
baseout kshetra

ganitat, bezout kshetra heey proofer kshetrache eka roop aahe. heey eka avibhajya kshetra aahe jyamadhye done pramukh aadarshanchi berries punha eka pramukh aadarsh aahe. yacha arth assa qee ghatakanchya pratyek jodisathi eka baseout olakh asate aanee pratyek maryaditpane tayar kelele aadarsh heey mukhya asate. konatehi pramukh aadarsh kshetra (pii. aay. dii.) heey eka bezout kshetra aahe, parantu bezout kshetra heey notherian ring asanyachi aavashyakta nahee, tyamule tyat antim naan tayar jhalelya aadarsh asoo shakatat (jeey spashtapane pii. aay. dii. asane vagalate); asey asalyas, teey eka adviteey factorization kshetra (yuu. eff. dii.) nahee, parantu tarihi teey eka jee. see. dii. kshetra aahe. bezout kshetrancha siddhant notherian malmattechi aavashyakta naan thevata, pidsche anek gunadharm rakhoon thevato. bezout kshetranna french ganitshastradnya etian bezout yanche naav denyat aale aahe.
malmatta.
ring heey eka baseout domain aahe, jarr aanee keval jarr teey eka avibhajya domain asel jyamadhye konatyahi done ghatakanche sarvat mothe samaan vibhajak asel jeey tyanche eka resheey sanyojan aselah heey yaa vidhanachya samatulya aahe qee done ghatkandware tayar kelele ideal dekhil ekach ghatkadvare tayar kele jaate aanee induction heey darshavite qee serve maryaditpane tayar kelele ideals heey mukhya aahet. pii. aay. dii. chya done ghatakanchya sarvat mothya samanya vibhajkachya resheey sanyojan mhanoon abhivyaktila anekada bezoutchi olakh mhanatat, jithun shabdavali yete.
vareel jee. see. dii. sthitee hii keval jee. see. dii. chya astitvapeksha adhik majboot aahe heey lakshat ghya. eka avibhajya kshetra jethe konatyahi done ghatkansathi jee. see. dii. astitvat asate tyala jee. see. dii. kshetra mhanatat aanee asha prakare baseout kshetra heey jee. see. dii. kshetra asatat. visheshtah, bezout domenmadhye, irreducibles heey avibhajya asatat (parantu bijganitiy purnankache udaharan darshavite tyapramane, teey astitvat asanyachi aavashyakta nahee).
bezout domain "aar" sathee, khaleel serve atty samatulya aaheth
(1) aanee (2) chee samanata varr namood kelee gelee hotee. bezout domain heey jee. see. dii. domain asalyane, teey lagech asey darshavate qee (3), (4) aanee (5) samatulya aahet. shevati, jarr "r" heey notherian nasel, tarr maryaditpane nirman jhalelya idealschi eka anant chadhati sakhali astitvat aahe, mhanoon bezout domenmadhye mukhya idealschi eka anant chadhati sakhali aahe. (4) aanee (2) asha prakare samatulya aahet.
baseout domain heey eka proofer domain aahe, i.eae.eka kshetra jyamadhye pratyek maryaditpane tayar kelele aadarsh ulat karata yenyajoge asate, kinwa dusya prakare mhatale tarr, eka parivartaneey ardh-vanshparamparagat kshetra.)
parinami, ekhadi vyakti 'bezout domain iff proofer domain aanee gcd-domain' hii samanata adhik parichit 'pid iff dedeckind domain' shhee samaan mhanoon pahoo shakate aanee 
yuu. eff. dii. ".
proofer domain heey avibhajya domain mhanoon darshavile jaau shakatat jyanche sthanikikaran serve prime (samatulya, serve kamal) ideals heey moolyankan domain aahet. tyamule prime idealver baseout domenche sthanikikaran heey moolyankan kshetra aahe. sthanik ringmadhil invertible ideal heey principle asalyane, jarr teey moolyankan kshetra asel tarr sthanik ring heey baseout kshetra aahe. shivay, noncyclic (samatulya naun-discreet) moolya gut asalele moolyankan kshetra notherian nasate aanee pratyek purnapane krambaddha kelela abelian gut haah kahee moolyankan kshetracha moolya gut asato. heey naun-noetherian bezout kshetranchi anek udaharane dete.
gair-sampreshaniy bijganitemadhye, yogya bezout kshetra hii ashee kshetra aahet jyanche maryaditpane tayar kelele yogya aadarsh heey mukhya yogya aadarsh aahet, mhanje "r" madheel kahee "x" sathee "xr" swarupat. eka ullekhaneey parinam mhanje yogya baseout kshetra heey yogya dhatoo kshetra aahe. heey tathya parivartanatmak prakaranat manoranjak nahee, kaaran "pratyek" parivartaneey kshetra heey dhatoo kshetra aahe. ujavya bezout kshetranmadhye dekhil ardh-vanshparamparagat rings aahet.
baseout kshetravaril modules.
pii. aay. dii. vareel modulabaddal kahee tathye baseout domenvaril modulesparyant vistartat. r" heey eka baseout kshetra asoo dya aanee "r" varr "m" heey maryaditpane tayar kelele module asoo dya. jarr aanee fakt jarr teey tortion-mukt asel tarach 'm' sapat asel.