mukhya aadarsh kshetravaril maryaditpane tayar kelelya modulesathi sanrachana pramey

ganitamadhye, amoort bijganitachya kshetrat, mukhya aadarsh kshetravaril maryaditpane tayar kelelya modulesasathiche sanrachana pramey heey maryaditpane tayar kelelya abelian gatanchya moolbhut prameyanche samanyikaran aahe aanee andaje asey namood karate qee mukhya aadarsh kshetravaril (pii. aay. dii.) maryaditpane tayar kelele modules purnankanchya mukhya ghatakatvapramanech advitiyapane vighatit kele jaau shakatat. nikal kshetranvaril varg matrixsathi vividh pramanit swarupache parinam samajoon ghenyasathi eka sopee chaukat pradan karato.
vidhan.
jevha 'f' kshetravaril vector spacemadhye maryadit generating sanch asato, tevha ekhadi vyakti tyatoon vectorchya maryadit sankhya 'n' asalela aadhar kadhu shakate aanee tyamule jaaga 'f' n 'sathee samaroop aahe. f" saha sambandhit vidhan, jeey mukhya aadarsh kshetra "r" laa samanyikrut kele aahe, teey aata kharey nahee, kaaran "r" varr maryaditpane tayar kelelya modulesathi aadhar astitvat nasel. tathapi, asey module ajunahi kahee modulechya bhagankasathi samaroop aahe, jyamadhye 'aine' maryadit aahe (heey pahanyasathi modulechya generatorna 'rn' chya vihit aadharatil ghatak pathvinare aanee tyachya karnaldware bhagank ghenare rupankan tayar karane purese aahe.) sanch tayar karanyachi nivad badaloon, pratyakshat ekhadya vishisht sadhya upaghatkadvare kahee "aar. aine." chaa bhag mhanoon moduleche varnan kele jaau shakate aanee heey sanrachana pramey aahe.
mukhya aadarsh kshetravaril maryaditpane tayar kelelya modulesathi sanrachana pramey samanyatah khaleel done swarupat disun yeto.
aparivartaniy ghatak vighatan.
mukhya aadarsh kshetravaril pratyek maryaditpane tayar kelelya modulesathi, yogya aadarsh sootra _ 1 chaa eka adviteey ghatanara kram aahe joou chakriy moduleschya berijashi samaroop aaheh
aadarshanche generator sootra _ 3 heey eka ekkane gunakar karnyaparyant adviteey asatat aanee tyanna 'm' chey aparivartaniy ghatak mhanatat. aadarsh yogya asavet mhanoon, heey ghatak swatah ulat karata yenyajoge nasavet (yamule bereejmadhye kshullak ghatak talhata yetat) aanee aadarshancha samavesh mhanje aekhadyache vibhajyata sootra 4 aahe. mukt bhag ghatak sootra 5 shhee sambandhit vighatan bhagat drishyaman aahe. asey ghatak, jarr asateel tarr, anukramachya shevati udbhavtat.
thet berries advitiyapane nirdharit kelee jaat asatana, swatah vighatan denari samarupata sarvasadharanpane 'adviteey nahee'. udaharanarth, jarr pratyakshat eka kshetra asel tarr serve serve ideals shoonya asane aavashyak aahe aanee ekhadyala maryadit aayami vector jaageche vighatan eka-aayami upasthananchya thet bereejmadhye milate; asha ghatakanchi sankhya nishchit aahe, mhanje jaageche pariman, parantu upasthan swatach nivadanyasathi barech swatantrya aahe (jarr).
shoonya nasalelya sootra _ 3 ghatak, shoonya asalelya sootra _ 3 chya sankhyes, modulesathi aparivartanancha sampoorn sanch tayar karatat. spashtapane, yacha arth assa aahe qee aparivartanancha samaan sanch samayik karnare konatehi done modules anivaryapane samaroop aahet. 
kahijan 'emm' chaa mukt bhag swatantrapane lihinyas pradhanya detatah
jethe drishyaman sootra _ 3 heey shunyetar aahe aanee "f" hii mool kramatil sootra _ 3 chee sankhya aahe jee 0 aahe.
prathamik vighatan.
sootra _ 14 yaa ghatakanna 'm' chey 'prathamik vibhajak' mhanatat. pii. aay. dii. madhye, shunyetar prathamik ideals heey avibhajyanchya shakti asatat aanee mhanoon sootra 16. jevha sootra 17 asate, tevha parinami asanyojit module sootra 18 asate aanee heey "m" chya bhagachya aat asate jeey eka mukt module aahe.
saransh sootra _ 19 heey apghatanaksham aahet, mhanoon prathamik apghatan heey apghatansheel modulesmadhye apghatan aahe aanee asha prakare pii. aay. dii. varr pratyek maryaditpane tayar kelele module heey purnapane apghatansheel module aahe. pii. aay. dii. heey notherian ring asalyane, heey laskar-nother prameyache prakatikaran mhanoon pahile jaau shakate.
purvipramanech, mukt bhag (jethe sootra _ 17 aahe) swatantrapane lihine aanee 'm' asey vyakt karane shakya aaheh
jethe drishyaman sootra _ 22 shunyetar aahe.
purave.
eka purava khalilpramane pudhe jaatoh
yamule aparivartaniy ghatak vighatan hote aanee smith normal formchya karn pravishtya aparivartaniy ghatak aahet.
puravyachi aanakhi eka rupareshah
parinam.
yamadhye eka vishesh prakaran mhanoon maryadit-aayami vector spacesche vargikaran samavisht aahe, jethe sootra _ 27. fieldsna kshullak ideals naslyamule, pratyek maryaditpane tayar keleli vector space mukt aahe.
sootra _ 28 ghetalyane maryaditpane tayar jhalelya abelian gatanche moolbhut pramey milate.
tii" haah "keey" varr maryadit-aayami vector space "vhee" varr eka resheey operator asoo dya. sootra _ 29 ghetalyas, 'tii' varr moolyankan kelelya 'keey' madheel gunank asalelya bahupadiche bijganit, 'tii' baddal sanrachana mahitee dete. "v" kadey sootra _ 30 varr maryaditpane tayar kelele module mhanoon pahile jaau shakate. shevatcha aparivartaniy ghatak mhanje kiman bahupadi aanee aparivartaniy ghatakanche gunakar heey vaishishtyapurn bahupadi aahe. sootra _ 31 sathee pramanit matricks forms ektrit karoon, heey vividh pramanit swaroop deteh
advitiyata.
jaree aparivartak (shreni, aparivartaniy ghatak aanee prathamik vibhajak) adviteey asale, taree 'm' aanee tyachya vihit swarupatil samarupata adviteey nahee aanee thet berries vighatan dekhil jatan karat nahee. yache kaaran asey qee yaa modulesmadhye kshullak nasalele automorphism aahet jeey berries jatan karat naheet.
tathapi, ekat eka canonical tortion submodule "tii" aanee pratyek (vishisht) aparivartaniy ghatkashi sambandhit samaan canonical submodules asatat, jyamule canonical kram midatoh
jorden-holder prameyatil rachana malikechi tulana qaraa.
udaharanarth, jarr sootra _ 33 aanee sootra _ 34 heey eka aadhar asel, tarr
sootra _ 35 haah aanakhi eka aadhar aahe aanee aadhar matricks sootra _ 36 chaa badal yogasutra _ 37 jatan karat nahee. tathapi, heey sootra _ 38 yog jatan karate, kaaran heey tortion submodule aahe (yethe samatulya, 2-tortion ghatak).
samanyikaran.
gut.
jorden-holder pramey haah maryadit gatansathi (kinwa aniyantrit ringvareel modules) adhik samanya parinam aahe. yaa sarvasadharantemadhye, ekhadyala thet berijaivaji rachana malika milate.
krull-schmitt pramey aanee sambandhit parinam asha atty detat jyaa antargat modulemadhye prathamik vighatanasarkhe kahitari asate, asanghatit moduleschee thet berries mhanoon vighatan jyamadhye berries kramanusar adviteey asatat.
prathamik vighatan.
prathamik vighatan heey parivartaneey notherian ringjavar maryaditpane tayar kelelya modulesmadhye samanyikaran hote aanee yaa parinamala laskar-nother pramey mhanatat.
aparivartaniy modules.
yaulat, 'asanghatit' upghatkanmadhye adviteey vighatan heey aataparyant samanyikaran hott nahee aanee apayash aadarsh varg gatadware mojale jaate, jeey pidssathi nahise hote.
jyaa rings mukhya aadarsh domain naheet, tyanchyasathi done ghatkandware tayar kelelya ringvareel modulesathi adviteey vighatan dekhil aavashyak nahee. r" = z [σ−5] yaa ringsathi, 2 aanee 1 + σ−5 dware tayar kelele module "r" aanee tyache submodule "m" heey donhi avibhajya aahet. r" heey "m" chey samaroop nasate, tarr "r" ± "r" heey "m" ± "m" chey samaroop asate; asha prakare "m" chya yoganchya pratimanmadhun "l" 1 ", l" 2 <"r" ± "r" hii asanghatit upghatak midatat, jee "r" ± "r" chey vegale vighatan detat. r' ±r 'laa abhikarmaniy moduleschya thet bereejmadhye advitiyapane ghatak banavanyache apayash heey' r 'chya ghatakanche' r 'chya aparivartaniy ghatakanmadhye adviteey ghatak banavanyachya apayashashi thet sambandhit aahe (aadarsh varg gatadware).
tathapi, dedeckind domenvar aadarsh varg gut haah ekmev adthala aahe aanee sanrachana pramey kirkol badalansah dedeckind domenvar maryaditpane tayar kelelya modulesmadhye samanyikrut hoto. ajunahi eka adviteey tortion bhag aahe, jyamadhye tortion free purak (isomorphismparyant adviteey) aahe, parantu dedeckind domenvaril tortion free module aata aavashyakpane mukt nahee. dedeckind domenvaril tortionfree modules heey rank aanee stenitz vargadvare (jeey aadarsh varg gatamadhye moolya ghete) (samarupateparyant) nirdharit kele jaatat aanee 'aar' chya pratinchya thet bereejmadhye (rank vann free modules) vighatan thet berries karoon rank vann projective modulesmadhye badalale jaateh vaiyaktik berries advitiyapane nirdharit kelee jaat naheet, parantu stenitz varg (berries) aahe.
naan-antimaritya tayar kelele modules.
tyachpramane maryaditpane tayar naan jhalelya modulesathi, asha changlya vighatanchi apeksha karata yet naheeh ghatakanchi sankhya dekhil badaloo shakate. shakate chey zed-up-modules aahet jeey ekach velee done asanghatit moduleschee thet berries aanee teen asanghatit moduleschee thet berries aahet, jeey prathamik vighatanche analogue anant utpann kelelya modulesathi, agadi purnankanvar dekhil, zed darshavitat.
naan-antimaritya tayar kelelya modulesmadhye udbhavnari aanakhi eka samasya mhanje tethe tortion-mukt modules aahet jeey mukt naheet. udaharanarth, purnankanchi ring z vicharat ghya. mug q heey eka tortion-free zed-module aahe jeey mukt nahee. asha moduleche aanakhi eka shastriy udaharan mhanje bayer-spicker gut, joou shabdanihay jodani antargat purnankanchya serve kramancha gut aahe. sarvasadharanpane, konate anant utpann jhalele tortion-mukt abelian gut mukt aahet haah prashna konatya mothya cardinals astitvat aahet yavar avalambun asato. yacha parinam assa hoto qee, anant utpann jhalelya modulesathi konatehi sanrachana pramey heey sanch siddhantachya svayamsiddhatechya nivadivar avalambun asate aanee vegalya nivadikhali avaidh asoo shakate.

