Adagba O Henry /International Journal Of Computational Engineering Research / ISSN: OPERATION ON IDEALS

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1 Adagba O Hry /Itratoal Joural Of Computatoal Egrg Rsarh / ISSN OPERATION ON IDEALS Adagba O Hry, Dpt of Idustral Mathmats & Appld Statsts, Eboy Stat Uvrsty, Abakalk Abstrat W provd bas opratos o dals suh as addto, trsto, multplato, th formato of dal quotts, radals, ad th xtsos ad otratos of dals Kywords Idals, Commutatv Algbra, Itgral Doma, Fld, Commutatv Rgs, Extsos of Idals, Cotrato of Idals Mathmats Subt Classfato 46H0, 5B0, 3Gxx, 6S70 0 Itroduto Throughout ths work w shall arry out rta xrss st by Profssor MA Atyah hs Nots o Commutatv Algbra Ths moograph was frst publshd mmograph by Mathmatal Isttut at Oxford Uvrsty 965 Th whol of our prstato rls havly o Atyah s prototyp whh was latr publshd as our rfr [] by Addso Wsly 969 W dsuss gral mthod by whh o a dtrm th oprato o dals that s, bhavor of th dals a ommutatv utal rg By opratos, w ma bas opratos o dals suh as addto, multplato, trsto, th formato of dal quotts, radals, th xtsos ad otrato of dals If 0 b a rg, th has a maxmal dal ad a mmal prm dals s th maor obtv of ths work I spf ass whh hav b xtsvly studd ths qustos ar xtrmly hard to aswr Th ltraturs ovrd by ths study ar farly xtsv, s for xampl [3], [4], or [6] W osdr th formato of radals of dals whh s a atural osdrato th otxt of soluto of quatos ad th fatorzato of lmts ommutatv rgs Lt a b a dal of Th radal of x a for som tgr a, r a s th st of all x, suh that (or quvaltly, t s th st of lmts x whos mag x th fator rg a s lpott) Rtly Lpma [5], Eak t-al [8] ad Sally t-al [9] hav rmovd th assumpto o haratrst W a rovr ths rsult Idd, w fd osdrably mor Johso [7] has oturd that maxmal dals rdus th tralzrs ad oprators ad Eago t-al [] has oturd that dals dfd by matrs ad rta omplx assoatd to thm hav a uqu proprts W ar abl to show Dfto Wh w say that s a rg, w shall ma that multplato s ommutatv ad that th multplatv dtty, dotd by, also blogs to Morovr, 0, whr 0 s addtv dtty Also; f, ar rgs, a rg homomorphsm f Is a mappg suh that whvr x, y, w hav f x y f x f y f xy f x f y f W shall dot th dal of multpls of a lmt x by x That s x ax a I gral, w doat a dal of by otato a, b, p, m t I ths work, w dsuss th bas oprato o dals, suh as addto, trsto, multplato, th formato of dal quotts, radals, ad th xtsos ad otratos of dals W start wth th lass of dals whh ar by far th most mportat Commutatv Algbra By dfto, a dal p s to b th prm dal of f IJCER Mar-Apr 0 Vol Issu No 7-77 Pag 7

2 Adagba O Hry /Itratoal Joural Of Computatoal Egrg Rsarh / ISSN () p,, ad () xy p xp or y p Part of th raso for th mporta of prm dals ls th followg proposto whh w stat wthout proof Proposto A dal of s a prm dal f ad oly f ts assoatd quott rg A dal m s sad to b maxmal f m ad () a s a tgral doma () a s a dal suh that m a ; th thr a m or a W prov that vry maxmal dal s a prm dal by obtag th followg rsults 3 Proposto A dal a of s maxmal f ad oly f ts assoatd quott a s a fld Suppos a s a maxmal dal of, th a ad so a 0, th zro rg For ay x, w wrt x x a x y y a ad suppos that x 0 a To fd ts vrs, w ot that a x a a x Morovr, w hav a a x ; h ax Ad so thr xsts y suh that xy mod a H x y a Ths provs that a s a fld Covrsly, suppos that s a dal suh that a s a fld ad a b for ay dal b of, th frst luso bg strt Lt x, yb, y a Th y 0 a S a s a fld, o a fd suh that x y z x yz a b x b, s y b, by hypothss Ths provs that b b H a s maxmal Q E D prm Combg () ad (3), t s lar that vry maxmal dal s prm Th ovrs s obvously fals, s 0 s, th rg of tgrs, wthout bg maxmal Nxt, to dmostrat th abuda of prm dal, w prov th followg 4 Proposto Lt 0 b a rg, th has a maxmal dal Lt S b th st of dal a of th rg By hypothss, 0 ad so S s o-mpty W a thrfor b I S, so that for ay, I thr b b or b b ordr S by luso Cosdr ay asdg ha Cosdr th st b b, w lam that b s a dal I Idd, f x b, th x b for som I H, y b for som I ad wthout loss of gralty, w may assum, a ax b b b Furthrmor, f, b b x y b x y b b x y b, th Thus, our lam has b stablshd Morovr, s I b by hypothss, w ddu that b H b S Thus, ay asdg ha S has uppr boud S ad so by zor s lmma S has a maxmal lmt, m say Ths provs our proposto 5 Proposto If 0 s a rg, th has a mmal prm dal IJCER Mar-Apr 0 Vol Issu No 7-77 Pag 7

3 Adagba O Hry /Itratoal Joural Of Computatoal Egrg Rsarh / ISSN Lt b th st of prm dals s o-mpty by trsto s a dal a p p for som I I pp 4 Lt p b a ha of prm dals Thr To prov that a s dd a prm dal, suppos that xy a, y a, th y p for som I S xy p, I, by th supposto that xy a ad by hypothss p s prm, t follows that x p Morovr, f p p, th y p y p, ad by th argumt, w hav ust usd, w ddu that x p Thus xa p p p Clarly a, s I p Thus, a s a prm dal Morovr, s p p p for all, t follows that ay ha has a lowr boud ad by zor s lmma, has a mmal lmt Ths provs our proposto Nxt w tur to som rsults rlatd to th formulato of dals quotts Lt, a b b dals a ommutatv rg, th dal quott of a by b wrtt a b s dfd by a b x xb a 6 Proposto Lt ab, ad b dals of rg, th () a a b () a bb () a a b a b a b (v) a b a b (v) a b a b () a a b By dfto of dal, xa xb a (baus a s a dal) x a b a b a a b () By dfto a bb s gratd by produts of th form xy whr x a b ad y b xa b, yb xy a H, ah grator of a bb ls a ad so () Lt x a b ad osdr ay grator yz of b Th z zxa b yzx a, s y b b to a lmt of a xb a x a b a b a b H Nxt, lt u a b For ay lmt v, w b, w hav Lt sa b, t b, r Th a b a b provg qualty of th gvg dals I by dfto of But th a b b a as rqurd H multplato by x trasforms vry grator of vw b b uvwa uw a u a b a b a b st a str a srb a s a b s a b IJCER Mar-Apr 0 Vol Issu No 7-77 Pag 73

4 Adagba O Hry /Itratoal Joural Of Computatoal Egrg Rsarh / ISSN (v) Lt x a b ; th xb a a b a b for ah x a b for ah x a b Covrsly, y a b y a b for ah yb a b y a b a b a b Ths provs qualty (v) Lt xa b b a I partular, b b xb a x a b x a b H a b a b b u y y y, whr y b, th for ah S a lmt of s of th form z a b zy a z a b a b a b By Axom of Extso a b a b Nxt, w osdr th formato of radals of dals, whh s a atural osdrato th otxt of soluto of quatos ad th fatorzato of lmts ommutatv rgs Lt a b a dal of Th radal of a, r a s th st of all x, suh that quvaltly, t s th st of lmts x whos mag x th fator rg 7 Proposto Lt a, b b dals of a rg ad p b a prm dal of Th () r a a () rra r a () rab r a b r a r b (v) r a a (v) f p s a prm, r p (v) ra b r ra r b p for som 0 () f x a, th takg, w hav () By (), ra r r a Covrsly, H a r a x x r a m for som 0 m m That s, rra r a x r r a x r a for som m 0 x a for 0 () ab ab rab ra b Also, lt x a b x a, th x a s lpott) x a b for som a for som tgr (or m m a ad y y y ab y r ab H, 0 x a, x b xr a, xr b xr a r b r a b r a r b Fally, lt y ra rb, th y, 0 rarb rab ad w hav th ha of luso rab ra b ra rb r a b By axom of xtso, w ddu that IJCER Mar-Apr 0 Vol Issu No 7-77 Pag 74

5 Adagba O Hry /Itratoal Joural Of Computatoal Egrg Rsarh / ISSN rab r a b r a r b (v) ra a for som 0 a a a a a ra by a ra ra (v) a b ra rb by ra b rra rb Covrsly, lt yr r a r b, th y ra r b Suppos y u v t 0, s 0 H smultaously H (v) Lt x r p (), whr u r a, v r b ts ts k y u v u v k u v a b Thus y r a b, th x p for 0 r p r p p p Q E D for som 0 t s wth ds ts;, that s, u a, v b for whr t s mpossbl for s ad k t x p, s p s prm H p r p p Thus r p p by Nxt, w osdr xtsos ad otratos of dals Idd, lt, b ommutatv utal rgs ad lt f b a rg homomorphsm Th xtso of a dal a of rlatv to f dotd by gratd by f a th mag of a udr f That s, dotd by a f a y y f x x a, y Covrsly, f b s a dal, th th vrs mag, f b b ad s alld th otrato of b dud by f That s, 8 Proposto Lt a, a b dals of, b, b (), (), (), a a a a b b b b a, s th dal b of s asly vrfd to b a dal Ths dal s b f b dals of ad f b ay rg homomorphsm a a a a b b b b a a a a b b b b (v) a a a a, b b b b (v) r a r a, r b r b (v) Th st of xtsos s losd udr th oprato of sum ad produt whl th st C of otratos s losd udr th rmag thr () (a) Lt xa a, th x uf x so that f x f x f x, x u f x u f x u f x a a H a a a a whr x a a Lt x x x for som,, whr x a x a But th, w hav Th w u f x v f y x a a a, y a a a H wa a That s, a a a a, whr IJCER Mar-Apr 0 Vol Issu No 7-77 Pag 75

6 Adagba O Hry /Itratoal Joural Of Computatoal Egrg Rsarh / ISSN , whr f x b ad f x Thus, f x f x x f x f x b b Wh xb b That s, b b b b (b) Lt x b b, th x x x b ()(a) Lt xa a, th xu f x x a a But th x a, x a x u f x a ad x uf x a That s, x uf x a a H a a a a (b) If xb b f xb b f x b,, Covrsly, lt y b, b th f yb, f y b H f y b b y b b b b b b f x b xb xb xb b b b b b It follows that b b b b ()(a) Lt x a a Th x u f x, whr, x aa for ah S x v w, v a, w a ad u f x u f v w u f v f w a a for ah It follows that x a a H a a a a Covrsly, lt y a a, th y uv whr u a ad v a For ay, w hav u z f x, x a, z, v wk f sk, sk a, wk ad uv z f x wk f sk z wk f x f sk z wk f x sk, x a, sk a uv z wk f x sk aa for ah That s, y aa Thus a a a a Fally, a a a a H, whr f x b ad f y b (v)(a) Lt x a a, th x u f x whr u ad x a a (b) Lt x b b, th x xy f x f x y f x f y bb x bb b b b b all Lt y a, th y v f y whr v, y a xy a, for all ad t follows that xy u f x v f y S Thus xy u v f x y a a a x a a a a a a (b) Lt y b b f y b b Lt x b, th f x b f y f x b f xy b yx b b y b y b b b b b b (v)(a) Lt y r a, th y u f y whr y r a IJCER Mar-Apr 0 Vol Issu No 7-77 Pag 76 for ah That s, Thus x a a, for

7 Adagba O Hry /Itratoal Joural Of Computatoal Egrg Rsarh / ISSN y H y uf y r a Thus r a r a a for som 0 f y f y a f y r a for ah (b) yr b f y r b f y b for som 0 y b yr b r b r b m Covrsly, f x r b, th x b for som m 0 Thus f x m b f x m b f x r b x r b r b r b H r b r b (v)(a) Th st of xtsos s losd udr th opratos of sum ad produt Idd, ()(a) a a a a ()(a) a a a a ad ths surs that th sum of two xtsos s tslf a xtso Morovr shows th produt of two xtsos s tslf a xtso Th st C of otratos s losd udr trsto by vrtu of ()(b) b b b b hav a ad s losd udr th formato of radals by vrtu of vb, r b r b To prov that C s losd udr th formato of dal quott, w frst ot that for ay dal a, w a ad also for b, w hav b b H, w hav th qualty, b b b b Coluso Our dsusso of oprato o dals (ad dals of ommutatv rgs as a spal as), hlps to xpla th suprm mporta of prm dals ommutatv Algbra Itutvly w osdr th formato of radals of dals, whh s a atural osdrato th otxt of soluto of quatos ad th fatorzato of lmts ommutatv rgs Th xtso of a dal a of rlatv to f dotd by That s, dotd by a, s th dal b of gratd by a f a y y f x x a, y Covrsly, f b s a dal, th th vrs mag, f b b ad s alld th otrato of b dud by f That s, f a th mag of a udr f s asly vrfd to b a dal Ths dal s b f b Our rsults (4) ad (5) shows that th dals of o-trval utal rg form a omplt latt Ths s a proprty whh A- modul dos ot shar Rfrs Atyah, M F(969), Itroduto to Commutatv Algbra Addso-Wsly Eago J ad Northoth D (96), Idal dfd by matrs ad a rta omplx assoatd to thm, pro, Royal so A69, Esbud D ad Evas EG(976), A gralzd prpal dal thorm, Nagoya Math J 6, Hrks, M(953), O th prm dals of th rg of tr futo Paf JMath 3, Lpma J (97), Stabl dals ad Arf rgs, Amr J Math 93, Rs J (96), Trasforms of loal rgs ad a thorm o multplts of dals, pro Combrdgs phlosso57, Johso BE (968), tralzrs ad oprators rdud by maxmal dals, J Lodo Math So 43, Eak P ad Sathay A (976), prstabl dals, J of Alg 4, Sally J D ad Vasolos W V (975), Flat dal, omm Alg 3, IJCER Mar-Apr 0 Vol Issu No 7-77 Pag 77