Vol 5, o 9 September 04 ISS 079-8407 Joural of Emergg Treds Computg ad Iformato Sceces 009-04 CIS Joural All rghts reserved http://wwwcsouralorg Prme ad Sem Prme Subb-Sem Modules of (R, R) Partal B-Sem Modules M Srvasa Reddy, V Amaredra Babu (Assstat Professor, Departmet of S & H, D V R & Dr H S MIC College of Techology, Kachacherla- 580, Krsha(Dstrct), Adhra Pradesh, Ida (Assstat Professor, Departmet of Mathematcs, Acharya agarua Uversty, agarua agar-550, Gutur (Dt), Adhra Pradesh, Ida ABSTRACT A partal semrg s a structure possessg a ftary partal addto ad a bary multplcato, subect to a set of axoms The partal fuctos uder dsot-doma sums ad fuctoal composto s a partal semrg I ths paper we troduce, the otos of prme ad sem prme subb-semmodules of (R, R) - partal b-sem modules, obta prme avodace theorem for subb-semmodules of (R, R) - partal b-sem modules ad exted the results o prme ad sem prme sub sem modules over partal sem rgs by P V Srvasa Rao [8] to (R, R) - partal b-sem modules Keywords: Prme subb-semmodule, sem prme subb-semmodule, multplcato (R, R) - partal b-sem module, closed subset, sem closed subset, -closure, -subb-semmodule ITRODUCTIO The study of pf ( D, (the set of all partal fuctos of a set D to tself), Mf ( D, (the set of all mult fuctos of a set D to tself) ad Mset ( D, (the set of all total fuctos of a set D to the set of all fte mult sets of D ) play a mportat role the theory of computer scece, ad to abstract these structures Maes ad Beso[6] troduced the oto of sum ordered partal sem rgs (so-rgs) Motvated by the wor doe partally-addtve sematcs by Arbb, Maes[4] ad the developmet of matrx theory of so-rgs by Martha E Streestrup[7], G V S Acharyulu[] 99 studed the codtos uder whch a arbtrary so-rg becomes a pf ( D,, Mf( D, ad Mset ( D, Cotug ths study, P V Srvasa Rao [9] developed the deal theory for sorgs ad partal sem modules over partal sem rgs I [], [3], [0], we obtaed the b-deal theory of so-rgs ad troduced the oto of partal bsem module over the partal sem rgs R, S I ths paper we troduce the otos of prme ad sem prme subbsemmrodules of (R, R) - partal b-sem modules, characterze them terms of prme ad sem prme partal b-deals of partal semrg R ad obtaed the prme avodace theorem for subb-semmodules of (R, R) - partal b-sem modules PRELIMIARIES I ths secto, we dscuss mportat deftos, results ad examples for our use the ext sectos Defto : [6] A partal mood s a par ( M, ) where M s a o empty set ad s a partal addto defed o some, but ot ecessarly all famles ( x : I) M subect to the followg axoms: a Uary Sum Axom: If ( x : I) s a oe elemet famly M ad I = { }, the ( x : I) s defed ad equals x b Partto - Assocatvely Axom: If ( x : I) s a famly M ad If ( I : J ) s a partto of I, the ( x : I) s summable f ad oly f ( x : I ) s summable for every J ad ( ( x : I ) : J ) s summable, ad ( x : I) = ( ( x : I ) : J ) Defto : [6] The sum orderg o a partal mood ( M, ) s the bary relato such that x y f ad oly f there exsts a h M such that y = x + h, for x, y M Defto 3: [6] A partal semrg s a quadruple ( R,,, ), Where ( R, ) a partal mood wth partal addto s, ( R,,) s a mood wth multplcatve operato ad ut, ad the addtve ad multplcatve structures obey the followg dstrbutve laws: If ( x : I) s defed R, the for all y R, ( y x : I) ad ( x y : I ) are defed ad y [ x ] ( y x ),[ x ] y ( x y) 690
Vol 5, o 9 September 04 ISS 079-8407 Joural of Emergg Treds Computg ad Iformato Sceces 009-04 CIS Joural All rghts reserved http://wwwcsouralorg I addto to ths, f (R, ) s a commutatve mood the the partal semrg ( R,,, ) s called a commutatve partal semrg Throughout ths paper R stads for a commutatve partal semrg Defto 4: [6] A sum-ordered partal semrg (or so-rg, short), s a partal semrg whch the sum orderg s a partal orderg Defto 5: [] Let R be a so-rg A subset of R s sad to be a deal of R f the followg are satsfed: (I ) f ( x : I) s a summable famly R ad x for every I the x, (I ) f x y ad y the x, ad (I 3 ) f x ad rr the xr, rx Defto 6: [] A subset of a so-rg R s sad to be a b-deal of R f the followg are satsfed: (B ) f ( x : I) s a summable famly R ad x for every I the x, (B ) f x y ad y the x, ad (B 3 ) f x, y ad rr the xry Defto 7: [] A subset of a so-rg R s sad to be a partal bdeal of R f the followg are satsfed: f ( x : I) s a summable famly R ad x for every I the x, ad If x, y ad rr the xry Example 8: [] Cosder the partal semrg R = {0, u, v, x, y, } wth defed o R by x f x 0, for some x udefed, otherwse Ad defed by the followg table: 0 u v x y 0 0 0 0 0 0 0 u 0 u 0 0 0 u v 0 0 v 0 0 v x 0 0 0 0 0 x y 0 0 0 0 0 y 0 u v x y Defto 9: [0] A proper b-deal of a so-rg R s sad to be prme f ad oly f for ay b-deals A, B of R, ARB P mples A P or B P Defto 0: [0] A proper b-deal I of a so-rg R s sad to be sem prme f ad oly f for ay b-deal H of R, HRH I mples H I Defto : [7] Let ( R,,, ) be a partal semrg ad ( M, ) be a partal mood The M s sad to be a left partal sem module over R f there exsts a fucto : R M M : ( r, x) r x whch satsfes the followg axoms for x, ( x : I) M ad r, r,( r : ) R J f x exsts the r ( x ) ( r x ), f r exsts the ( r ) x ( r x), r ( r x) ( r r ) x, ad v x x R Aalogously, oe ca defe rght partal sem modules over R Defto : [3] Let R, S be partal sem rgs ad ( M, ) be a partal mood The M s sad to be a ( R, S) - partal b-sem module f t satsfes the followg axoms: M s a left partal sem module over R, M s a rght partal sem module over S, ad () for ay r R, x M, s S, r x s M Defto 3: [3], Let ( M, ) be a ( R, S) - partal b-sem module The a o empty subset of M s sad to be a subb-semmodule of M f s closed uder ad Defto 4: [3] Let be a subb-semmodule of a ( R, R) - partal b-sem module M The we defe ( : M ) as ( : M ) { r R rmr } 69
Vol 5, o 9 September 04 ISS 079-8407 Joural of Emergg Treds Computg ad Iformato Sceces 009-04 CIS Joural All rghts reserved 3 PRIME AD SEMIPRIME SUBBI- SEMIMODULES I ths secto we characterze the prme ad sem prme subb-semmodules of (R, R) - partal b-sem modules Defto 3: Let M be a (R, R) - partal b-sem module ad be a proper subb-semmodule of M The s sad to be a prme subb-semmodule of M f for ay r R ad M, rr mples r ( : M ) or http://wwwcsouralorg Remar 36: If M s a multplcato (R, R) - partal b-sem module ad s a partal subb-semmodule of M the = (:M)MR Sce M s a multplcato (R, R) - partal bsem module, a partal b-deal I of R such that = IMR I ( : M ) IMR ( : M ) MR Hece = (: M) MR ote 3: If A s a b-deal of R ad K s a subbsemmodule of a (R, R) - partal b-sem module M the s prme subb-semmodule of M f ad oly f AKR mples A ( : M ) or K Theorem 33: Let M be a (R, R) - partal b-sem module ad K be a proper subb-semmodule of M If K s prme subbsemmodule of M the ts assocated partal b-deal ( K : M ) s a prme partal b-deal of R Suppose K s a prme subb-semmodule of M Let A, B be ay two b-deals of R such that ARB ( K : M ) The ( ARB) MR K A( RBM ) R K A ( K : M ) or RBM K If RBM K the BM K BMR KR BMR K B ( K : M ) or M K Sce K s proper, M K Therefore A ( K : M ) or B ( K : M ) Hece ( K:M ) s a prme partal b-deal of R The followg s a example of a (R, R) - partal bsem module M whch the coverse of the above theorem s ot true Example 34: Let R be the partal semrg wth fte support addto ad usual multplcato The M = s a (R, R) - partal b-sem module by the scalar multplcato : ( x,( a, b), x) ( xax, xbx) ad K = 0 4 s a subb-semmodule of M Here (K: M) = { r R rmr K } = { 0 } whch s a prme partal bdeal of R Sce{ } (0,) R K, ( K : M ) ad ( 0,) K Hece K s ot a prme subb-semmodule of M Defto 35: Let M be a (R, R) - partal b-sem module The M s sad to be a multplcato (R, R) - partal b-sem module f for all subb-semmodules of M there exsts a partal b-deal I of R such that = IMR ow we show that the coverse of the theorem 33 s true for the multplcato (R, R) - partal b-sem modules Theorem 37: Let M be a multplcato (R, R) - partal b-sem module ad be a subb-semmodule of M The s a prme subb-semmodule of M f ad oly f (: M) s a prme partal b-deal of R By the theorem 33, we get the ecessary part For the suffcet part, suppose (: M) s a prme partal bdeal of R Let I be a partal b-deal of R ad K be a subbsemmodule of M IKR Sce M s a multplcato (R, R) - partal b-sem module, a partal b-deal J of R K JMR ow IKR I ( RKR) R I ( RJMRR) R ( IRJ ) MR IRJ ( : M ) I ( : M ) or J ( : M ) I ( : M ) or JMR I ( : M ) or K Hece s a prme subbsemmodule of M ote that M cosdered the example 34 s ot a multplcato (R, R) - partal b-sem module Theorem 38: A (R, R) - partal b-sem module M s a multplcato (R, R ) - partal b-sem module f ad oly f there exsts a partal b-deal I of R such that RmR = IMR for each m M Suppose M s a multplcato (R, R) - partal bsem module Let m M The RmR s a subbsemmodule of M a partal b-deal I of R RmR IMR Coversely suppose that a partal b-deal I of R RmR = IMR m M Let be a subb-semmodule of M The for ay, a partal b-deal I of R RR = I MR I s a partal I b-deal of R such that RR I MR ( I ) MR IMR 69
Vol 5, o 9 September 04 ISS 079-8407 Joural of Emergg Treds Computg ad Iformato Sceces 009-04 CIS Joural All rghts reserved Hece M s a multplcato (R, R) - partal bsem module Defto 39: Let M be a multplcato (R, R) - partal b-sem module ad be a subb-semmodule of M The we say that I s a presetato partal b-deal of f =IMR for some partal b-deal I of R, for short, a presetato of We deote the set of all presetato partal b-deals of by Pr ( ) Defto 30: For partal b-deals I ad J of R, we defe the relato ~ as follows: I ~ J f ad oly f IMR = JMR Remar 3: ~ s a equvalece relato o the set of all partal b-deals of R http://wwwcsouralorg for ay subb-semmodules U, V of M, UMV U or V, for ay m, m M, m Mm or m m () : Suppose s a prme subb-semmodule of M ad let U, V be subb-semmodules of M UMV Sce M s a multplcato (R, R) - partal b-sem module, partal bdeals I, J of R U IMR adv JMR UMV ( IRJ ) MR IRJ ( : M ) By the theorem 37, (: M) s a prme partal b-deal of R I ( : M ) or J ( : M ) U IMR or V JMR Defto 3: Let M be a multplcato (R, R ) - partal b-sem module ad, K be subb-semmodules of M such that = IMR ad K = JMR for some partal b-deals I, J of R The the multplcato of ad K s defed as K = (IMR)M (JMR) = (IRJ)MR Theorem 33: The product of two subb-semmodules s depedet of ts presetatos Let M be a multplcato (R, R ) - partal b-sem module ad, K be subb-semmodules of M Suppose I, I ad J, J are two presetatos of ad K respectvely The IMR I MR ad K JMR J MR ow K = ( I MRM ) ( JMR) ( IMRM ) ( JMR) ( IRJ ) MR= (J RI )MR= J RMM ) ( I RM) ( J RMM ) ( I RM) ( J RI ) MR ( I RJ) MR= ( ( I MR) M ( J MR) Hece the theorem Defto 34: Let M be a multplcato (R, R) - partal b-sem module ad m, m M such that RmR IMR ad RmR I MR for some partal b-deals I, I of R The the multplcato of m ad m s defed as m I MR M I MR I RI MR m ) ( ) ( ) ( Theorem 35: Let M be a multplcato (R, R ) - partal b-sem module ad be a subb-semmodule of M The the followg codtos are equvalet: s a prme subb-semmodule of M, () ( ): Suppose for ay subb-semmodules U, V of M, UMV U or V Let m, m M m Mm Sce M s a multplcato (R, R ) - partal b-sem module, partal b-deals I, J of R Rm R =IMR ad Rm R =JMR mmm ( RmR) M ( RmR) = ( IMR ) M ( JMR) = ( IRJ )MR Rm R or Rm R or m m Hece () (): Suppose for ay m, m M, m Mm m or m ow we prove ( : M ) s a prme partal b-deal of R Let I, J be partal b-deals of R IRJ ( : M ) The ( IRJ ) MR Suppose I ( : M ) ad J ( : M ) IMR ad JMR I, J, m, m M, a, br m aimr\ ad m b JMR \ m a) M( m b) ( IMRM ) ( JMR) ( IRJ) MR ( m a or m b whch s a cotradcto Hece ( : M ) s a prme partal b-deal of R Hece by the theorem 37, s a prme subbsemmodule of M Defto 36: Let M be a multplcato (R, R) - partal b-sem module A subset S of M s sad to be multplcato closed subset ( short closed subset) f for ay m, S, ( mm) S 693
Vol 5, o 9 September 04 ISS 079-8407 Joural of Emergg Treds Computg ad Iformato Sceces 009-04 CIS Joural All rghts reserved Remar 37: Let be a subb-semmodule of a multplcato (R, R) - partal b-sem module M The s prme f ad oly f M \ s a closed subset of M Suppose s prme for ay m, m M, m Mm mples m or m m ad m mples m Mm m, m M \,( mmm ) ( M \ ) M \ s a closed subset of M Theorem 38: Let M be a multplcato (R, R ) - partal b-sem module Let A be a subb-semmodule of M ad S be a closed subset of M such that A S The there s a subb-semmodule of M whch s maxmal wth respect to the property that A ad S Furthermore, s a prme subb-semmodule of M http://wwwcsouralorg let H I H The H s a subb-semmodule of M ad H ow we prove H s prme Let m, m M, m Mm H ad m H The m H for some I m H For ay, H H ad hece m H For ay >, H H m H ad hece m H Hece m H H C The by Zor s lemma, C has a maxmal elemet Hece the theorem Defto 30: Let M be a (R, R) - partal b-sem module ad be a subb-semmodule of M The s sad to be sem prme f ad oly f ts assocated partal b-deal ( : M ) s sem prme Clearly every prme subb-semmodule s sem prme The followg s a example of a partal b-sem module M whch a sem prme subb-semmodule s ot prme Tae C = { B B s a subb-semmodule of M, A B ad B S } Clearly A C Moreover (C, ) s a partally ordered set whch every smply ordered famly has a upper boud By Zor s lemma, C has a maxmal elemet Let t be e, s a subbsemmodule of M whch s maxmal wth respect to the property that A ad S To prove s prme, let a, bm amb Suppose a adb The ara ad brb ( ara) S ad ( brb) S s, t S s ara ad t brb Sce S s closed subset of M, ( smt) S Moreover smt ( ara) M ( brb) = M + (ara )M + M (brb) + (ara)m(brb) Sce s a subb-semmodule of M, M, (ara)m, M (brb) are subsets of Sce amb, (ara)m(brb) Hece sm t Thus S, a cotradcto Hece s a prme subb-semmodule of M Theorem 39: Every prme subb-semmodule of a multplcato (R, R) - partal b-sem module M cotas a mmal prme subb-semmodule Tae C = { H H s a prme subb-semmodule of M, H } Sce C, (C, ) s a oempty partally ordered set Let { H I } be a decreasg cha of subb-semmodules of M H I ad Example 3: Let M be the (R, R ) - partal b-sem module as the example 34 Tae K = 0 4 The ( K : M ) = {0} s a sem prme partal b-deal of R ad hece K s a sem prme subb-semmodule of M But K s ot a prme subb-semmodule of M Theorem 3: Let M be a multplcato (R, R ) - partal b-sem module ad be a subb-semmodule of M The the followg codtos are equvalet: () s a sem prme subb-semmodule of M, () for ay subb-semmodule U of M, UMU mples U, ( ) for ay m M, mmm mples m Suppose s a sem prme subb-semmodule of M The (: M) s a sem prme partal b-deal of R Let U be a subb-semmodule of M UMU Sce M s a multplcato (R, R ) - partal b-sem module, a partal b-deal I of R U = IMR UMU = ( IMR)M(IMR) = (IRI )MR IRI ( : M ) U = IMR () ( ): Suppose for ay subb-semmodule U of M, UMU mples U Let m M mmm Sce M s a multplcato (R, R ) - partal bsem module, a partal b-deal I of R RmR = IMR mmm = (RmR )M(RmR) RmR Hece m 694
Vol 5, o 9 September 04 ISS 079-8407 Joural of Emergg Treds Computg ad Iformato Sceces 009-04 CIS Joural All rghts reserved M ( ) ( ): Suppose for ay m, mmm mples m Let I be a partal b-deal of R IRI ( : M ) The (IRI )MR Suppose I ( : M ) The IMR I, m M ad a R m a IMR \ ( ma) M( ma) ( IMRM ) ( IMR) ( IRI) MR m a, a cotradcto Hece s a sem prme subb-semmodule of M http://wwwcsouralorg closed, we have S s sem closed subset of M We deote the set of all prme subb-semmodules of M cotag the subb-semmodule of M by V( ) Theorem 37: Let M be a multplcato (R, R ) - partal b-sem module ad be a subb-semmodule of M The s sem prme f ad oly f V () Defto 33: Let M be a multplcato (R, R) - partal b-sem module A subset S of M s sad to be multplcato sem closed subset ( short, sem closed subset) f for ay m S, ( mmm) S Clearly every closed subset of a multplcato (R, R) - partal b-sem module M s sem closed subset The followg s a example of a multplcato (R, R) - partal b-sem module M whch the sem closed subset s ot a closed subset of M Example 34: Cosder the partal semrg R = {0, u, v, x, y, } as the example 8 The M: = R s a multplcato (R, R) - partal b-sem module ow S = {u, v} s a sem closed subset of M Sce {0, u}, {0, v} are partal bdeals of R such that RuR = {0, u}mr ad RvR = {0,v}MR, we have umv = {0, u}m{0, v}= {0} (umv) S Hece S s ot a closed subset of M Remar 35: Let M be a multplcato (R, R ) - partal b-sem module ad be a subb-semmodule of M The s sem prme f ad oly f S = M \ s a sem closed subset of M Theorem 36: Let M be a multplcato (R, R ) - partal b-sem module ad S be a subset of M The S s sem closed subset f ad oly f t s the uo of closed subsets of M Suppose S s a sem closed subset of M ad let m 0 S Sce S s sem closed, ( m0 Mm0 ) S m S m m0mm0 Sce S s sem closed ad m S, ( m Mm ) S Cotug le ths we get a set B 0 { m0, m,, m,} of elemets of S such that m mmm m0mm0 The t ca be proved that B 0 s closed subset of M cotag m 0 Hece S B, the uo of closed m0s subsets of M Coversely suppose that S 0 B I, the uo of closed subsets B of M Sce every closed subset s sem closed ad uo of sem closed subsets s aga sem Suppose s a sem prme subb-semmodule of M The S = M \ s a sem closed subset of M S where each S s a closed subset of M Sce S I S, we have S I Tae C = {H H s a subb-semmodule of M, H ad H S }, I Clearly C ad hece C The by Zor s lemma, C has a maxmal elemet Let t be K, I The by the theorem 38, K s a prme subb-semmodule maxmal wth respect to the property that K ad K S, I ow I K ( M \ S ) M \ S Hece I V () Coversely suppose that V () The M \ M \ ( V ( )) = { M \ H H s prme subb-semmodule of M ad H} M \ s a uo of closed subsets of M M \ s the sem closed subset of M by the theorem36 Hece s sem prme subbsemmodule of M by the remar 35 4 PRIME AVOIDACE THEOREM I ths secto we obta the prme avodace theorem for subb-semmodules of (R, R) partal b-sem module Defto 4: Let L, L,, L, L be subb-semmodules of a (R, R ) - partal b-sem module M The () we say that a coverg L L L cotag L s effcet f there exsts o {,, 3,, } such that L L L L L L () we say that L = L L L s a effcet uo f there exsts o {,, 3,, } such that L L L L L L 695
Vol 5, o 9 September 04 ISS 079-8407 Joural of Emergg Treds Computg ad Iformato Sceces 009-04 CIS Joural All rghts reserved Ay coverg of L cosstg of subbsemmodules of M ca be reduced to a effcet oe by deletg all uecessary terms Defto 4: The -closure of a subb-semmodule of a (R, R ) - partal b-sem module M s defed by { a a a a for some a, a } Remar 43: Let M be a complete (R, R ) - partal b-sem module ad be a subb-semmodule of M The s a subb-semmodule of M whch cotas ad Frst we prove s a subb-semmodule of M: Let ( x : I) be ay famly M x I The x x x for some x, x I x x x for some x x ad hece r R The x x x http://wwwcsouralorg ad x Let x ad for some x, x ad r R r x r r x r r x r for some r x r r x r, r x r ad hece Hece s a subb-semmodule of M For ay x, x + 0 = x Therefore To prove, let x The x x x where x x x ad x x x for some x x x x,,, x ( x x ) x x for some x x x x, ad hece x Hece the remar Defto 44: A subb-semmodule of a (R, R ) - partal bsem module M s sad to be -subb-semmodule f = Theorem 45: Let L, L be subb-semmodules ad L be a subb-semmodule of a complete (R, R ) - partal b-sem module M If L L L the L L or L L or L L Suppose L L L ad L L ad L L l L \ L ad l L \ L Sce, L L L, we have l L ad l L ow l l l l Sce L, L are subbsemmodules, ether cotradcto Hece the theorem l L or l L whch s a Theorem 46: Let L, L,, L be subb-semmodules ad L be a subb-semmodule of a complete (R, R ) - partal b-sem module M If L = L L L s a effcet uo the for ay {,, 3,, }, L L, Let {,, 3,, } ad let x L Sce L = L L L s a effcet uo, L L y L y L y L ow x + yl= L L L Suppose x + yl for some {,, 3,, -, +,, } The x + y = z for some x, zl where {,, 3,, -, +,, } y L for {,, 3,, -, +,, }, a cotradcto Hece x + yl Sce L s a subb-semmodule, x L Hece x L Hece the theorem Theorem 47: Let L L L L be a effcet coverg cosstg of subb-semmodules of a complete (R, R ) - partal b-sem module M, where > If ( L : M ) ( L : M ) for ay {,, 3,, }, the L s ot a prme subb-semmodule of M Sce L effcet coverg, L L L s a L ( L L ) ( L L ) ( L L ) s a effcet uo The by above theorem 46, ( L L ) ( L L ) L L Sce L L, l L \ L Suppose L s a prme subbsemmodule of M (L : M ) s a prme partal b-deal of R Sce ( L : M ) ( L : M ), s ( L : M ) \ ( L : M ) s s ( L : M ) \ ( L : M ) s l s L L ad 696
Vol 5, o 9 September 04 ISS 079-8407 Joural of Emergg Treds Computg ad Iformato Sceces 009-04 CIS Joural All rghts reserved s l s L L cotradcto Hece the theorem ( L L ) L, a L Theorem 48 (Prme avodace theorem): Let M be a complete (R, R ) - partal b-sem module, L, L,, L be a fte umber of subb-semmodules of M ad L be a subb-semmodule of M such that L L L L Assume that almost two of L s are ot prme, ad that for ay, ( L : M ) ( L : M ) The L L for some Let L L L L be a effcet coverg of the gve coverg, where,,, m {,, 3,, } m Suppose m = The L L L s the effcet coverg L L L or L by the theorem 46 L L or L L, a cotradcto Suppose m > The at least oe L, whch s prme By the theorem 47, ( L : M ) ( L : M ) for some {,, 3,, m}, a cotradcto ad hece m = Therefore L L for some {,, 3,, } Theorem 49: Let P be a prme subb-semmodule ad,,, be subb-semmodules of the multplcato (R, R) - partal b-sem module M The t P f ad oly f P for some wth t If Coversely suppose t t ( : P for some the t http://wwwcsouralorg P P t ( : M ) ( P : M ) : M ) ( P : M ) ( M ) ( P : M ) for some Hece P for some wth t Corollary 40: Let M a complete multplcato (R, R ) - partal b-sem module, be a subb-semmodule ad at least ( ) of P, P, P are prme - subb-semmodules such that P P P the P for some wth 5 COCLUSIO I ths paper we troduced the otos of prme ad sem prme subb-semmodules of ( R, R ) - partal bsem modules ad characterzed them terms of prme ad sem prme partal b-deals of R We studed the equvalet codtos of prme ad sem prme subbsemmodules of ( R, R ) - partal b-sem modules ad obtaed the prme avodace theorem for subbsemmodules of ( R, R ) - partal b-sem modules REFERECES [] Acharyulu, GVS: A Study of Sum-Ordered Partal Sem rgs, Doctoral thess, Adhra Uversty,99 [] Amaredra Babu, V, ad Srvasa Reddy, M: Bdeals of Sum-Ordered Partal Sem rgs, Iteratoal Joural of Mathematcs ad Computer Applcatos Research(IJMCAR), Vol3, ssue, pp 57-64, Jue 03 [3] Amaredra Babu, V, Srvasa Reddy, M, ad Srvasa Rao, P V: Partal b-sem modules over partal sem rgs, Mathematcs ad Statstcs (3), pp 0-9,04 [4] Arbb, MAad Maes, EG: Partally Addtve Categores ad Flow-dagram Sematcs, Joural of Algebra, vol 6, pp 03-7, 980 [5] Le Roux, HJ: A ote o Prme ad Sem prme Bdeals, Kyugpoo Math J, Vol 35, pp 43-47, 995 [6] Maes, EG, ad Beso, DB: The Iverse Sem group of a Sum-Ordered Semrg, Sem group Forum, vol 3, pp 9-5, 985 [7] Streestrup, ME: Sum-Ordered Partal Sem rgs, Doctoral thess, Graduate school of the Uversty of Massachusetts, Feb 985 (Departmet of computer ad Iformato Scece) [8] Srvasa Rao, PV: Partal sem modules over Partal Sem rgs, Iteratoal Joural of Computatoal Cogto (IJCC), Vol 8, o 4, pp 80-84, Dec 00 [9] Srvasa Rao, PV: Ideal Theory Of Sum-Ordered Partal Sem rgs, Doctoral thess, Acharya agarua Uversty, 0 [0] Srvasa Reddy, M, Amaredra Babu, V, ad Srvasa Rao, P V: Prme ad Sem prme bdeals of so- rgs, Iteratoal Joural of Scetfc ad Iovatve Mathematcal Research (IJSIMR), Vol-, ssue-, Octomber-03, pp34-43 697