﻤﻭﺯﺤﻤﻟﺍ ﺔﻴﻬﺘﻨﻤﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺍ ﻰﻠﻋ ﺹﺭﻟ ﺔﻴﻟﺍﺩﺒﺍ ﺭﻴﻐﻟﺍ ﺕﺎﻘﻠﺤﻟﺍ   ، ﻡﺜ ﺘﻨﻤﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺎﺒ ﺔﻘﻠﻌﺘﻤﻟﺍ ﺞﺌﺎﺘﻨﻟﺍ ﺽﻌﺒ ﻥﻫﺭﺒﻨ ﺹﺭﻟﺍ ﺔﻤﻭﺯﺤﻤﻟﺍ ﺔﻴﻬ ﺔـﻴﻟﺍﺩﺒﻻﺍ ﺭﻴﻏ ﺕﺎﻘﻠﺤﻟﺍ ﻰﻠﻋ  . ﻭ ﺭـﻴﻏ ﺕﺎﻘﻠﺤﻟﺍ ﻰﻠﻋ ﺔﻓﺭﻌﻤﻟﺍ ﺱﻨﻭﺩ ﻁﻤﻨﻟﺍ ﻥﻤ ﺔﻴﺌﺍﺩﺘﺒﻻﺍ ﺔﻴﺌﺯﺠﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺍ ﻡﻭﻬﻔﻤ ﺔﻴﺎﻬﻨﻟﺍ ﻲﻓ ﺡﺭﻁﻨﺔﻴﻟﺍﺩﺒﻻﺍ  ، ﺱﻨﻭﺩ ﻁﻤﻨﻟﺍ ﻥﻤ ﺔﻴﻟﻭﻷﺍ ﺔﻴﺌﺯﺠﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺍ ﻥﻴﺒ ﻭ ﺎﻬﻨﻴﺒ ﺔﻗﻼﻌﻟﺍ ﺩﺠﻨ ﻭ ﺔﻴﺌﺯﺠﻟﺍ ﺕﺎﺴﺎﻘﻤﻟﺍ ﻭ  ﻰﻠﻋ ﺔﻓﺭﻌﻤﻟﺍ  ﺔﻴﺌﺍﺩﺘﺒﻻﺍ ﺔﻴﻟﺍﺩﺒﻻﺍ ﺭﻴﻏ ﺕﺎﻘﻠﺤﻟﺍ.  Arwa  Eid  Ashour  561.  Introduction Let  R  be an arbitrary ring.  A nonzero submodule  N  of an  R-module  M is primary  ,   where   (  )   (  )  rann N rann N=    ,N NÍ if  for every nonzero submodule       for some positive integer  n }.  This definition  0=Nr n ( ) {rann N r r R= Î  of a primary submodule over noncommutative rings was introduced  in [1].  In [2],      we  generalized the definition of primary compactly packed modules over any arbitrary ring.  We define a proper submodule submodule N of an R-module M to be  primary compactly packed (pcp) if for each family { }Pa a lÎ of primary submodules of M with   ,UlaaÎÍ PN    lb Î$    such that .N PbÍ  A module  M is called pcp if every proper submodule of M is pcp.  We generalize the concept of pcp modules over noncommutative rings to the concept of  primary finitely compactly packed (pfcp) modules over noncommutative rings. Thus  we say that  a proper submodule N of an R-module M is pfcp if for each  family { }Pa a lÎ of  primary submodules of M with  ,UlaaÎÍ PN     laaa Î$ n,...,, 21   such that 1.iniN Pa=Í U    A module M is  said to be pfcp if every proper submodule of M is pfcp. In Section 2 of this paper we give some examples of pfcp modules and find the relation between pcp modules and pfcp modules. We also find the  conditions that make a pfcp module pcp.  In Section 3 we investigate some properties of pfcp modules. We also find the necessary and sufficient conditions for any R-module M to be pfcp.  Finally,  in  Section 4,  we introduce the definition of Dauns-primary submodules  over noncommutative rings,  which is a generalization to the concept of Dauns-prime submodules,  over noncommutative rings. We study the relation between Dauns-primary submodules over noncommutative rings  and the concept primary submodules over noncommutative rings. "2000  Mathematics  Subject  Classification" :  Primary 16D25,  Secondary 16D80,  16N60 Primary Rimary Finitely Compactly Packed Modules  572. RELATION  BETWEEN  PRIMARY COMPACTLY PACKED   MODULES  AND  PRIMARY  FINITELY  COMPACTLY  PACKED MODULES        We recall first the following definitions, see[2].  Definitions 2.1     A proper submodule N of an R-module M is primary compactly packed (pcp) if for each  family { }Pa a lÎ of primary submodules of M with ,N Paa lb lÎÍ $ ÎU  such that .N PbÍ    A module M is said to be pcp if every proper submodule of M is pcp.           Now we give the following definition Definitions 2.2. A proper submodule N of  an  R-module  M  is  primary finitely compactly packed (pfcp) if for each family { }Pa a lÎ  of  primary submodules of  M  with N Paa lÎÍ U ,  1 2, ,..., na a a l$ Î  such that 1 iniN Pa=Í U .  A module M is said to be pfcp if  every proper submodule of M is pfcp. Remark 2.3. It is clear from the definitions that every pcp module is pfcp module, however the converse is not true as illustrated in the following first example. Examples 2.4    1)  Let V be a vector space with dimension  greater than 2 over the field  / 2F = Z Z .  Then every submodule of  V  is prime,  so every submodule of  V  is primary.   Let 1e  and 2e  be distinct vectors of a basis for  V.  Let   ,11 FeV =     ,22 FeV =    ,)( 213 FeeV +=  and let 1 2L V V= +  .   Then  1 2 1 2{0, , , }L e e e e= + .  Thus  ,1V    ,2V   and  3V  are primary submodules of V with the property that 31iiL V=Í U ,   but  , {1, 2,3}iL V iË " Î . Thus L is pfcp, however L is not pcp.      2) If M is an R-module that contains only a finite number of primary submodules, then M is pfcp module.    Theorem 2.5   Let M be an R-module in which every  finite family of primary submodules of M is  totally ordered by inclusion, then M is pcp if and only if M is pfcp. Arwa  Eid  Ashour  58Proof. ( ® ) Trivial ( ¬ )  Let N Paa lÎÍ U , where Pa is primary submodule for each a .  Since M is pfcp, there exist 1 2, ,..., na a a such that  1iniN Pa=Í U .  Since the family 1{ }iniPa =   of primary submodules of M  is totally ordered by inclusion, there exists 1 2{ , ,..., }nb a a aÎ  such that   1iniP Pa b==U .  Thus M is pcp. 3. IMPORTANT  RESULTS  ON  PRIMARY  FINITELY        COMPACTLY  PACKED  MODULES        Theorem 3.1.   If M is pfcp module which has at least one maximal submodule, then M satisfies the ACC on primary submodules.  Proof. Let 1 2 ...N NÍ Í     be an ascending chain of primary submodules of M and let iiL N= U . If  L=M and H is a maximal submodule of M,  then iiH NÌ U .  Since M is pfcp,  1 2, ,..., kn n n$  such that 1jknjH N=Í U .   Since   1 2 ...N NÍ Í  is an ascending chain, {1, 2,..., }m k$ Î  such that rnkjn NNN mj ===U1  for some {1, 2,3,...}.r Î   Since H is maximal, .rH N=   Since , 1, 2,...r r i iiN N N i+Í Í " =U  and rN  is maximal,  either , 1, 2,...,r r iN N i+= " = thus r iiN N M= =U  which is impossible,  or r i iiN N M+ = =U  which  is also impossible.  Thus L must be a proper submodule of M.  Now since M is pfcp, 1 2, ,..., sn n n$  such  that  1jrnjL N=Í U .   Since 1 2 ...N NÍ Í  is an ascending chain,   {1,2,..., }m r$ Î  such that  knrjn NNN mj ===U1  for some {1, 2,3,...}.k Î Hence   Primary Rimary Finitely Compactly Packed Modules  591 2 1... ...k kN N N N +Í Í Í = = .  Therefore the ACC is satisfied for primary submodules.       Since every finitely generated module and every multiplication module has a proper maximal submodule ,  see[3],  then we have the following Corollary. Corollary 3.2.   Let M be a  pfcp R-module.   If  M  is a finitely generated or a multiplication  R-module, then M satisfies the ACC on primary submodules. Theorem 3.3.   If  M  is an R-module with the property that  every nonempty family of primary submodules of M is totally ordered by inclusion and if M  satisfies the ACC on primary submodules, then M  is pfcp. Proof.  Let N  be a submodule of M with the property that N Paa lÎÍ U , where Pa is primary submodule of  M  for each a .  Then by the hypothesis { }Pa  is totally ordered by inclusion and satisfies the ACC on primary submodules.  Therefore  there exists b lÎ  such that .P Pa bÍU   Hence  N PbÍ  for some b lÎ .  Thus  M  is pcp.  Hence M  is pfcp.   Theorem 3.4  Let  MM ®F :    be an  R-module isomorphism.   If  M  is pfcp module,  then  M   is pfcp module. Proof.   Let  M  be primary compactly packed,  and suppose that  UlaaÎÍ KN   where N  is a proper submodule of M  and  aK is a primary submodule of  M  for  each  la Î .   Since F  is an  R-module isomorphism,  then  ).()( 11 Uga aÎ-- FÍF KN    Thus   )).(()( 11 Uga aÎ-- FÍF KN    Since aK is a primary submodule of  M  for  each  la Î ,    by [ 1 ] ,    )(1 aK-F  is a primary submodule of  M   for  each  la Î .    But  M  is pfcp.   Thus      Arwa  Eid  Ashour  60there exist  1 2, ,..., na a a lÎ  such that   )()(111 Uni iKN=-- FÍFa.   Therefore   Uni iKN1=Ía  for some  la Îi ,  and hence  N  is pfcp.   Thus  M   is pfcp.  4. Dauns-Primary Submodules.    In this section,  we introduce the definition of Dauns-primary submodules and investigate the relation between this concept and the concepts of Dauns-prime submodules and primary submodules.       The following definition was introduced by  J.Dauns; [4],[5]. Definition 4.1  Let  R  be any ring and  M  any  R-module.   A submodule  N  of  M  is called Dauns-prime if,  whenever   NrRm Í    for  NMm -Î   and  ,Rr Î   we have  .NrM Í         Scott Annin in his Ph.D. thesis,  proved the following result,  see[6]. Proposition 4.2    Let  R  be any ring and  M  any R-module.   A submodule  N  of  M  is  Dauns-prime if and only if   NM |   is prime submodule of  M,   where   M | N = ,|{ Rrr Î  }NrM Í .            Now,  we introduce the definition of  Dauns-primary submodules. Definition 4.3     Let  R  be any ring and  M  any module.   A submodule  N  of  M  is called Dauns-primary  if,  whenever   NrRm Í    for  NMm -Î   and  ,Rr Î   we have  ,NMr n Í   for some positive integer  n.       The relationship between the definition of primary submodules and Dauns-primary submodules is given by the following proposition. Proposition 4.4    Let  R  be any ring and let N  be a  submodule of  the  R-module M.  Then  N is a Dauns-primary submodule of  M  if and only if   NM |   is primary submodule of  M.   Proof.   For the  "if" part,   assume that  NM |  is primary submodule of  M,  and suppose NMm -Î  with  NrRm Í  for some   Rr Î .   Then  NNRm |)( +  is a nonzero submodule of  NM |  that is killed by r.   Since  NM |   is primary submodule of  M,   we conclude that  ,NMr n Í   for some positive integer  n.    Primary Rimary Finitely Compactly Packed Modules  61Conversely,  Suppose  N  is  Dauns-primary submodule of  M and consider a submodule  .|/0 NMNN Í¹    Suppose that  ,NNr Í   for some   Rr Î .   Choose an element  NNt -Î .   Then  NrRt Í .   Since  N  is a Dauns-primary submodule of  M,  we conclude that  ,NMr n Í   for some positive integer  n.   Thus  NM |   is primary submodule of  M.  REFERENCES  [ 1 ] Ashour, A. E.: Primary  Ideals and Primary  Modules  over  Noncommutative  Rings,  Journal of the Islamic University of Gaza,  18(1),   (2010). [ 2 ] Ashour, A. E.: On Primary Compactly Packed Modules over non commutative rings , Approved  on The Second International Conference of natural and Applied Science,  Al Aqsa  University,  Palestine, ( 30-31), May  2011.  [ 3 ]  Athab, E. A.: Prime And Semiprime Submodules,  M.Sc. Thesis, College of   Science, University of  Baghdad, 1996. [ 4 ] Dauns, J. : Prime  Modules,  Journal fur die reine und Angewandte Mathematik,  298, 156-181 (1978). [ 5 ] Dauns, J. : Prime  Modules and one-sided ideals,  Lectures notes in Pure and Applied Mathematics,  55, 301-344 (1980). [ 6 ] Annin, S. :  Associated Primes Over Noncommutative Rings,  Ph.D.  Thesis,  University Of  California,  Berkely,  2002. 

Primary Rimary Finitely Compactly Packed Modules And Dauns-Primary Submodules Over Noncommutative Rings

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Primary Rimary Finitely Compactly Packed Modules And Dauns-Primary Submodules Over Noncommutative Rings

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