o hekotopblx k/iaccax hekommytathbhhlx h.n ...math.nsc.ru/~alglog/01-10/10/10n4-6.pdf2010/10/10 ·...
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Anre6pa H norHKa, 10, N°4(1971),407-448.
519.48
O HEKOTOPblX K/IACCAX HEKOMMYTATHBHhlX
ftOPflAHOBhlX KO/IEU
H.n.lIlECTAKOB
B 1948 r. A.A.An6epx s £ 6 3 onpenennn Knacc cxaHnapxHbjx anre6p,
see accounaxnBHbie H ftopnaHOBbi anre6pbi. flna KOHeiHOMep-
HHX CTaHnapxHbix ajire6p xapaicxepHcxHKH s4 2 6biJiH aoK aaaHbi ocHOBHbie
crpyKTypHbie TeopeMH (C63>ClOj, L12J ), o6o6inaiomHe KnaccniecKHe re—
opeMw BeaaepSopna mis accounaTHBHwx ajireSp.
B 1988 r. P.A.ma<J)ep B [7l33 pacnpocrpaKHji STH xeopeMbi Ha BBe -
neHHbift HM xaM >Ke K^acc o6o6meHHO-CTaHAapTHbix anreSp,
ace anfaTepnaTHBHbie H CTanaapTHwe anreSpbi, PeayjifaTarbi LUaiJiepa
aareM o6o6meHH B pafioxax P.E.Bjioxa [I 8j H A.T3flHLl43.
B npennaraeMoft pa6ore paccMaxpHBaeTca KJiacc HeKOMMyxaxHBHbix
KO/ieu, onpeAenaeMbifi
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2 nH6o anbxepHaTHBHa, JinSo flopnanoBa . (3rox peaynbxax
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n p o c T a a H e K O M M y r a T H B H a a ftopaaHO-
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aEOHcupoaaH BnoxoM B H 73 .
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— O. (8)
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OnPEflEHEHHE 7. HeKOMMyrarHBHoe ftopnanoBo Koni>uo (X,
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, HTO uiacc o6o6meHHO— cxaHflapTHbix Konen
ace craHnapTHbie H ajifaTepnaTHBHbie Kcuibiia, npHH^M ocHOBHbie CTpyKTypnue
reopeMbi, cnpaBeoiiHBbie B KOHeiHOMepHbix craHnapTHMx H anbxepHaTHBHbix
anre6pax, ocTarorcs BepHbiMH H ana KoneiHOMepHbix o6o6meHHo-cTaHnapT -
Hbix anre6p.
n.2. B [^8J P.E.Bnox onpeaenmi HOBbifi icnacc HeKOMMyxaTHBHbix
ftopnaHOBfaix Koneii co cnenyroraHMH
o. ( i i )
412
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BJIOKOM ana KOHeiHOMepHbix HeKOMMyxaTHSHfaix ftopnaHOBbix anre6p yno-
ToacaecTBaM (10) a (11).
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TOHcaecTBaM (10) H (11). Eojiee roro, nerxo npoaepHTb,
H BCHKOS craHnapTHoe Konbuo yaoBneTBOpaer STHM TO«necTBaM.
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H O B O K O J i b u o ^ . y f l o ' s n e T B o p a r o m e e T O > K -
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= £7,
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B cany (14) H ( r ) nonynaeM
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1. K.A.JKEB/IAKOB, PaapeuiHMocrfa H HmibnoTeHTHOCTi.Koneu, AnreSpa H norHKa, 5, N> 3 (1986), 37-58.
2. E.H.KySbMHH, O6 aHTHKOMMyTaTHBHwx anre6pax, yaoaneTBOpaio-max ycnoBHio Snrejia, CH6.MaT.jKypHan, 8, Nfe 5 (1967), 1026-1034.
3. A.H.llIHPIilOB, O HeKOTopwx HeaccounaTHBHbix xonsuax H anre6pa-HiecKHx anre6pax, Max. c6., 41, N? 3 (1957), 381-384.
4. H.n.llIECTAKOB, KoHeiHOMepHwe anre6pw c Hmib-6a3HCOM, Anre6-pa H jiorraca, 10, N? 1 (1971); 87-99.
448
5. A.A.ALBERT, A structure theory for Jordan algebras, Ann.of Hath.,48,5*6-567 (19*7).
6. A.A.ALBERT, Power-associatiYe rings, Trans. Ainer.Math. Soc.,6*(19*8).552-593 .
7. R.S. BLOCK, Determination of A™ for sinple flexible al -gebraa, Proc.lat.Aead.Sol.USA.,61(1968),39*-397.
8. R.B.BLOCK, A unifioation of the theories of Jordan andalternatiTe algebras, lot.of the Aner.Math.Soc.16,V5(1969),803.
9. H.BRAU1,M.KOBCHEB, Jordan-Algebren. Berlin-Heidelberg -lew Torks Springer,1966.
10.B.KLBIBTELD, Standard and accessible rings. Canadian J.•ath. ,8 (1956).335-3*0.
11.K.MCCRIBX01, Structure and representation of non-cosiiuta-tlre Jordan algebras. Trans.Aaer.Math.Soe. ,121 (1966) ,187-199.
12.R.D.SCHAFK8, Standard algebras, Pacific J.Matb..29,11,203-223 (1969).
13.B.D.SCHAFBR, Generalised Standard algebras,J.of Algebra ,12 (1969), 386-417.
14.A.THBDT, Zwa Wedderbumisolien Zerlegangssats,Math.Z. ,113(1970),173-195.
21 anpena 1971 r.