
naan commutative ring

ganita saastramlo, naan commutative ring anedi oka ring, dheeni gunakaaram commutative kaadu; antey, ringlo "a" mariyu "b" unnaayi, antey "ab" mariyu "b" bhinnamgaa untaayi. samanamga, "naan commutative ring" anedi commutative ring kaani ring. 
naan commutative beejaganitam anedi naan commutative ringula lakshanaala adhyayanaaniki ankitam cheyabadina ring siddhaantamlo bhaagam, indulo commutative ringulaku kooda vartinche lakshanaalu unnaayi.
konnisaarlu "naan commutative ring" anae padaanni "ring" kuu badulugaa paerkonabadani ringnu suuchimchadaaniki upayogistaaru, idhi tappanisarigaa commutative kaadu, anduvalla commutative kaavacchu. saadhaaranamgaa, idhi adhyayanam chesina lakshanaalu commutative ringulaku parimitam kaadani nokkicheppadaaniki, endukante, aneka sandarbhaallo, "ring" nuu "commutative ring" kosam sankshiptalipigaa upayogistaaru.
kontamandi rachayitalu valayalaku gunakaara gurtimpu undani bhavinchanappatiki, eee vyaasamlo vaerae vidhamgaa perkonakapothe aaa oohanu chestaamu.
udaaharanalu.
naan commutative ringulaku konni udaaharanalu
saadhaaranamgaa commutative kaani valayalaku konni udaaharanalu (conny saadhaarana sandarbhaallo commutative kaavacchu):
charitra.
jyaamiti nundi utpannamayye vibhajana valayaalatoe praarambhinchi, naan commutative valayala adhyayanam aadhunika beejaganitamlo pradhana praantamgaa perigindi. 19va mariyu 20va sataabdaalaloo aneka mandhi rachayitalu naan commutative rings siddhaantam mariyu vivarananu vistarinchaaru mariyu suddhi chesaaru. atuvanti sahakaarula asampoorna jaabitaalo ii. artin, richard bruyer, p. emm. kohn, doubleu. aarr. hamilton, I. enn. herstein, enn. jacobson, keey. morita, ii. noether, oo. oor, zhe. wedderburn mariyu itarulu.
commutative mariyu naan commutative beejaganitam madhya thedaalu.
shaastreeya aasakti gala naan commutative rings commutative rings kante chaalaa klishtamgaa untaayi kabatti, vaati nirmaanam, lakshanaalu mariyu pravartana baga ardham kaaledu. commutative rings nundi naan commutative rings varaku konni phalitaalanu saadhaaraneekarinchadamlo chaalaa pani vijayavantamgaa jarigindi. commutative mariyu kaani ringula madhya pradhana vyatyaasam emitante, kudi ideals mariyu edama idealsnu vidigaa pariganinchaalsina avasaram undhi. naan commutative ring siddhaantakartalu eee rakamaina aadarsaalalo okadaanipai oka sharatu amalu cheyadam saadhaaranam, ayithe adhi vyatiraeka vaipu pattukovaalsina avasaram ledhu. commutative ringula kosam, edama-kudi vyatyaasam ledhu.
mukhyamaina taragatulu.
vibhajana valayalu.
vibhajana ring, dheenini vakra kshetram anie kooda pilustaaru, idhi vibhajana saadhyamayye ring. pratyaekamgaa, idhi sunna kaani ring, dheenilo prati sunna kaani moolakam "a" gunakaara vilomam, i kaligi untundi.ii., oka moolakam "x" thoo. vaerugaa perkonnatlayithe, daani unitla samuuham anni sunna kaani muulakaala samiti ayithe matrame ring anedi vibhajana ring.
vibhajana valayalu kshetraala nundi bhinnamgaa untaayi, vaati gunakaaram parivartanaatmakamgaa undavalasina avasaram ledhu. ayithe, wedderburn yokka chinna siddhaantam prakaram anni parimita vibhajana valayalu parivartanaseelamainavi mariyu anduvalla parimita kshetraalu. chaaritraatmakamgaa, vibhajana valayalanu konnisaarlu kshetraalugaa suuchistaaru, ayithe kshetraalanu "parivartana kshetraalu" anie pilustaaru.
ardha saralamaina valayalu.
saralamaina (irreducible) sub modules yokka pratyaksha mottam ayithe, aikyatato kuudina (tappanisarigaa commutative kaadu) ring meeda unna module semissiplel (ledhaa puurtigaa tagginchadaginadi) anie cheppabadutundi.
oka ring daani paina edama modulega semee simple ayithe (edama)-simple anie cheppabadutundi. aascharyakaramgaa, edama-sadhikarika valayam kooda kudi-saadhikarikamgaa untundi mariyu dheeniki viruddhamgaa untundi. anduvalla edama/kudi vyatyaasam anavasaram.
semipremative rings.
semipremative ring ledhaa jacobson semisimple ring ledhaa zhe-semisimple ring anedi jacobson radikal sunnaga unde ring. idhi semee simple ring kante marinta saadhaaranamaina ring rakam, conny ikkada saadhaarana modules ippatikee ring gurinchi taginanta samacharanni andistaayi. puurnaamkaala valayam vanti valayalu ardha-bhedatmakamainavi, mariyu oka artinian ardha-bhedaatmaka valayamu kevalam ardha-saralamaina valayamu. semi premitive ringulanu aadima ringula yokka upa-pratyaksha utpattulugaa ardham chesukovacchu, veetini jacobson saandrata siddhaantam vivaristundi.
saadhaarana ungaraalu.
oka saadhaarana ring anedi sunna kaani ring, idhi sunna ideal mariyu daanikade kakunda rendu vaipula ideal kaligi undadu. oka saadhaarana valayanni ellappuduu oka saadhaarana beejaganitam gaa pariganinchavacchu. valayaluga saralamaina conny modules vale laeni valayalu oka kshetrampai puurti maatrika valayaniki yea chinnapati aadarsaalu undavu (endukante m ("n", r ") yokka edainaa aadarsam m (" n ", i") roopamlo untundi, "i" anedi "r" yokka aadarsam), conny chinnapati edama aadarsaalanu kaligi untundi (antey, konni sthiramaina sunna niluvu varusalanu kaligi unna maatrikala samitulu).
artin-wedderburn siddhaantam prakaram, edama ledhaa kudi artinian ayina prati saadhaarana ring anedi vibhajana ring meeda oka maatrika ring. mukhyamgaa, vaastava sankhyalapai parimita-dimensional vector space ayina ekaika saadhaarana valayalu vaastava sankhyalu, sanklishta sankhyalu ledhaa chatushkonalapai maatrikala valayalu.
garishta ideal dwara ring yokka edainaa bhaganni oka saadhaarana ring antaaru. mukhyamgaa, oka kshetram oka saadhaarana valayamu. daani vyatiraeka ring "r" o saralamgaa unte matrame ring "r" saralamgaa untundi.
vibhajana valayampai maatrika valayamgaa laeni saadhaarana valayaniki udaaharana wheel beejaganitam.
mukhyamaina siddhaantaalu.
wedderburn yokka chinna siddhaantam.
wedderburn yokka chinna siddhaantam prati parimita domine oka kshetram anie perkondi. maroe maatalo cheppalante, parimita valayalaku, domainlu, vibhajana valayalu mariyu kshetraala madhya vyatyaasam ledhu.
artin-jorn siddhaantam siddhaantaanni pratyaamnaaya valayalaku saadhaaraneekaristundih prati parimita saadhaarana pratyaamnaaya valayamu oka kshetram.
artin-wedderburn siddhaantam.
artin-wedderburn siddhaantam anedi semee simple rings mariyu semee simple algibralaku vargeekarana siddhaantam. siddhaantam prakaram, oka (artinian) semee simple ring "r" anedi vibhajana valayalu "dii" paie parimitamaina aneka "nii"-bye-"nii" maatrika valayala gunanaaniki isomorphic, konni puurnaamkaalu "nii" kosam, rendoo suuchika "i" yokka kramaparivartana varaku pratyaekamgaa nirnayinchabadataayi. mukhyamgaa, edainaa saadhaarana edama ledhaa kudi artinian ring anedi vibhajana ring "d" paie "n"-bye-"n" maatrika ringku isomorphicga untundi, ikkada "n" mariyu "d" rendoo pratyaekamgaa nirnayinchabadataayi.
pratyaksha paryavasanamga, artin-wedderburn siddhaantam vibhajana valayampai (oka saadhaarana beejaganitam) parimita-dimensional ayina prati saadhaarana valayanni oka maatrika valayamgaa suuchistumdi. idhi joseph wedderburn yokka asalu phalitam. emil artin taruvaata dheenini artinian valayala vishayamlo saadhaaraneekarinchaadu.
jacobson saandrata siddhaantam.
jacobson saandrata siddhaantam anedi oka valayampai saadhaarana maadyuullaku sambandhinchina siddhaantam.
edainaa aadima valayanni vector space yokka sarala parivartanala valayaniki "dattamaina" upa valayamgaa chudavachani choopinchadaaniki siddhaantaanni vartimpajeyavachu. eee siddhaantam modatisaarigaa 1945loe nathan jacobson rachinchina prasiddha kaagitam "parimitata oohalu laeni saadhaarana valayala nirmaana siddhaantam" loe saahityamlo kanipinchindi. dheenini saadhaarana artinian valayala nirmaanam gurinchi artin-wedderburn siddhaantam yokka mugimpu yokka saadhaaraneekaranagaa chudavachu.
marinta adhikarikamga, siddhaantaanni eee krindi vidhamgaa perkonavacchuhu
nakayama yokka lemma.
zhe ("aarr") anedi "aarr" yokka jacobson radikal gaa undanivvandi. u" anedi oka ring meeda saraina module ayithe, "r", mariyu "i" anedi "r" loe saraina ideal ayithe, appudu "u"· "i" nuu "u"· "i" roopamlo unna muulakaala mottam (parimita) mottaala samitigaa nirvachinchandi, ikkada "u" anedi kevalam "u" paie "r" yokka charya. tappanisarigaa, "u" · "i" anedi "u" yokka upa module.
v" anedi "u" yokka garishta upa module ayithe, appudu "u"/"v" anedi saralamainadi. kabatti "u"·j ("r") anedi tappanisarigaa "v" yokka upasamiti, idhi "j" ("r") yokka nirvachanam prakaram mariyu "u"/"v" anedi saralamainadi. anduvalla, "u" loe kaneesam oka (saraina) garishta upa module unte, "u"·j ("r") anedi "u" yokka saraina upa module. ayithe, idhi "r" paie "u" ekapaksha maadyuullaku pattukovalasina avasaram ledhu, endukante "u" yea garishta upa maadyuullanu kaligi undavalasina avasaram ledhu. sahajamgaane, "yuu" anedi notherian module ayithe, idhi vartistundi. r" anedi notherian ayithe, mariyu "u" anedi parimitamgaa utpatti cheyabadithe, appudu "u" anedi "r" paie notherian module, mariyu mugimpu santrupti chendutundi. kontavaraku visaeshamainadi emitante, balaheenamaina ooha, antey "yuu" anedi "aarr"-modulega (mariyu "aarr" paie parimitata ooha ledhu) parimitamgaa utpatti cheyabadutundi, idhi nirdhaaranaku haamii ivvadaaniki saripotundi. idhi tappanisarigaa nakayama lemma yokka prakatana.
khachitamgaa, okadaaniki eee krindivi unnaayi.
lemma yokka oka version naan-commutative unitary rings "r" paie kudi maadyuulla kosam kaligi untundi. phalitamgaa vacchina siddhaantaanni konnisaarlu jacobson-ajumaya siddhaantam anie pilustaaru.
naan commutative localization.
sthaanikeekarana anedi oka valayaniki gunaka vilomaalanu jodinche kramabaddhamaina paddhati, mariyu saadhaaranamgaa parivartana valayalaku vartinchabadutundi. ring "r" mariyu upasamiti "s" ichinatlayithe, kontamandi ring "r" * mariyu ring homomorphismnu "r" nundi "r" * varaku nirminchalanukuntaru, antey "s" chitram "r" * loe "unitlu" (invertible elements) kaligi untundi. inca, dheenini cheyadaaniki "r" * anedi 'saadhyamainanta uttamamainadi' ledhaa 'atyanta saadhaarana' maargamgaa undaalani korukuntaru-saadhaarana paddhatiloo dheenini saarvatrika aasti dwara vyakteekarinchaali. r" yokka sthaanikeekarananu "s" dwara saadhaaranamgaa "s" −1 "r" dwara suuchistaaru; ayithe konni mukhyamaina pratyeka sandarbhaallo itara sangnaamaanaalu upayoginchabadataayi. s" anedi oka samagra domine yokka sunna kaani muulakaala samiti ayithe, appudu sthaanikeekarana anedi bhinnala kshetram, anduvalla saadhaaranamgaa frac ("r") anie suuchimchabadutumdi.
naan-commutative ringulanu sthaanikeekarinchadam marinta kashtam; bhaavi unitla prati sett "laku sthaanikeekarana ledhu. sthaanikeekarana unikilo undani nirdhaarinche oka sharatu dhaatuvu paristhiti.
sthaanikeekaranaku spashtamaina aasakti unna naan-commutative ringulaku oka kesu differential operatorla ringulaku. udaaharanaku, deferentiation operator "d" kosam adhikaarika viloma "d"-1 nii prakkane unchadam yokka vivarana dheeniki undhi. idhi avakalana sameekaranaala paddhatulalo aneka sandarbhaallo jarugutundi. ippudu daani gurinchi microlocalisation anae pedda ganita siddhaantam undhi, idhi aneka itara shaakhalatho anusandhaaninchabadi undhi. micro-" tagg anedi mukhyamgaa forier siddhaantamtoe anusandhaanaalaku sambandhinchinadi.
naitika samaanatvam.
morita equivalence anedi aneka ring-theoretic lakshanaalanu samrakshinche ring la madhya nirvachinchabadina sambandham. 1958loe samaanatvam mariyu dvandvatvam yokka saaruupya bhavananu nirvachinchina japanese ganita shaastravetta keeti morita paeru meeda dheeniki eee paeru pettaru.
r", "r-mode" paie (edama) maadyuulla vargam mariyu "s", "s-mode" paie (edama) maadyuulla vargam samanamga unte rendu valayalu "r" mariyu "s" (anubandha, 1 thoo) samanamga (morita) anie cheppabadutundi. kudi module vargaalu "mode-aarr" mariyu "mode-ess" samanamga unte matrame edama module vargaalu "aarr-mode" mariyu "ess-mode" samanamga untaayani chuupavachchu. inca, samaanatvaanni ichhe "r-mode" nundi "s-mode" varaku edainaa functor svayamchaalakamgaa sankalitam avtundani chuupavachchu.
bruyer group.
k" kshetram yokka bruyer samuuham anedi abelian samuuham, dheeni muulakaalu "k" paie parimita rank kaligina kendra saadhaarana algibrala yokka morita equivalence taragatulu mariyu adanamgaa algibrala tensor utpatti dwara prerepinchabadutundi. idhi oka kshetrampai vibhajana algibralanu vargeekarinche prayatnaala nundi udbhavinchindi mariyu beejaganita shaastravetta richard bruyer paeru pettabadindi. eee samuuhaanni galois cohomology paramgaa kooda nirvachinchavacchu. marinta saadhaaranamgaa, oka pathakam yokka bruyer samuuham ajumaya algibras paramgaa nirvachinchabadutundi.
oric paristhitulu.
dhaatuvu paristhiti anedi ayasteen dhaatuvu pravesapettina paristhiti, idhi bhinnala kshetram yokka nirmaanaanni, ledhaa marinta saadhaaranamgaa oka valayanni sthaanikeekarinchadaanni, parivartana valayalaku minchi vistarinche prasnaku sambandhinchi. ring "r" yokka gunakaara upasamiti "s" kosam "saraina dhaatuvu paristhiti" anedi mariyu, khandana. saraina dhaatuvu paristhitini santruptipariche domaine saraina dhaatuvu domine antaaru. edama kesu kooda adhe vidhamgaa nirvachinchabadindi.
goldie siddhaantam.
ganita saastramlo, goldie siddhaantam ring siddhaantamlo praathamika nirmaanaatmaka phalitam, dheenini 1950laloe alfred goldie niroopinchaaru. ippudu kudi goldie ring anie pilavabadedi oka ring "r", idhi parimita ekareeti parimaanaanni kaligi untundi ("parimita rank" anie kooda pilustaaru), idhi kudi modulega untundi mariyu "r" yokka upasamitula kudi vinaasanaalapai aarohana golusu paristhitini santruptiparustundi.
goldie siddhaantam prakaram, semee prime rait goldie rings khachitamgaa semee simple artinian rait classical ring off quotients kaligi untaayi. eee bhagala valayam yokka nirmaanam puurtigaa artin-wedderburn siddhaantam dwara nirnayinchabadutundi.
mukhyamgaa, goldie siddhaantam ardha praadhamika kudi notherian valayalaku vartistundi, endukante nirvachanam prakaram kudi notherian valayalu "anni" saraina aadarsaalapai aarohana golusu sthithini kaligi untaayi. kudi-netherian ungaramu saraina goldie anie haamii ivvadaaniki idhi saripotundi. dheeniki viruddhamgaa undaduh prati saraina dhaatuvu domine saraina goldie domine, anduvalla prati commutative integral domine kooda.
goldie siddhaantam yokka paryavasanamga, goldie kaaranamgaa, prati semee prime principle rait ideal ring anedi prime principle rait ideal rings yokka parimita pratyaksha mottaaniki isomorphicga untundi. prati prime principle rait ideal ring anedi rait oor domine paie oka matricks ringku isomorphicga untundi.