PRIMARY DECOMPOSITION, ASSOCIATED PRIME IDEALS AND GABRIEL TOPOLOGY
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1 Orietal J. ath., Volue 1, Nuber, 009, Page Orietal Acadeic Publiher PRIARY DECOPOSITION, ASSOCIATED PRIE IDEALS AND GABRIEL TOPOLOGY. EL HAJOUI, A. IRI ad A. ZOGLAT Uiverité ohaed V aculté de Sciece Départeet de athéatique B.P 1014 Rabat, aroc e-ail: hajoui4@yahoo.fr Abtract Let R be a coutative rig with a uit, N be a R-ubodule of a give R- odule ad be a o-trivial Gabriel topology o R. I thi paper, we give ufficiet coditio for the cloure Cl ( N ) of N i to be priary R- ubodule ad we how that it reduce to or N if R i Noetheria. oreover, thi will be ued to deduce a priary decopoitio of Cl ( N ) fro the oe of N whe i of fiite type (ad R i Noetheria). I uch cae, we tudy A ( ) ad we give eceary ad ufficiet coditio to have A ( N ), where A ( ) deote the et of the aociated prie ideal. 1. Itroductio ad Preliiarie Let R be a coutative rig R with uit. A prie ideal P of R i aid to be aociated to a give R-odule if there exit x \ ( 0) uch that P = A( x). The et of all thee ideal i deoted by A ( ). Throughout thi paper, we will ue ubodule N to ea that N i a ubodule of. The, a N i aid to be priary i if N ad for all a R ad x \ N with ax N there exit N uch that a N. I 010 atheatic Subject Claificatio: Priary: 16D10; Secodary: 13C05, 13C11. Keyword ad phrae: priary ubodule, priary decopoitio, aociated prie ideal, Gabriel topology, ijective odule. Received Deceber 0, 009
2 . EL HAJOUI, A. IRI ad A. ZOGLAT 10 thi cae the ideal P = A( N ) i prie ad N i aid to be P -priary (ee [5]). I a give Gabriel topology o R, we deote by Cl ( N ) the cloure of N i, i.e., Cl ( N ) = { x : I Ix N}. (or ore detail o Gabriel topology ee [1,, 3, 4]). Hece, the ubodule N i called -cloed if N = Cl ( N ). Alo, we ay that N ha a uppleetary i if there exit L uch that L N =. Defiitio 1.1. A R-odule i ijective if for all R-odule hooorphi f : 1 ad θ : 1 where f i oe-to-oe, the there exit a hooorphi ρ : atifyig θ = ρof. Defiitio 1.. Let be a R-ubodule of E. We ay that E i a eetial exteio of (or iply i eetial i E ) if N ( 0) for ay ozero R-ubodule N of E.. Priary Decopoitio of Cl ( N ) I thi ectio, we give ufficiet coditio for the R-ubodule Cl ( N ) to be priary. If R i Noetheria, a eceary coditio i alo give. Thee reult will be ued to obtai a priary decopoitio of Cl ( N ) fro a priary decopoitio of N. Propoitio.1. Let P -priary. The we have: (i) If P, the Cl ( N ) = N. (ii) If Cl ( N ) =, the P. Proof. (i) Suppoe that N be two R-odule ad aue that N i P ad Cl ( N ) N. The N Cl ( N ). Now, let x Cl ( N )\ N. The there exit I uch that I x N. Hece, for all λ I we have λ x N. Sice x N ad N i P-priary,
3 PRIARY DECOPOSITION, ASSOCIATED PRIE IDEALS 103 there exit Coequetly N uch that λ N. Therefore λ P = A( N ). I P ad P, a cotradictio. (ii) N i P -priary, thu N, let x ice Cl ( N ) =, the x Cl ( N ), therefore there exit I uch that I x N. Next, proceedig i a iilar way a i (i) above we deduce that P. Reark.. The covere of (ii) i the above propoitio i alo true whe R i aued to be Noetheria. See the ext ectio for the proof. or itace, ote that if P i a prie ideal of R, the P = { I ideal of R : I P} i a topology of Gabriel defied o R. Therefore, we have the followig Corollary.3. If N i P -priary, the Cl ( N ) = N. Theore.4. Let P N be two R-odule. If N i P-priary, the Cl ( N ) = or Cl ( N ) i Q -priary with P Q. Proof. Suppoe that N i P -priary ad ad Cl ( N ). Let x \ Cl ( N ) ad a R with ax Cl ( N ). The there exit I uch that I ax N. But I x N for x Cl ( N ). Therefore we ca fid λ I uch that λ x N ad aλ x N. Hece there i a N Cl ( N ) which iplie that Cl ( N ) i priary. N with Let Q = A( Cl ( N )). The Cl ( N ) i Q -priary ad becaue N Cl ( N ). P Q Theore.5. Let R a Noetheria rig, ad N be a P -priary ubodule of. The Cl ( N ) = if ad oly if P. Proof. Accordig to Propoitio.1, we eed to prove oly that P iplie Cl ( N ) =. Let P ad aue the exitece of x \ Cl ( ). Sice R i Noetheria, P i of fiite type ad therefore 0 N
4 there exit. EL HAJOUI, A. IRI ad A. ZOGLAT 104 1,,..., R uch that P = R1 + R + + R. Sice,,..., P A( ), there exit ( 1,,..., ) ( N ) uch 1 = N k that k N, k = 1,,...,. Thu, we have k 0 N for every k = 1,,...,. Now, et = k = 1 k ad let λ P. Hece, we ca write λ = Therefore, we have k x ( λk11 + λk + + λk ) fiitek= 1 = β( ) 1,,..., 1 1. fiite = x N with =. 1 Ideed, there exit k { 1,,..., } uch that k k (becaue if ot k < k for ay 1,,...,, k k, a k= 1 k= 1 k = ad o = < = cotradictio). Next, uig k x 0 N k, we obtai k k x0 N ad coequetly 1 1 x0 N. Uig thi fact, it follow fially that λx0 = β( ) 1 1,,..., 1 fiite = ad therefore P x 0 N. But, ice P x0 Cl ( N ), a cotradictio. Thi coplete the proof. x0 N for P, we coclude that Corollary.6. If R i Noetheria ad N i a P -priary ubodule of, the Cl ( N ) = N or Cl ( N ) =. ro ow o, we uppoe that R i Noetheria ad i of fiite type (i.e., there exit x 1, x,..., x eleet i uch that = Rx 1 + Rx + + Rx ). Hece, accordig to Theore 6.8 i [5], we ee that ay R- ubodule N of ha a priary decopoitio. Let N = Q i, where Q i i i= 1 ad J = { j : 1 j adp j }. Pi -priary, a priary decopoitio of N,
5 PRIARY DECOPOSITION, ASSOCIATED PRIE IDEALS 105 Theore.7. (i) If J =, the Cl ( N ) =. (ii) If J, the Cl ( N ) = Q j i a priary decopoitio of j J Cl ( N ) extracted fro that of. Proof. ollow fro Theore.5 ad Corollary.6. Exaple.8. (1) Let p, q be two prie uber i N, ad = { Z : Z pz}. We have: Z p (i) q Cl ( q Z) = q Z. Z p (ii) = q Cl ( q Z) = Z. () Let = p p p r 1 1 r be a priary decopoitio for N \{ 0, 1}, be a Gabriel topology defied o Z, ad J = { j : 1 j ad p j Z }. Z (i) If J =, the Cl ( Z) = Z. Z j (ii) If J, the Cl ( Z) = Z, with = pj. j J (3) Let R = K[ X, Y ], with K a coutative field. ( X, XY ) = ( X ) ( ax + Y, X ) i a priary decopoitio of ( X, XY ) ( for ay a K ), becaue ( X ) i a prie ad the i priary, ad the radical of ( ax + Y, X ) i ( X, Y ) which i axial, the ( ax + Y, X ) i priary. Next, let = { I ideal of R : I ( X )}. The Cl R R R (( X, XY )) = Cl (( X )) Cl (( ax + Y, X )) = ( X ) R = ( X ). 3. Aociated Prie Ideal ad Gabriel Topology I thi ectio, we uppoe agai that R i Noetheria ad i of fiite type ad we will reproduce oe of our reult obtaied i the previou
6 . EL HAJOUI, A. IRI ad A. ZOGLAT 106 ectio uig the otio of the aociated prie ideal. Theore 3.1. Let N be a P -priary R-ubodule of. The, we have: (i) P if ad oly if Cl ( N ) = N. (ii) If Cl ( N ), the Cl ( N ) i P -priary. Proof. (i) Suppoe P. Theore 6.6 i [5] give rie to the followig equivalece: N i P -priary if ad oly if A ( N ) = { P}. However, ice N i P -priary, P A( N ) ad o we ca fid x \ N uch that P = A( ) which iplie P x. Thu, we have x Cl ( N )\ N ad x 0 0 N 0 0 therefore Cl ( N ) N. Thi how that P wheever Cl ( N ) = N. The covere i give by Propoitio.1. (ii) I view of Theore.4, we have Cl ( N ) i a Q -priary ad P Q. Alo, by Theore 6.6 i [5], we ee that A( / Cl ( N )) = { Q}. The there exit x \ Cl ( N ) uch that Q = A( ) ad hece 0 Qx0 Cl ( N ). or every Q, there exit I uch that I x 0 Cl ( N ). But I x0 N ( becaue x0 Cl ( N )), the there exit β I with βx 0 N ad β x. O the other had, for N beig P-priary, we ca fid 0 N N atifyig N. Therefore P = A ( N ). Thi prove that Q P ad hece P = Q. Theore 3.. Let Cl (( 0 )) = ( 0) the Cl ( N ) i ijective. N be two R-odule. If N i ijective ad Proof. Suppoe the cotrary, i.e., Cl ( N ) i ot ijective. The, uig Theore B.4 i [5], we ee Cl ( N ) ha a proper eetial exteio L. But ice N i ijective, N doe ot have ay eetial exteio ad the N i ot eetial i L. Therefore, there i L L atifyig L ( 0), L N = ( 0) ad L Cl ( N ) ( 0). Now, for L0 = L we have: ( 0) L Cl ( N ) L = L0 ad L0 N = ( L ) N = ( L N ) = ( 0). Next, becaue L 0 i a ubodule of L ditict of (0), the x 0
7 L PRIARY DECOPOSITION, ASSOCIATED PRIE IDEALS 107 Cl ( N ) ( 0). urtherore, ice Cl (( 0) ) = ( 0 ) we have ( 0) L 0 0 Cl ( N ) Cl ( L0 ) Cl ( N ) = Cl ( L0 N ) = Cl (( 0) ), a cotradictio. Propoitio 3.3. If i a R-odule, the A ( ) = A( Cl (( 0) )). Proof. If P A( ), there exit x0 \ ( 0) uch that P = A ( x 0 ) ad o P x 0 ( 0). x0 Cl (( ) ad P A( Cl (( 0) )). Coverely, let P A( Cl (( 0) )), the there exit x Cl (( 0) )\ ( 0) uch that P = A( ). Thu there exit I uch that I x 0 ( 0). Hece I P = A ( ) ad therefore, P A( ). x 0 x 0 Propoitio 3.4. Let N be a R-ubodule of. The, we have A ( N ) = if ad oly if Cl ( N ) = N. Proof. Cobiatio of Theore 6.1 i [5] ad Propoitio 3.3 yield the ollowig N A( N ) = A( Cl (( 0 ))) = Cl (( 0)) = ( 0) N 0 Cl ( N ) N = ( 0) Cl ( N ) = N. Corollary 3.5. If Cl (( 0 )) = ( 0) ad N i a R-ubodule of havig a uppleetary, the N i -cloed. Proof. If N ha a uppleetary i, the by Theore 6.8 of [5], it follow A( ) = A( N ) A( N ). Next, ice Cl (( 0 )) = ( 0), A ( ) = A( Cl (( 0) )) = A( ( 0) ) = ad alo A ( N ) =. Therefore Cl ( N ) = N. Propoitio 3.6. If N i a ijective R-ubodule of ad A ( ) =, the Cl ( N ) = N. Proof. Suppoe that Cl ( N ) N ad N i ijective. The, N doe ot
8 . EL HAJOUI, A. IRI ad A. ZOGLAT 108 have ay proper eetial exteio. Thi follow by applyig Theore B.4 of [5]. Therefore, there exit L Cl ( N ) atifyig L ( 0) ad L N = ( 0). Let x0 L \ ( 0). The x0 Cl ( N ). urther, we ca fid I uch hat I x0 N. Hece, I x0 Ix0 N L N = ( 0) ad the I x 0 = ( 0). O the other had, ice Rx 0 ( 0) ad R i Noetheria, the A ( Rx 0 ) ad there exit R uch that Q = A( x0 ) A( Rx 0 ). Coequetly, we have Q ( becauei A ( x0 ) Q) ad Q A ( ) =, a cotradictio. Referece [1] J. Ecoriza ad B. Torrecilla, ultiplicatio odule relative to torio theorie, Co. Algebra 3(11) (1995), [] J. Ecoriza ad B. Torrecilla, Relative ultiplicatio ad ditributive odule, Coet. ath. Uiv. Carolia 3() (1997), [3] J. Ecoriza ad B. Torrecilla, Divioriel ultiplicatio rig, Note i ath. 63, arcel-dekker, 004. [4] B. Stetrö, Rig of Quotiet, Spriger, Berli, [5] H. atuura, Coitative Rig Theory, Cabridge Uiverity Pre, 1986.
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Aerica Joural of Matheatic ad Statitic 03, 3(): 6-3 DO: 0.593/j.aj.03030.04 Left Quai- ArtiiaModule Falih A. M. Aldoray *, Oaia M. M. Alhekiti Departet of Matheatic, U Al-Qura Uiverity, Makkah,P.O.Box
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