Joural of Egeerg ad Natural Scece Mühedl ve Fe Bller Derg Sga 25/2 FACTORIZATION PROPERTIES IN POLYNOMIAL EXTENSION OF UFR S Murat ALAN* Yıldız Te Üverte, Fe-Edebyat Faülte, Mateat Bölüü, Davutpaşa-İSTANBUL Gelş/Receved: 27..25 Kabul/Accepted: 2.4.25 ABSTRACT We vetgate the factorzato properte o the polyoal exteo A[ X ] of A where A a UFR ad how that A[ X ] a U-BFR for ay UFR A. We alo coder the rg tructure A + XI[ X ] where A a UFR. Keyword : Factorzato, Polyoal rg. MSC uber/uaraı: 3F5, 3A5. TÇA HALKALARIN POLİNOM GENİŞLEMELERİNDE ÇARPANLARA AYIRMA ÖZELLİKLERİ ÖZET A br TÇA hala (tetürlü çarpalara ayrılable hala) ola üzere A ı polo geşlee A[ X ] üzerde çarpalara ayıra özelller araştırıyoruz ve herhag br TÇA hala A ç A[ X ] U-KÇA hala (U-Kııtlı Çarpalarıa Ayrılable Hala) olduğuu göteryoruz. A br TÇA ola üzere A + XI[ X ] yapııda halaları da göz öüe alıyoruz. Aahtar Sözcüler : Çarpalara ayıra, Polo halaları.. INTRODUCTION Oe of the a proble rg theory to detere whether the polyoal exteo of ay coutatve rg R poee or ot the properte belogg to R. The purpoe of th paper to vetgate whch factorzato properte exactly atfed the polyoal exteo R[ X ] of R where R a UFR. Let R be a coutatve rg wth detty. Ay eleet ab, R are aocate, deoted by a b, f ab ad ba, that, ( a) = ( b). A out a R a rreducble (or a ato) f a = bc a b or a c. Hece rreducble f ad oly f R a tegral doa. R atoc f each ozero out eleet of R a fte product of rreducble eleet. A prcpal deal rg (PIR) called a pecal prcpal deal rg (SPIR) f t ha oly oe pre deal P R ad P lpotet, that, P = () for oe teger >. R ad to e-pota: ala@yldz.edu.tr, tel: (22) 449 7 8 95
M. Ala Sga 25/2 be preplfable f x = xy x = or y U( R). SPIR ad tegral doa are exaple of preplfable rg. Clacal rreducble decopoto ay caue the bad behavor of factorzato for oe eleet a coutatve rg becaue of otrval depotet. For exaple 5 = 5 for every potve teger 2. A U-decopoto whch troduced by Fletcher [9] elate th bad behavor of a factorzato. A U-decopoto of r R a factorzato r = ( p... p )( p... p ) uch that () () the ' p ad p are rreducble ' p ( p... p ) = ( p... p ) for =,..., ad () p ( p... p p... p ) ( p... p ) for =,...,. + Let r = ( p... p )( p... p ) be a U-decopoto of r R. The the product p... p called the relevat part ad the other the rrelevat part. Ay rreducble decopoto ca be rearraged to a U-decopoto. Two U-decopoto r = ( p... p )( p... p ) = ( q... q )( q... q ) l are aocate f () = ad () p ad q are aocate for =,...,, after a utable chage the order of the factor the relevat part. R called uque factorzato rg (UFR) f each out ha a U-decopoto, that, R atoc ad ay two U- decopoto of a out eleet of R are aocate. Fletcher how that R a UFR f ad oly f R a fte drect product of UFD ad SPIR []. There are a faly of factorzato properte whch are weaer tha uque factorzato. For a detaled tudy of thee factorzato properte tegral doa ee Ref. [4,5,6]. I th paper we coder thee factorzato properte for a coutatve rg wth zero dvor. Ay coutatve rg R called a bouded factorzato rg (BFR) f for each ozero out a R, there ext a atural uber N( a ) o that for ay factorzato a = a... a of a where each a a out we have < N( a). If we replace the codto ' that p ad p are rreducble by p ' ad p are out the defto of U- decopoto, we have the defto of U-factorzato. A coutatve rg R a U-BFR f for each ozero out a R, there ext a atural uber N( a ) o that for each U- factorzato of a, a = ( a,..., a )( b,..., b ), < N( a) []. R a UFR R a U-BFR. R called a U-half-factoral rg (U-HFR) f R U-atoc (that, every ozero out ha a U- factorzato whch all the relevat dvor are rreducble) ad f a = ( a,..., a )( b,..., b ) = ( c,..., c )( d,..., d ) are two U-factorzato wth b, d rreducble, the l =. It clear that UFR U-HFR. For thee factorzato properte ee [,7,8]. For a tegral doa D t well ow that D a UFD DX [ ] a UFD. But f zero dvor are preet the tuato ot o clear. I [2] Adero ad Marada how that R[ X ] a UFR f ad oly f R a fte drect product of UFD'. I other word R[ X ] ot a UFR f R a SPIR. I fact R ot eve a U-HFR f R a SPIR. For exaple the eleet 2 2X ha two dtct rreducble decopoto (ad U-decopoto) whch a UFR, 4 aely 2 2 2X = 2(2 + X ) = 2. X. X. 96
Factorzato Properte Polyoal Exteo I here we cotue to vetgate thee factorzato properte R[ X ] where R a UFR ad gve a potve reult for the cocept a U-BFR ad a BFR. More precely we how that f R a SPIR the RX [ ] a BFR ad f R a UFR the RX [ ] a U-BFR. We alo coder thee factorzato properte the rg R + XI[ X ] for ay deal I of R where R a UFR. I [2] Gozalez, Peler ad Robert how that A + XI[ X ] a HFD (half-factoral doa) f ad oly f I a pre deal of A for the doa cae where A a UFD. I here we vetgate whch factorzato properte are atfed the rg A + XI[ X ] for ay deal I of A where A a UFR ad how that A + XI[ X ] alway a U-BFR for ay deal I of A. For ay udefed terology or otato, ee []. 2. BOUNDED FACTORIZATION PROPERTIES ON A[ X ] Suppoe A a SPIR ad P = ( p) the uque pre deal of A, where P = () ad the allet teger whch atfe P = (). If = the A a feld. So fro ow o, we aue that > ad A, P ad wll be a above ule otherwe tated. Propoto 2.: Let A be a SPIR. If f = a + a X +... + a X A[ X] ay rreducble eleet A[ X ] the f ( X ) oe of the followg for up to aocate: () f = p () f = X + () f = a + a X +... + a X + X + a X +... + a X for oe where + a, a,..., a P + + (v) f = + a X +... + a X + X + a X +... + a X for oe where + a,..., a P. + Proof : Suppoe a P for all. We ca wrte a = pa a A. The f = p( a + a X +... + a X ). Sce f ( X ) rreducble A[ X ], f p or f f where f = a + a X +... + a X. f p p= f. c cx ( ) AX [ ] f = f. c f. Sce A[ X ] preplfable ad f ( X ), cx ( ) f a ut A[ X ], ad hece f ( X ) a ut A[ X ]. So we ay tae f = p up to aocate. If f f the f = f. d d A[ X] f = p. f. d pd. U( A). But th a cotradcto ce p ot a ut. Therefore f all a P the f = p up to ut. Now uppoe f P[ X]. Coder the cotat ter of f ( X ) : Cae I: Suppoe a =. The f = X. f f A[ X]. So ether f X or f f. f f gve X a ut a above whch a cotradcto. So f X ad f ( X ) a ut A[ X ]. Hece we tae f = X up to aocate. Now a. The ether a P or a U( A). 97
M. Ala Sga 25/2 Cae II: Let a P. The for oe a P ce f P[ X] U( A). So we ay tae a =. a. So a P Cae III: Let a U( A). We ow that ay eleet of A[ X ] a ut A[ X ] f ad oly f the cotat ter a ut A ad all other coeffcet are lpotet A. Ay out eleet of A a lpotet. So a ut be a ut A for oe ce f( X ) ot a ut A[ X ]. So aga we ay tae a =. Propoto 2.2: If A a SPIR the A[ X ] a BFR. Proof: Let f = a + a X +... + a X A[ X] be a ozero out eleet of A[ X ]. Frt ote that A[ X ] preplfable ce prary A [3]. Collectg all the factor p α each coeffcet a, we ca wrte uquely each a a a = p u where α ad α α α U( A). The f = p u + p u X +... + p u X A[ X]. Sce f the uber of u rreducble factor p ay rreducble decopoto of f ( X ) at ot. Now we how that ay factorzato of f ( X ) to a product of rreducble eleet the uber of rreducble factor a Propoto 2. at ot deg f =. It obvou that the uber of rreducble factor X ' are at ot. Let f = p... px... Xf... f be a rreducble decopoto of t f ( X ) where f rreducble a Propoto 2. other tha p ad X. Let f = a +... + a X + X + a X +... + a X, f b b X X a X a X + l + l 2 2 2 = +... + + + + +... + 2 2 2+ where a ad b 2 are the lat coeffcet of f ad f 2 whch equal to, repectvely. 2 Coder the coeffcet of X + of the product f f : 2 So + 2 ab. = + 2 > a P a b P, If + 2 < b P a b P, f + 2 + 2 = a = b = a b =. f + 2 + 2 + 2 a = + pr, r A. = + 2 Sce + pr P, + pr a ut A, ad hece ot a zero eleet. Hece by... t ducto o t we ca ee that the coeffcet of X + + the product f... f ot zero t where. So f t > the deg( f ) < deg( p... px... Xf... f ) t whch a cotradcto. Thu t ad hece A[ X ] a BFR. Followg theore []. For a coutatve rg R, a R called U-bouded f up{ a= ( a,..., a )( b,..., b ) a U-factorzato of a} <. Theore 2.3: Let R, R,..., R be coutatve rg, >, ad let R = R... R. The R 2 a U-BFR each R a U-BFR ad U-bouded. Hece R U-bouded. R Now we ca tate the a reult of th paper a a corollary. 98
Factorzato Properte Polyoal Exteo Corollary 2.4: If A a UFR the A[ X ] a U-BFR. Proof : Sce A a UFR, A a fte drect product of UFD ad SPIR, ay A = A... A. The A[ X] = A[ X]... A[ X]. If A a UFD the clearly A [ X ] UFD ad hece a U-BFR. If A a SPIR the A [ X ] a U-BFR by Propoto 2.2. Hece by Theore 2.3, A[ X ] U-BFR. Lea 2.5: Let A B be a exteo of coutatve rg. If B a BFR ad U( B) A= U( A) the A alo a BFR. Proof: Let a A be a ozero out ad let a = a... a be ay factorzato of a to out. The a U( B) for =,...,. Hece a = a... a a factorzato of a B. Sce B a BFR, N( a) for oe potve teger Na. ( ) So A a BFR. Propoto 2.6: If A a UFR the A + XI[ X ] a U-BFR for ay deal I of A. Proof : Let A = A... A, a fte drect product of UFD' ad SPIR'. The I of the for I = I... I where each I a deal of A A. So XI[ X ] = ( A... A ) + X( I... I )[ X] = ( A + XI [ X ])... ( A + XI [ X ]) + If A a UFD the by Lea 2.5 A + XI [ X ] a BFR ce A + XI [ X ] A [ X ] ad U( A + XI [ X]) = U( A[ X]) = U( A). If A a SPIR the A + XI [ X ] aga a BFR ce A [ X ] a BFR by Propoto 2.2 ad U( A[ X]) ( A + XI [ X]) = U( A + XI [ X]). Clearly A alway U-bouded each cae. Hece A + XI[ X ] a U-BFR for every deal I of A by Theore 2.4. 3. RESULTS AND DISCUSSION I th paper we cocetrate o polyoal exteo of a UFR wth zero dvor to vetgate factorzato properte related to t. We how that f A a UFR the A[ X ] alway a U-BFR ad A + XI[ X ] alway a U-BFR for ay deal I of A. We do ot ow thee reult are rea vald f the ter U-BFR ubttuted for UFR. REFERENCES [] Ağargü A.G., Adero D.D. ve Valde-Leo S., Factorzato I Coutatve Rg Wth Zero Dvor, III, Rocy Mouta J. of Math., 3, -2, 2. [2] Adero D.D., Marada R., Uque Factorzato Rg Wth Zero Dvor, Houto J. Math.,, 5-3, 985. [3] Adero D.D., Valde-Leo S., Factorzato I Coutatve Rg Wth Zero Dvor, Rocy Mouta J. of Math., 2, 2, 439-48, 996. [4] Adero D.D., Adero D.F., Zafrullah M., Factorzato I Itegral Doa, J. of Pure ad App. Algebra, 69, -9, 99. [5] Adero D.D., Adero D.F. ve Zafrullah M., Factorzato Itegral Doa II, J. Algebra, 52, 78-93, 992. [6] Adero D.F. ve El Abde D.N., Factorzato I Itegral Doa, III, J. of Pure ad App. Algebra, 35, 7-27, 999. [7] Axtell M., U-Factorzato Coutatve Rg Wth Zero Dvor, Co. Algebra, 3, 22. 99
M. Ala Sga 25/2 [8] Axtell M., Fora S., Roera N. et. al., Properte of U-Factorzato, It. J. of Co. Rg, 3, 23. [9] Fletcher C.R., Uque Factorzato Rg, Proc. Cab. Phl. Soc., 65, 579-583, 969. [] Fletcher C.R., The Structure Of Uque Factorzato Rg, Proc. Cab. Phl. Soc., 67, 535-54, 97. [] Gler R., (972), Multplcatve Ideal Theory, Marcel Deer, New Yor. [2] Gozalez N., Peler, Robert R., Elatcty Of A+XI[X] Doa Where A I UFD, J. of Pure ad App. Algebra 6, 83-94, 2.