AJITEEPAHJIOrHKA CeuHnap. <&AKTOPH3AUHOHHblE TEOPEMbI fljifl HEKOTOPUX KJIACCOB KOIffiU EE3flEMTEJffiflH7JIH. I. JI.A.EOKyTB. I.

Transkrypt

1 AJITEEPAHJIOrHKA CeuHnap Tou 4 BunycK 4 PyKOBoaHieJii A.H.UaJimeB 1965 r. <&AKTOPH3AUHOHHblE TEOPEMbI fljifl HEKOTOPUX KJIACCOB KOIffiU EE3flEMTEJffiflH7JIH. I. JI.A.EOKyTB I. BseaeHHe /? - HeKoiopoe accouhaihbhoe KOJIBUO des aejihtejien ny- JIH a c 1. Iloa $akioph3aifliohh00 Teopeiioa «nn /^ B 3108 padoie uu dyaeu nohmiaxb orwcahhe (B HCKOTOPHX TepuHHax) unoxectba ^^ nehyjiebhx BJieueHioB (fl B, C, 2)) KOJiBoa R,fljw KoiopHX coothomehhe Hanpraiep, ewih ^ - Teno, TO HB (I) caeayei coothonerae: (2) X e /?. B nocjiehhhe row $&KTopH3amiOHHbie teopeuu (z npraeaehhoy BHme ouucjie) WK aekotopux mapokhx KJISCCOB KOJISK <5es zewieneft Hyjm KOHOM (CM. [I], 4, CJiegciBHe I; [2], 4 ).HiieHHO,KoH HBHHfi BHA B06X 16TBepOK HS ^^ fljifl HOJIHHX $HJH>TpOBaHHEIX c HHsepCHMi ajiropmtmom [I] H ajifl KOJiei^ co cjiafiboi ajiropnthou [2]. HepBuM KB STHX KJiaccos coaepsht, B qacthocth, ajiredph cie - PHSOB OT HeasBeoiHux, Biopofl - CBotfoflHHe ajiredpa., ITO HS onpesejiehhh icojieii co cnadmi asrophtmoii Heno- BHieKaei, MTO B HHX BHnojraHeicH ojiezyvmn $aktopmsa - reopeua.

2 26 JI.A.BoKyTB TEOPEMA A, n p o H 3 B O J i B H H B s Ji e M e H T UHOse e c T B a 'ffit, n o c T p o e H H o r o a ji H K O j i t nonyqaeich us leisepkh c n o M o m B K> c j i e f l y r a m a x o n e p a i j H f t: 2) T a e X, - HeHyjiesHe ajiemehtu as j 3) y>a (A, B,C,Z»= (#+ *,&,, + XB), p a e Xey?^ fl+cx^o, + XB*O. Ms TeopeuH A HeipyaHO BusecTH OTueqeHHyio BHme Teopeay KoHa. Teopeua A BHnojiHHeTCH He B Kaawou KOJIBUS. Hanpioiep, ona He B KonBi^e R, KOTOpoe ecu. ajtrcopa c 3. (eaa HCKOTOnoaeM p ) c CHCTeuofl odpasyroiuhx 1^ ^Li^i^ij *-!,n on- JhieHHO, B 3Tou nocjieflheii KOJIBUC R. Buno^flneTCH TEOPEMA B. IlpoH3BOJiBHafl letsepica M H O x e c T B a??t nojiyiaeicfl MS qeibepkh Qd H q e i s e o K C O O T B e T C T B y I O ^ H X H C X O f l H H U C O O T H O ra e H H H ii (3), c n o M o m B n onepai^htl: I) 3 ) r a e (w^.u,^) -oflha H3 qeisepok (4), X(wB-uZ)), d(^w+/j)>cl(ff),, ^^^ ^ 0 > ^O, &(/,+ ) >

3 leopeuu HJIH HeKOTopax KJiaccos des aejihtejiejt Hyan. I. B $opmyjmpobke TeopeuH anaic ctffl} odoahaiaei cienehb (B ecie - CTBBHHOM cuhcae) ajieuehia /7 Kontaa ft (CM. 2 n. 2). MOKHO nociabhtb aaaaiy o <3 on/cannh K Ji a c c a Ko^ea, K O T O p u e H B U H B T C H ajirefipaun c ^ Has > c chcieuoa odpaaynnhx /-\ U H O x e c T B o u onpeae^hiomhx coothome- H H fi (3), r a e Wd-;/?,-, ui>:fle ^, H a Ji H K o T o - p H X BHIIOJIHHeTCfl $ a K T O p H 3 a a H O H H a H T e o p e u a B. OcHOBHoK iiejibb HacTOHieH padora HBnfleiCH pemehhe nociasjieh» HOB Btrae saaayn npw cjieayioinhx aonojiehtejibhiix orpaiwqehhhx Ha CHcooTHomeHHfl (3): 82) Cjiosa X-/ft, ^//t-, i><^i * nonapho paaraieh Meawy co - dott. OCHOBHOR pesyjiliai padoth MOSHO c$ophynhpobat cjiesyjonum oopasom: Knacc anreop cy Has J?, KOTOp u e s a a a H H co ox H o m e H H H M H (3), y a o B J l e T B O p H D m H M H y C J I O B H H M SI) H 82), H B K O T O P U X B l i n O J I H H e T C H $aktoph3ai!h o H H a H T e o p e u a B,coBnaaaeT c K Ji a c- c o H 5/V - K o n e ij (CM. 2). PaccMOTpnu dojiee nospodno cosepsahhe Haciosmei} padoxu. B 2 aaeich onpeaenehhe SN -KOJieu. KoJIB^a KJiacca 5yV HBJIHBTCH ajiredpamh c I, aaflahhhuh nekotopoh CHCXSMOR oopaaymmnx t"\ OnpeaeJIHBlMUH COOTHOmeHWHMH BHaa (3). IIpH 3TOM COOTHOme - HHfl (3) SOJI2CHH yxobjietbopfltb, KpOM6 ycjiobhb SI) B S2), HeKOTOpoMy ycjiobhio S3) (CM. onpe^ejiehhe I), Koiopoe BapaiaeT ipedosahhe "HecyiwHpyeMocTH" cooihomehhl! (3). Ho onpeflenehhd.csoajiredpa OTHOCHTCH K qacay S^/V -Koneq. flajiee, as B 2 Teopewu I cjieayei, qio nopmajibhiie am-edpu B A.H.MajiBiieBa [3], [*], cootbeictbyroipe HOpManLmoi cuciemah pa- B6HCTB des nebicc, npasux a cpeahhhbk UHIUIOB %mmi 2 (CM. 2, n. 3), a TaKJte ajiredph/^y,??^^ HS padora astopa [5] nphhaanekar K KJiaccy SN -KOJieu. B KOHue 2 aaetch onncahne scex coothomehhti BH^a (I) OflHOpOflHHX 3JieM6HTOB ^B^CfJ) JIBdOFO SN -KOJIBIja. B 3 flokasubaeicfl ochobhaa TeopeMa padoih (TeopeMa 3), KO- Topan yibepacaaet, ^TO B S/V -KonBijax BunojrafleiCH $aktoph3aipi - TeopeMa B. OdpaiHoe ytbeps^emie o TOM, vro ycjiosae ZJIK

4 28 JL.A.EoKyiB cneactbheu ycjiobhtt 81), 82) H TeopeuH B, noqia oie- BHflHO. B 4 yciahabjihbadtch HeKotopae cjieactbhh reopens 3. HiteHHO, aoicashbaetcfl, mo S/Y -Kojn>n.a yaobnetbopjnoi ycnosrao e a H H - C T B e H H O C T H s o n o j i H e H H H (leopeua 4). Oicioaa (cjieactbhe I), ito B <5yy -KojiBijax BtraojiHHeTCH cseayjnodhx HeHyaeBHX aaeuehtob ^ o HS S// -KOJisqa /? cy- TaKHe anew en TH c H o^,?to aftn6ft = cftj Hana-g=ftd., (S) rje, Hanpmiep, a.r- rjiabhhfl npabhit Haean, nopoawehhhii Kojrtiia ($ea aejimtejieil Hyjw H o 1, yaobjietbopnioiime (5), HaasaHH Haua B pafioie [5]?/? -KOTOEtaira ( gwazi Bezout ) ). Kmcc QB -KOJieu HBaneTCH Cojiee mhpokrai, qeii KJiacca Besy-KOjueu H WB -ROJICU ( weak Bezout rings ), paccuaxphbasnuiech KOHOM B padoie [6}. KiieHHO, Beay-KOJiBua, Kpoue ycjiqbmft (4), yao- BJIOTBOpHBT eme ycjiobhb aft + a B V/B -Kojimax npeanaymee pasehctbo BunojiHHeTCH B ecra a.ftn R.*O. KB nocjieflhhx pesyjiiiatob, BBHay Toro, ^TO cymectbyri 3/V - Koxbiia, He BJiosmiue B TeJia, noayqaen yrbepsgehne: CynecTByBi Q 3 -KOJima, He BJiosrauie B lena. OTM6THM, ITO Eesy-KOjiBua yaobjietbopirot ycjiobfflo Op9 u nostoiiy BJIOSHUH B TeJia ( [6], ieopema3.3). flnfl W3-KOJieu sonpoc o<5 me BflOKHMOCTH B T6Jia OCTaeTCH OTEpUTUM. B KOHue 4 (n. 3) nokashsaetch, KHK H3 xeopeuu A $aktoph3ai(hohhafl Teopeua Kona ann KOJICU co cjiadm OTMeTHii, q?o c nomonttb pesyjiitatob 3ToM patiora UOHCT Curs oco<5^eho ochobhoe ytsepsaenhe pafioih asiopa [5].HUSHHO, ecjw 5 - noayrpynna c I i co CTeneHHOB $yhkunefi d(oz) (c OOIWHMIH CBOflciBaMM :d&)=q> -f, 2 _,...; d(x) = O TOJIBKO npa x= /, d.(ja/)=c( x:)-t- ctfy))* TO B rpynnobom KOJiti^e P^Sj, rae P -nojie, MOKHO paccmotpeib MHOieCTBO OflHOpOSHHX 3Jiei«eHTOB 77 S, of (Srj = y J / ^ - V ~ n ae aueei aejiaiejiea Hynn, TO aio unoaectbo oopaayei noayrpynny no yuhoaehhb.kotopyb MH Cyaew o<3o3ha^aib qepes

5 $akiopn3aiyiohhue leopeuti ZUIH HBKOTOPHX KJiaccos KOJieu. dea aejiarejiefl HynH. I. Ododman saaaiy, nociabjiehhyio B [5], MOKHO yicasaib sapaaht npodaemh Majimesa, o KOTOpofi rosophjiocb B [5]; CymecTByei JIH KOJIBU.O sa^a P(S), rae S - nojiyrpyraia co cienehhoft $yhkn.heft, TaKan, ITO: 1) P(S) des aejihtejiefl Hyjin; 2) P(.S)hcrm BJiosmia B rpyunyj 3) P(S) B Tejio He BJioaraio. OKasusaeTCH ( a STO yctanab^hbaetch c noiiombd pesyjibiaiob arok pafiotu), TOO raicae KOJiBi^a cymecibyiot H iro TaKHMH dyayi, nanpawep, KOJIB^a P(Qn), n> 3, noctpoehhhe astopom (CM. [5] ). 2. SN -KOJiBua 1. B STOW naparpa$e uu nphbeaew onpesejieahe S/V -KOJICU H aokaaem asa ytbepasekhh, oaho as Koiopttx npeaciabjihet co6oii ao - CTaioiHoe ycjiobue nphaafljiekhocth KOJiBi^a K^accy S/Y -KOJieij. 2. IlyciB S - nojiyrpyraia c I, savannas B HeKoiopOM AL =BL (eel), (6) ycjiobhhm: SO) 51) HHKaKan HaqaJiBHan dyksa JJIH CJIOB UHoacecTBa i 5 ^^/} H6 HBJIfleiCH KOHeiHOa <5yKBOft fljifl CJIOB atoro se uhomectba; 52) Bee CJIOBB, sxossipie B coothoinehhn (6), nonapno pasjih^hh ueny codod. ycjiobhmch B KaieciBe MHOKecisa I sonyckaib H nycioe BO. 3a$nKCHpyey TenepB Jiesue H npasue yacth pasenctb (6). CJIOBO X aj[$abhia /^ dyaeii HastiBaTB KaHOHMiecKim, ecrai OHO He coflepkht B KaieciBe noac^os HH OHHO HB CJIOB /7t-. OdosHaqaii qe - pes n COBOKynHOCTB scex CJIOB nojiyrpynna S BKJiBiaH H I,HMe- KaHOHnyecKHfi BIW. B cmiy leopeuu HeJliiaH ( [7] ; CM. [5], 2 ), qio SJIH paccmaipubaeuoa nojiyrpynnn S Ka^oe CJIOBO anpasho Oflnowy H TOJIBKO oflhouy cjiosy MS MHOKecisa /7. IlyciB TenepB P - noae, PC$) - noayrpynnosah aaredpa c I noayrpyraih S naa nojcu P. OnpeaeJiHii B anredpe P(S) $yhkiuib CTeneHH d(x), X&P(S). HiieHHO, cienehbb cjiosa naaosem ero flmhy. IloJioacHU npw SXOM, yio d(d) O. OflHopoamoj CTeneHH 77 HasoseM nhhehhyn ROHdHHaiimo CJIOB S^HHBI n HJIH ECJM TenepB X^PCS; H "

6 30 Jl.A.BoKyiB ^ - oflhopoflhhie 3JieMeHiu cienehm, ^^O, TO nojiosuni, al(x)-&- BJieMeHT Xd B flajibheftmem dyaem HasusaTB ciapmea BJieMeHTa X H odoshatjaib X^ = X. HocTpoHM Teneps, HCXOTIH MS chciemh coothomehhfi (6),HeKOTOpyw cucteuy pasehctb ^. HueHHo, nycib HJIH aekotoporo t W}h}u,-f. +*. Torsa B cnpaseanhbo pasehctbo: (8) HOB ^ uu CyaeM nohhmat cobokynnocib BceBosuoKHbix pa B6HCTB (8), noctpoehhhx fljih BC6X L. J '. PaccMOTprai Tenepi. cneayiomee ycjioshe: /f^bjcjj)- HenyjieBbie osnopo^hhe P(S) H Toraa 3TO pasehctbo HMeeT OBHH HS cneaymmax BUZIOB 1) =/7X, 8=XD, X^P(S), (9) 2) fl TenepB Ha KOJII.UO P(S) cjieflyid«ee ipedosahhe: S3) ycsobhe (9) BHnO^HHeTCH fljifl BCeX fl,b,c,q cienehh I. MOXHO ckaaaib, ITO ycjiobme (10) BupaacaeT MCXOHHOii CHCTeMbI COOTHOineHHK (6). OIIPEIEJIEHME I. S/V -KO^BUOM Hasosew KOJIEH.O saaa P(S),ysoycjioBUHM so) - S3). SAME^HKE. Cpasy BHSHO, ITO K uacjiy SJY -KOJICU OTHOCHTCH ajiredpti, TaK Kan ycjiobhh (7) H (9.2) B STOM cjiyqae HH - ijero He supakaiot, a ycjiosne (9.1) JIJIH 3neM6HTOB cienehh I HO. Bseaeu nekotopue aonojihmtejibhue onpeaejiehmh. Byaew TOBOPHTB, ITO CHCTeua coothomehha (6) He coaepatht nesicc UHKJIOB SJIHHU 2, ec- JIH seshk rpa$ aiok CKCTeiiu coothoinehhft [8] He coaepkhi jiesax UHKflJiHHti 2 (B qacthocth, IJHKJIOB anhhh I). AHaJiorH^HO nohiuiaetch ycjiosae a^h npasux M cpeflhhx lynoios HJIMHH 2. (noa cpeflhhhmi rpa$oii Mbi dy^eu HOHUMSTB rpa$, BepiiiHHauH Koioporo HBJIHIOTCH see

7 <DaKTOpn3amioHHHe TeopeiiH <5es HeKoiopux KJiaccos Hyjin. I. 31 H Bee KOHeiHH.e fiykbh CJIOB cacteiih (6) a ase dykbu or, KOTOporo coeflhhehu CTOJIBKMMH pedpawh, CKOJIBKO pas B (6) BCTpeia»- TCfl COOTHOmeHHH aajiee, CJIOBO ^JIHHU 2 yctoaqhbtai, BCJIH OHO He BxoflHT B CHCTeiay pasehctb (6) B Kaqecise Jiesofl HJIH npasoa qacieft. IIpesnojioEiui TenepB, ITO Hcxoanafl chciema coothoinehn8(6)nonyrpynnu S yaobjietbophei ycjioshjai: 1) CHCTeua (6) He coaepacht jiebtix,npabbk, cpeflhhhtix UHKJIOB fljihhh 2; 2) IlyCTl Wh = Uf - H6KOTOpO COOTHOffleHMe H3 (6), x - HeKOTOpan <3yKBa aji$abhia " t. Toraa OflHO as CJIOB 3:^ -2T/" HBJIH6TCH yctoftqhbhu fljih CHCT6MH (6). 3) IlycTi HMeew flsa paam^hhoc pasehctba HB (6) Toraa oaho HS cnos HBJIH6TCH yctoflqhbhm fljih CHCT6UH (6). SaueiHU, ITO ycjiobme (II. I) HBJifleTCfl HeodxoaHMHM neshocth ajirefipbi P(S) K KJiaccy S/V -Konei^. TEOPEMA I. ECJIH C H C i e w a coothoraehhjl (6) nonyrpynnu yflobjieibopflei ycjio - B a H u (7) H (II), TO ajiredpa P (S}, r a &P - n o- jie, n p M H a s J i e K H T K KJiaccy 5/V - a Ji r e d p. JOKA3ATEJIBCTBO. Dyci* - HeKOTOpuft BJieweHT ajiretfph /7t e /7. tocjio A, yiactbyxhuee B BTOM Be, (5yaeM HasusaTB B H c o T o ft ajieuehia /7 K ofioshaqaib pes Kpoiie Toro, B 3ioa se cjiy^ae SJIH jaofioro /7^ MU dyaeu 7^ e/7. PaccuoTpHii lenept B ajiredpc P(S) pasehctbo

8 32 J.A.EoKyiB B, C, 3) - HeHyjieBHe oaaopoflhue sjieiaehth cienehu I,n Kaaceu, ITO axo pasehctbo mieet BHJ (9). flokasaiejibctbodyseu sec- TH HHflyicmiea no ihcay n(fl) + n(c). "&S3as.7i(fl)-f-7i(C)-2^.s.ti(fl)= = n(c)=di TO name yrbepsaehhe OIGBHSHO BBHBJ ycoiobhh (II.I). IlyeTB TenepB rz(r/7)*-77( J=/<>2.. PaccuoipHM asa cjiyqah. I) /?3Vv; 2? 9,6 H CJIOBO W/i Hueei ce<5e nojodhoe B Jtesofl pasehca?ba (12). Toraa cymectbydi TaKHe Uf f. ^, & 3f H W = ^/y - OSHO us ooothoraehhfii (6). z^a, ctohmae B Jiesofi qacih pasehctba (12), B cray yc- (II. 2) HBJMBTCH yctofl*fflbhmh. 003TOMy C3W, U,%3-f, h. Ec- JIH reneps /^3^r, TO OHHTB B cajiy (II. 2) nojiyjaeu, «JTO O^HO us CJIOB Xf, JT/? HB3IH6TCH yctoatobiflj H no^tomy C 33C.. 3lO HO- KasuBaei, yio see <5yKBU, ExoanmHe B /^, EXOBHI H B B pasehctse #(B - a '4 ' } =(c- d'40) (13) C-a &fl nueet MeHBiiiyn Bacoiy, yew sjieueht C. Ho npeapasehctbo (13) Hueei sna (9). Ho Toraa H Hoe pasehctbo (II) HMeei BHJ (9). 2) IlycTB TenepB HHicaKHe asa CJIOBB us JiesoK wacim pabeaciba (12) He noaoshh uescay cofioft. B CHJiy CHMiieTpHM, 310 KB MOKHO npea- HOJIOKMTB H SJIH cjiob npsbofl qacth paschctba (12). flajiee, ec^m see <5yKBH, Bxoafliiwe B fl, BXO^HT H B C, TO paccywaehhh npehtwymero cjiywah nokasubaiot, ITO (12) nueet HeodxoflHMbiii BH^. tloatouy MO- KHO cihtaib, yio B /7 BXOSHT O'yKBa, ne BxoflHman B ^, a B C (5yKBa, ne Bxoaninafl B /7. HaJiee, BCJIH n(b)= 77 (&)=*.!, TO Haiiie ytbepacaehiie JierKo CJIG- H3 (II.I). IIosTouy MoacHO CIHTSTB, WTO B^h,}h2 V(.h^^h2. IlyciB W - (SyKsa, BxosnmaH B fl, HO He BxoflHman B C. Tor^a C?",, "z, V/h^u.f,, OflHO MS CJIOB U,fz,UZ-/,, B CHJiy (II. 2), HB^fleTCH BHU. IlycTB, aanpmiep, lamai dyaes CJIQBO ^/y^ B CHxy (II. 2) H Hamiix npejnojioxehhlt, CHOBO wy2 * sxc^hniee B jie syio qacib pabehcisa (12), He HMCBT cede noaodhtix cpesa CJIOB as

9 TeopeMH JJIH HeKOTopux KJiaccos 6e3 aejihiejiek nyjih. I. 33 (12). STO nokastisaet, ITO paccmatphbaewhft Teopeua ^OKasaaa. Jlerico nposephtb, ITO ycjiobhhw sioft TeopewH ysobjietbopjnoi HOpiaajiBHHe ajirecpn A.M.MajiBueBa, cootbetctbyromhe HOpMaJiiHuii uaw pasehctb, ne coaepsamum Jiesux, npasux, cpeflhhhhx L^MIUIOB HU 2. Kpowe Toro, ycjiobhaw 3ioti Teopewu yaobjieibophiot, H3 (5). *. flokaseu saeci HeKoiopae ytbepaaehmh othochte^bho» - KO- JIEMMA I. S/V-KOfliija ne coaepacat a enefl nyjih. ZIOKASATEJIBCTBO. npeanojiosnu npoihbhoe. IlyciB. Mono u B - oflhopoflhbie Hbie ajieuehth. IlyciB /7 s> /70, /7=Xa ar,<f6^]. CoCepeu B pabehctse (14) see co cjiosa X H OKanqHsaioipecH Ha CJIOBO pasehctbo.y = o, T.e. Jf tffto, 8,* O- Jlerso BO npothbopeiht ycjiobhhu (7), (10). JIEMMA 2. II y c T B B eich p a s e H C T B O STO pasehct- - K o Ji B K e P(S) M M e - (15) rae ^^-HeHyneBiie oahopoflhhe 3Jie u e H T U cienehm 1,^^-HeHyneBue oflh poshiie ajieuehtu cienehh 7?/7?>^.Tor sa p a s e H C T B O (15) M M e e i BHJ (9). flokasatejilctbo. IIpeacTaBHu aneiyiehth /7 H ^ us (15) B ^hjet** ' ft ajieiiehtu HS hj Bj, hh B k npw He noflofihh apyr spyry. KpoMe Toro, cihiaem, ITO (cootbetctbchho, HI ) nonapho PaccMOTpMM cobokynaocib scex CJIOB VVt-/? I^L^Tn /4jnp (16)

10 34 JI.A.EoKyTB H soicaaem, UTO XOTH 6u O^HO HS STHX CJIOB Wi0hj0 He Hweei cecse noaodabix cpeflu octajibaux. fleiictbhtejibho, nycib, aanprcuep, 777 /o. PaCCMOTpHM CJIOBa EOJIH see OHM HMeroT noflodaue cpeflw CJIOB V/ihjj i^i, TO, TaK KBK CHCTewa COOTH01U6HH0 (6) H6 COSepSCHT JieBbtX IJHKJIQB JflHHH 2, 1JH flojl- KHU filiflh dbl HMeTB, 1TO 77?-/>/7, TO 6CTB TTJ^p+d t 1TO H6B03 - MOKHO. MiaK, nycib CJIOBO Vv^^ ne Mweei cede noaodhhx cpe^h octajib HHX CJIOB as (16). CodepeM B Jiesoft a npasoa lacinx pabeaciba (16) cjiosa, HaqnaaiouuiecH co CJIOB, pabhhx cjiosy W,/i/ Bah, 8', C3W,h,C' B--Q, Toraa MMeei Mecio pasehctbo npjmeu BunoJIHfleTCH coothoiuemie a?), ITO ecjih pasehctbo (17) HMeei BIW (9), TO H acxoa Hoe pasehctbo mieei BH^ (9). aemmbi 2 TenepB Jierno BHBeciH m npeannynthx, HcnojiBsyn HHayKmno no BHCOTC n(b} 3JieueHTa 3. Jleuua 2 flokasaaa. TEOPEMA 2. II y c T B B 5/Y - K o Ji B u e P(S) u u e ei uecto p a s e H C T B O (18) ~ n P H 3 B fl B H H e HeHyjiesne oahopoflhue ajienehth, ct(fl)^c((c). T o r a a BIO pasehctbo aweei BHH (9). flokasatejilctbo Oyaeu BBCTH HHflyKimefl no nape, TITO SIM napu ynopjmoiehh jiekchkorpa^h^eckh a ^TO BO scex paseactsax (18) $ukcnpobaau Jiesue u npasue uacth.. ct(/7)-ji cf(3))=j, TO BBiwy JieuiiH 2. IlyciB (<*(*),<*&})-(*, )>(*,*). (19) PaccMOTpHii MaoffiecTso Bcex pasenctb (18), ysobjietbophiomhx yc

11 $aktoph3amiohhue Teopeuu fljrn HOKOTOPUX KJiaccos 35 (5e3 sejihiejiem Hynn. I. JIOBHIO (19), H flokaxem c nomomwo. HHayKUHH no ihcay n(fl)+n(c)i 1TO OHH HM6DT BHfl (9). 77fCJ=3., TO yisepsaehhe, KBK aerko BnaeTL,cnpa- (CM., aanphuep, aajebheflmiie paccyaaehhfl). IlyciB /7 ^ /^0 /7, =X, W, rae W - Cyras. PaccMOTpmi B Jie- H npasoft qacthx pasekctba (18) see cjiosa, Hawiuiaioiptecfl Ha X,. rfle, (20) (21) B CHJiy npeanoboaehna HHayKiwa nojiy^aeu, 110 pasehctba (21) HM8DT BH«(9),6CJIH TOJIBKO /^ ^(9, <^ +O.flpM 3TOU OHH HO HM6IOT BHfl njf(50 (9.1) C OflHHM H T6M K6 3JI6M6HTOM A ; JIH(50 (9.2) C OflHHM H T6M «6 pab6hctbom \/M = U.F??.. 9lO IIOKa - 3biBaeT, qio pasenctbo (18) B pacciiatphbaemoh c;iy<iae nueei BHA (9) TeopeMa 2 aokaaaaa. SoKasaHHoe yibepsaehhe npe^ctasjiset coc5ofl leopeuy SJIH nojiyrpynmi oshopoflhia ajieuehtob SN - 3. QaKTOpHsauMOHHaH reopeua ann 5/V - I. IlyCTB P(S) - HCKOTOpOe S/V -KOJIBUO. PaCCMOTpHM COBO - Z scex qetbepok (/7fBjC,&) HeHyjueBic 3JieMeHioB KO^B- T3KHX, 1TO B 9TOM K03IBI16 BHHOJIHfleTOfl COOTHOieHHe Hepea /V ofioaaaihu no^mhoxectbo H3 fflz,, coctoflqee us cootbetctbynmnx HCXOSHHU coothomehhjai (6). BKJDOIHM B 3?o CTBO H qeisepky (22) Hueei TEOPEMA 3. (^artophsaqaoraafl TeopeHa SJIH 5/V - n p o H S B O ^ B H i i K a01 e u e H T uhoaecisa U O X B O no^yihtb H 3 djieiiertob M H O acectsa/y c nomomsb cjre^yiomhx one - p a K H it :

12 36 JLA.EotcyTB 3) fa (ft, 8, C-2J = cl(fl + CX) 2 d(fl), d(xb-<#) 2 SAME^AHHE. JlerKO HOHHTB, ITO ecjia B (7), BHnOJIHH6TCfl C$OpMyJIHpOBaHH peim, TO 3To KOJIBUO yflobjietbopnei ycjiobhn (10). B aiou cuiicjie yc- JIOBM6 (IQ) HBBH6TCH H6 TOJILKO flociatoqhhu fljifl BbOIOJIHHUOCTH T60pe- MH 3, HO H HeodXOAHMUM. 2. ycjiobhmch o<5 oflhom ocjoshayehhh. HH36HCH y <5yKB <3yayi os - mo paccwatphbaiotcfl OflHopoflHae 3Jie»ieHTH ynaaahhoa cienehh Hyjiesue sjieiiehtu. HojioacHM, Kpoue Tore, MTO BHpaaceHHH, i, odoahaqaiot HVJIB KOjiBita PCS). 3. Teopeuy 3 dyaeu TiOKasuBaiB c noiiombjo HHflyitaHH no (23) HaHdonBineH cie- ECJIH TO IlyciB // ^ 2.. HpMpaBHHBaH B (22) nehh, = C >., nanphuep, /? - ciapiaah lacib anemeaia Mti 2, pabehctbo (24) jaueer BUS (_9). IlyciB (9.1) H, Hanpnuep, fi = CX, ^=X^. Toraa, cooihomehhe. B cajiy reope OHO nwees, ITO Gf(/7 -CX)<tfp?)j d( -XB)<c((2)) nojiyqaeii HeodxoaHMoe nait nyctb TenepB RXR cooihomehhh (22) pasehctbo (24) HMeeT (9.2). B cany onepaimh _fa B <^ MOKHO cihiaib, d, yiactbybnmfi B pasehcrae (9.2), paben I, a 6CTB OflHO US HCXO^HiCC P3B6HCTB (6). Ilocjie 9THX npe^nojiokehhii pasehctbo (24) npauet B paccuaiphbaemom cjiyiae MU mieeu, iro dffl) = (25)

13 QaKTopasamiOHHHe reopens BJIH aekotoptcc luiaccos des aejunejiefl Hyjifl. I. 37 rae #;l ±,&i,q^ cootsetctbhh c IIPJIHHTHMH o<5o3haqehmhmh,ecti. BJieueHTH cienehh / HJIH HyjM. CpasHMBan B pasehctse (22) sjiemenra o^hofl n TOM %e cienehh, chcieiiy pssehctb: IIpH k<o pasehctba (26. k ) mieni sa (26. 77V-77Z) no JCJIOBIHO nojiyiaew: '777, (26- pasehct- *j(tyfyxj/)em., nepenameu ( ) B saae Mcnojitsya nocjieanae paseh- <.)Wm_, (27) (27) ectl coothoiiiehhe OHO H6 MOS6T HM6TB BHfl8 (9.2). II03TOMy O^HOPO^HUMH 3JieM6HTaMH. rje cf(x) = d.ilpa X *= <? 3TOT dun OH JiMHeflHOfl KOIIdHHaqnett KEK ROHeqHitx, TSK a 6yKB ooothoraehhtt (3). HO 3TO H6B03MOZHO. otfpasom, as pasehctsa (25) nojiyiaeu: =?n-<f (28) IIo npeflndjioxehab, nz 1, IlycTB, HanpHMep, n=l. TorpaseHcrao nokasusaet,?to x. = C_ =O.

14 38 JI.A.EoKyiB ocpasou, pasehctbo (22) aueet sioro fiyaeu c^matb, qio n~>-f} rn'>1 Ms pasenctb (28), BBHay Teopewa 2, joieeu: a, ^^_x =/ _a=6> HJIH ^^-2 = 0 He PaccMOTpuM TenepB TaKoe uakchwajibhoe IMCJIO A,, npn KOiopou HM6IOT M6CTO Bcex 1 i *i K,. Ms npeflbmymero BHSHO, ITO k,^ 2.. Byaeii CTOTBTL, ITO kt<7z+4 HJIH A", < 77? +1, TaK K3K MHaie pab6hctbo (22) HM6JIO 0H BHfl (9.2). BbincuiHeHHe pasehctb (29) ofiecneqwbaet BimojiHeHne nepshx &/ pasehctb CHCTeMH (26) : ( 26. n + m ) - ( 26.n+Tn-K,* 1 ).fleb- CTBMT6JIBHO, pab6hctbo ( T7-S ) t TflO O^= S ^ /</-/, B CHJIJ (29) BHnojiHfleicfl noqjiehho (TO ecit nepstib IJIBH Jiesofi ^acra paseh nepsouy qjieay npasog H T.a.). PacewoTpuM pasehctbo CK)fla SHaieHHH /7^' ^', B3HTH6 H3 pab6hctb (29), HO- )rn= O. H Q MS (29), (5yae«/?n^f,*JWfn.,. (30) MaKcwuajiBHocTH /rx, nocjieflhee coothoinehhe ect& coothomehhe HeHyjiesuiiK aneuehtaitm. II o Teopewe 2 OHO EOJISHO MMOTB (9.1). IIoaTouy (51) TenepB, wo as ooothoaehhft (26. n+7n-~/<, ) - ( /-77Z-.2A-, -> / ) cjiesykt

15 xeopemi BJIH Hexoiopia wiaccob 39 (teafleshteneflhysn. I. *n- ^^ s ~j nph -5=^ aih pasehciba y«e flokasaah. npeano- JIOXHM, ITO $opmyra (32) sephh flflh scex aaeudhtoa Vf,^ } W^-; * ^6^5 4/fx-/. flokameuec WIH V^^-,/ ^ PaccuoTpm pabehciao ( ^-777-^^-5 ): rae o ^ -(\*/8f> -, rae /? ^ 777. CrpynnHpyeii B pasehcrae (33) qaesh Bi^a /7t- Bj M O<^/' 3Toro, McnoflBsyn pasehcisa (29), HH4yKTHBHoe npennonomehie M KO ITO BseaeHHue odoahaiehhh, pasehctbo (33) HOZHO nepeniiean B (35)

16 JI.A.BoKyis - Ot n.- JlerKo saiiethtb, wo ocsesehkhe supaaeraih cootbeicibehho WOMUOTCH. flaaee, mieei iiecio pasehctbo, npa s=ff,-^ OHO oiebhflho. ECJIH s or, - cray pasehctb (29), <&m-k+2 = &m.-s = O CKasaHHoe, as pasehctaa (35) B (36) (31), B6HCTBO (36) 6CTB pabghctbo XY=ZT,.Vj 2, T -, TO, = /r,-, TO ecis Teopeiia 2, noayqaeu,.-k,. ECJIH (37), pa- rje cl(u,)=i. Ho Toraa ecib oanobpemehho HaqaaiBHioc dyra coothomehhfl (6) H Koneqioa dykb STHX coorao- (cu. (34)). BTO HCBOSUOXHO. TaKHM odpasoii, B pacciiaiphbae - cjiyiae Z=O, Y=0. nyctl TenepB s<k,-i, TO ecib ol(z)^ri-k,+i. PaseHCTBo (36) He uoaei MMOTB BHaa (9.2), TBK KBK OL,L-Kl+i OKUHqHBaeTCH Ha KOHeiiwe dykbu CHCTeim (6). IIoaTouy nojiyiaeu: z=xu} y=ur. Bo Bcex paccmotpehhhx cajnaflx, cymecibyet ajieuehi X K-S-z (B03MOXHO HyJIi), TSROtt, 1TO

17 HIf09 ' 0= "X ojoi awods 'WHKOICOU wt H lorehbheopo -uv<j/ Hdn *$ ' Vg^ ' VJ? ' "^ HHBH8 'OHHaaiOiaaiOOQ "Hlf^H lolbhbheopo 'J'' BHHasredira soa oih * WAX! ' 00* D '( S-zz/+z/'92) BaiOHgaed ojoniifoaenodii en oih 'sdaiiai Jf9KBMOn *( BMH9Hirouna BEIT onhoieiook H oiwfoxpcoh (j^) H (6$) aioneaed oih ' :9ffHa a sieojmsdsii OHKOK (6E)BaaoH9aed'BHH9hBH80po Hie (ot) f-s-^i.. _ -oil o ' ' 7-^5- xaoa Hdn (gj) BasoimBd 'noeedpo WKHBI '(6 ) if^«do$ oration = -M OJ903 8lT oie aooobira XBdoiOHOH Hirff Hriadosi

18 JI.A.BoKyTi B BTHX odoshaieanjix cnpabefljihbh cateaynipie -~><*coo PaseHCTBo (26. ^ ) UOKHO TenepB nepenhcatt B BHje: I riibj- 6+/-/C» 6+ IIOflCTaBJIHH B 3TO pab6hctbo 3HaieHHfl BCHCTB (43), nonywoi: i^j f B8HTH6 H3 pa- Z + i=-k dcj3 1+ L = K cnaraeiiue npasob a. nesofl ^aciefl»xorb paseboisa pasra.. Z K/i Kw ^iv^)- Z (^/S-^^"^- "-+>-" rf d*/»k ' SToro pasehctbo: (43) mieeu: K

19 *aktoph3auhohhhe TeopenH AJU HeKOToracc (tea aejuweaeft Byafl. I. Tawm ofipasoh, aeotixoano aokasati pasehcrao Z V;..,^ - Z Ot^i V/n l^m.k i ^K Hcnojn>3yH (4-3), nepemueu npejhayiee pasehctbo B -j^t, K-i~s, TO ec» j=k- tc npestmymero pabenciba npmiyi.bha. Torsa aesaa H (44), oc5pasom, pasehctbo (45) aonasaho. Hs aioro paserctba, 9ro H AOKasuBaei pasehctba (4i). PaseHCiBa (42) nokasmadt, wo mieei wee TO cooxaoieme: (22), B Biae: tenepi. ajiemeara, B Hcxogaoe eooiboiehie c - W47) Ha pasenotba (22) B cmiy (46) (48)

20 44 H.A.EoKyrs B nocjiesaeu pasehctbe cl(v}^cl(cf~flh} CTBO (48) HiieeT. nokaateii, ITO paseh-, us (43) PaseHCTBa (49) flokaaano, qio wcxoanoe pabehciso (22) ecit oneparthh w/i u f) K Teopeua 3 4. HeKOiopue P(S) - 5 A/ -KOJIBIJO. PaCCMOTpHM B P^ pab6hctbo (50) 1) PaseHCTBO (50} ne Hiieei saaa (9.2); 2) 3jieMeHra fl,b}c,2) raieiot BHS (25). J B CHJiy aokasaiejibctba TeopeMH 3 no;iy?aeii, ITO I. B alow nyhkie MU flokaseu nekotopoe scnomoratejibhoe yisep- B',C',2)' ' ' ' - ie xe ajiemehra, yio H B pasehctbe (47). B t JD3MMA 3. IlycTB H u e e M B KOJiLite P(S) p a- B e H c T B o (50), yaobjieibopflumee ycjiobii- H it (51), H p a s e H C T B O " : >*-~* (53) r a e. + O. T o rs a a e a j r e j i e H T t i Xfl' ' H l e i B e p K a (w,h.,uff) - e ae, ITO H B p a s e H C T B e (52).

21 3>aKTOpu3aipiOHHHe Teopeuu xnn HeKOTOpiK imaccob 45 des aejihtcjieii nyjin. I. flokasatejilctbo. IlyciB flokasateubctba c$opmyjmpobahhob neioi&i nposeaeu see pac Teopeubi 2 osuobpeiiehho.zpin o(5onx pasehctb (50) a (53). IIo ycjiobhk nojiy^aeu npesse scero, ITO fl-vn-i w, C=V^a 5-/z. U/^./, 2) mieeu PaccMoipuM IHCJIO /r^,- BbifipaHHoe nepea pabehctbawh (29) ^JIH COOT- HomeHHH (50), H TaKoe ace UHCJIO /r^*, HO cootbeictbyiomee pabehctsy (53). flokasem, ITO Kt=Ki*. IIpeanojio3mM, qio K,<K?. Torna B pasenctbe (30) MH raieem, qio Ot-^. f-tfn-k k=0 a, snaim ^ 3io osnaiaei, ITO pasehctba npothbopeiat sadopy incjia /r/. 3io ei, ITO /r,=/f(*. PaseHCTBO (30*), nocipoehhoe fljifl pasehotba (53), HMeei Oicioaa, KaK jokasaho, BIO pabencibo c pasehctbamh (31), MBI, Bsnay B KOJitue P($), TenepB, ITO H soodme ajieuehia Xff_^)...>X0, $nry B pasehctbax (52*$; nocipoehhhx arai coothomehhh (53),coB c ajieaehtauh XHr2t.--,X0 MS pasehctb (52). TOB ^_2 H ^-K,-Z uu BHoe npeanoaioacehhe, ITO scex I, d-^i^s, rae S^K-1.flOKaaceupaseHCTBO (55)

22 H H B H H 9 H i r O I I O t f H H N H H 9 a X O H M t f 9 0 H H B & - * ir o H BoxaBiraB B tt i ir o a - /V5" * -oaoj n9k p 01 'jjnaoirojc xnxe en iraopo i9bdoax9iraok -y Hirog; XHS9IT HIOOHH9ffIOHHK9 9MffOICO^ BOI9BHHHOU OHhHJOIfBHY 8HHBI OXh 'I9JCK91TO 'HIH9H9Ife 9HS9ltfH9H - ^ ' ^ '& *3 'ff ' b (95) ' ed oxo9r xa9ira y a oxh 'OJQX en Hiro9 'ghhshironok XHaedii HXO anaoico^ x9bdoaaairaoe/ ^/ oxh 'ixhdoaoj oiraroa aodoxoh9h - BaxoH9aBd oaioiirsibsbhoiz: xgbodgase oxg oih (ad9ii9x n9bhjciroii rat t H ^t axongabd oxh 'i9^k8iro '(01?) H (62) OXh 'H9Bh^KOII *8f) HItBaXOH9aBd 0 BaxoHaasd EH H HBH -9HH fbhh93eoifonk9dn OJ8IHBH jcltho a' 9J OaXOHSaBj * +5 = 9 Blfff 19

23 Teopeuu SJIH HSKOTOPHX KJiaccos (3fr3fleJiHTeJieft HJJIH. I. 20KA3ATEJIBCTBO. floeiatoiho aonasatb,?to S/V -KOJiwia BJieTBOpffiDT ycjiobhk) eflhhctbehhocth HpaBtCC floiiojihehhj!, T.K. KOJIBIJO, ahthh30mop$h06 S/V -KOJIBIiy, CHOBa HBJIH6TCH S/V - KOJIBUOJ. PaccuoTpuM pabenctba (56). HoKasaieJiLCTBo c$opmyjinpobahhoro BCCTK c nowoiubk) HHayKium no qjicjiy /f =d(tfb)*cl(4b*). (58) ECJIH /r=(9 HJIH /r=/, TO ytbep^aerae oqebhaho. npeanojiokhm re neps, qio yibepsaehhe TeopeuH aokasano JJIH scex /c<m-n nycii HeKOTOpHX COOTHOJUeHHfl BHfla (58) K = m f m.>±. B cajiy TeopeMH 3 qeibepkh (fl,b,c,z)) a BeTCTByraipe pasehctbam (56), Jinfio nphhafljieaai HHOxeciBy A^, JIHfio nony^amtch MS ajieuehtob sioro MHOseciBa c HOMOBIBK) onepaua^ I)- PaccMOipHM raec(.0ea7, ot^+. nony^aeich H3 oc^ c IIOMOIUBD HeKoiopofi onepaimh I) - 4). 0 TaKHx nocjieaobaie^bhocthx (59) dyaeu rosopntb, ITO OHH oopasyat aenoiny. lenepb, ITO naine ytbepx^ehne aokasano ^JIH scex (56), ajifl Koiopux K=rn H oana as 6CTB SaKJIKUHTeJIBHHft 3JieM6HT H6KOTOpO{t UCnOIKH BHfla (59) fl^hhh /T.. ECJIH AZ=/, H, HanpHuep, ^etbepka (fl.b.c^'d) Hiiee? TO ECJIH xe TOTsepKa (tfj&^tyiateet BHS (i,t,i,i),to (fl,bf,c,d*)ntw* BHS ^,5*/,Z?*;, i.e. B *=?)*. IlyciB xenepb leisepka (^ibfc,'d)qc'sj> sakjibihtejibhhfl nekoiopofl qeno^kh SJIHHH PacciioTpHii I) <*.+! = pc&n). 3TO 3H8WT, ^TO 0^^= (C.jft.fl, B). Ho yibepswehhe TeppeuH cnpabeajihbo #JIH pasehctb Oic»aa cjieayei, vso ytbepaweiuie reopeira cnpabefljihbo H RUB. HHX pabohctb.

24 48 JI.A.EoicyTB 2) y ECJCH cl (X)=d-(y)=0, TO flan flokaaaiejibctba Hyraoro Hau yr- HeOdXOflHMO IIpHMeHHTB HHflyKE[HK> HO fljihhe aetloikh. ECJIH K6 HJIH d-cf) OT^HMHO OT HyJIH, TO nphmehhetch HHflyKIWfl HO /r (CM. (!?8)). 3) ocn.+ i ctc^-n) STO oshaiaei, ITO npmmehhtb Tenept HHflyKimio no = C ' T0 ect (d, B, C,2>)= (d.+c^ Biopoe as pasehctb (56) nphnraiaet 113 npea^okehhs HH^yKUHH,noJiy^aeM, ^TO cymectbyiot TaKHe sne- 3DE W.? Ho Torga ECJIH nph aiou pasehctbo (fl,b,cj7))me6t #=axw, TO MS (56) cjieayei, BUZ? (9. id), TO BCTL pabenctbo flb=c'j) He raaeet saaa (9.2). Toraa B caay nojioaehhoro BHine, nojiyqaeu, wo nepsoe as pasehctb (56) yao- ST ycjiobhhu (51) a Toraa, B cajiy jieuwh 3, aona x7 ' D' S>'<T\' D *' TI*1 TaKHe sjieuehtu H^, ^,0, Z/. wo ^'C.'2?'). (60)

25 reopeuh win HeKoiopnx KJiaccos (3ea aejihteaieji HyjiH. I. K pasehctbau, 'W1). (si) MOKHO nphmehhtb nepboe npesnoaosehae HHflyKiiHii no B0,1)o,E,/-" A-.. noaioiiy E 7)*= [ H ^erko BURSTS, Teopeua CJIEflCTBHE I. S/V -KOJisaa H B J I H B T C H GLB 20KA3ATEJILCTBO. IlycTB ft=p(s) - nekoiopoe, C - ajieuehth as /2 Taraie, flp.ncr.to. 810 BHaiHT, qio cyiecibyor ajiemehiu BfDeR., yaob^etbopflioiniie co- OTHoraeHioo (62) Bcex ajieuehtob 6,2), ysobjietbopnioiptx pasenctby (62), BHdepew aneuehth.5*2)*, ann KOTOPHX TOCJIO

26 50 JI.A.EoKyrL nphhhmae? MHHnuajii>Hoe IlpHMeHHM K npohsbojibhomy H3 pabbhctb (62) H pabchctby ft B = =C.Z)*TeopeMy 4. Toraa nojiyyuw, ITO Ha npe;zwyiero BHSHO, ITO ct(_f) = 0, TO ectb / * - 9io CIEflCTBHE 2. CyneciByioT (? -KOJitua, He B J I O K H U i i e B TOJia. STO yibepsaehhe BtiieKaei MS npefli«p;ero B cmiy ioro $akia, ITO cymectbybt S/V -KOjaija, ne BJioauoiHe B iena. 3. B STOW nyracie UH nokaxeu, KBK us TeopeMH fl cnegyet $an- Teopeua Koaa \_2\H KOJiea co cjia<5mi ajirophtmom., npekae scero, ^phhaz^jieka^ee Kony [2} onpeaeaeraie KOHTHHyaHTHHX IIOJIHHOMOB. IlyCTB t,t ti)... - HeHBBeCTHite. fm- ^=fn-i (t, -..t^trsfn-zfr...^) (63) B-pacSoie 2 aokaaahh c^eayiomiie pasehctba: Teneps R, - KOJILUO co cjiaduu ajirophtuom (CM. [21 ). HoKaaeH, iro GCJIH (^,8,C,D P('S) y Ct(tf)>d(C), TO cymectbyiot Tanae R., (66) (67) B cmiy Teopewa /^ tieisepka (tf,b,c,1))i cootbetctbyomafl

27 TeopeiiH BJIH HeKOTOpnx KJiaccos ($63 seahtejieft HyjiH. I. 51 pasehctby (64), nonyiaetch us ajiejiema (X,/,j(,i)c noiiomw) onepa - mitt /), /^y, </>x. PasencTBa ^-jf =/ / uaieat BBS C6'0 o /z.= ^?. flanee, ecjm K pasehctby (66) BHaa (67) nphmehhtl onepaiyih p ana fa y,to nojiypasehctbo TOPO xe Toraa, 3,6,0,7) (67). 'C, =- HaKOHea.paccuoTprai pasehcibo Toraa (67). X y (67) B r. JlHiepaiypa I. Cohn P.M. 2. Cohn P.M. 3. A.E. A.M. Factorization of in non-commutative power series rings.-proc.cam.philos.soc., 58(1962), Rings with & weak algorithm. Trans.Amer. Math.Soc.,109 (1963), accoumaihbhba CHCT6U B rpyniui - Mai. C(5. 6 (1939), 2, BKjnoieHZH accoiinathbhux chcieii B rpyium II, Mai. c<3. 8 (I9*0>., 2,

28 52 JI.A.BoKyTB 5. BoKyTB JI.A, 6. Cohn P.M. HeKOTOpne npnuepu xoneu, 6es H JiorMKa, 3 (1964), 5-6, Noncommutative unique factor! eation domains. Trans. lmer.math.soc., 109 (1963), Neumann H. 8. ASHH C.H. On theories with a combinatorial definition of "equivalense".- Ann. of Math., (43(19^2) BJioiBHra nojiyrpynn B rpynnh. 3AH (I960), K> 2,