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
3xox KJiacc o6o6maex Knaccw Konea, BBaneHHtie BJIOKOM H8H H
B § 2 AJIH Koneu Ha nannoro miacca noKaabiBaexca xeopeMa o
HOCXH HH/itx-Konfaua c ycnoBHeM MaKCHManfanocTH mis nonKoneu, o6o6maio—
maa aHanorHHHbie xeopeMti K.A.>KeBnaKOBa ann fiopnaHOBbix KOJieuQlH H
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B § 3 H § 4 aoKasbiBaexcsi, HXO B asyx nocraxoHHo mupoKttx
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408 H.n.llIecTaKOB
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, HTO BcHKaa raKaa npocxaa anreSpa Han noneMf xapaK -
2 nH6o anbxepHaTHBHa, JinSo flopnanoBa . (3rox peaynbxax
6 MJI aHOHCHposaH B.IOKOM B C^l ).
§ 1. OnpeaeneHHH n npocxwe cneflcxBHH
n. 1. Ilycxb Oi - neaccomiaTHBHoe KOni.no. Mu SyaeM sc6
npeanojiaraTfa, HTO C/L - i - onepaxopHae KOJIHUO, r\ae T^. /
Hoe accomiaxHBHO-KOMMyxaTHBHoe Konfauo c eflummeft / n aneMeHXOM a ,
, HXO -g — 1- -5- = / . SaMexHM, HXO iipn xaKOM npennonoaceHUH B an-" r-rpynne KOJibiia t/i/ ae MO«eT 6wxfa sneMeHTos nopanxa 2, xo
ecxi. ecnH ^^C — 0 , TO me = 0 . Hop. HneanoM
^/ MM 6yneM noHHMarfa ^ -Hflean ( (f> -noaKonbuo) .
TO o6o3HaiHM- Hepea (,3C,fy,J&) = (^CU)Z — •&(yZ) accounaxop
TOB £C)JJtZi ,- nepea \lJC,l/~\ JCi/ — UOC ~ KOMMyxaxop
lo&^C H y , Hepea 3CoU = JCiJ -\- yx — JSopnaaoBo
SXHX aneMeHTOB H
flajiee, o6oaHainM nepea /^^ (^r^.) onepaxop npaaoro (neBoro) yM-
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OnPEflE/IEHHE 1. KOJIUIO (% HaatiBaexcH snacTHHHbiM, ecyin B
OC B bin on He HO xoxaecxBo.
B tacXHOCTH, BCHKOe KOMMyxaTHBHOe IKIH aHXHKOMMyxaXHBHOe KOJIb-
ao anacxHHHO.
/IEMMA 1 . B a j i a c T H H H O M K o n f a u e s e p -
H M c n e f l y r o m n e T o x a e c T B a ;
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O HexoTopwx juiaccax HeKOMwyTaTHBHwx ftopnaHOBtix naneu. 409.
(5)
flOKASATE/IbCTBO. .TlHHeapHsys ToacaecTBO snacTHHHocTH (1) , no-
nyiaeM
Aanee, B jiK>6oM Kont>ue cnpaBennHBo TO»aecTBO
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— (wz, ,
TO ecTb (3) noKaaaHO. BwiHraa renepb HS ro>KAecTBa (3) ynsoeHHoe TOK-
necTBO (6) , nonyiHM (2).
flanee, HMeeM B cwiy ( 1 )
- U (oc..zl = ( j c , , z)
TO ecTb (4) AOKaaano. SaMeTHM Tenepfa, HTO B snacTHHHOM
LL(3C,lf,Z^ - artfarepHaTHBHan 4>yHKUHH sneMeHTOB jct z/, . B
U,(jrty,Z~) + Z/(t/,3Z,JZ) = O . OrcioAa « HS (4) cneayer (S) .
noKaaana.
C^HACTBHE. n y c T f a ^ - s n a c T H H H o e x o n t
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flOKASATE/IbCTBO. HCHO, HTO 3 aenaeTca <P -nonMOnyneM
410 H.n.lUecTaKOB
flanee, nerxo BHAeri., HTO J7 = %• , me<
TaxHM oSpaaoM, ana noxaaaTejifaCTBa cnencTBHS nocxaTOiHO
HTO ana jiio6wx aneMeHTOB Ct,S,C «3 C%. (a°D)C, C((Zo&)€. %-,HM66M
B cony (5)
OFIPEflE/lEHME 2. KoMMyraTHBHoe Konfauo (X HasbiBaercfl ftopaa -
HOBbIM, eCJIH B £% BbinonneHO TO>KfleCTBO
(xz,y,sc) -_(?: (7)
OFlPEflE/IEHHE 3. SnacxHHHoe xonbuo, B xoxopOM BbinonHeHo TOJK-
necTBO (7), HaafaiBaerca HeKOMMyraTHBHbiM ftopaaaoBbiM xontuoM.
HanOMHHM, HTO KOJIfaUO HaSblBaeTCH HH^WlOTSHTHblM HHfl9KCa 71, , 90-
/IH npoH3BeneHHe nio6bix 7Z ero sneMeHTOB c npoHSBOJifaHoft paccraHOBKoft
CKo6ox pasno aynio; KOHWIO HasbiBaeTcn nunb-K.onbu.OM, OCJIH Kawabift ero
nopojKnaeT HHjibnoreHTHoe nonxonfauo.
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paMH. Cnenyioinaa xeopeMa aasepmaeT KnaccH^HxauHio npocrbix KoneiHo -
MepHbix HexoMMyTaTHBHbix ftopoaHOBbix ajireSp:*)TEOPEMA 1 . n y c ,T> fa £% - K O H e n H o ^ e p H a a
n p o c T a a H e K O M M y r a T H B H a a ftopaaHO-
B a H H n t - a n r e 6 p a j i a a n o n e M & x a p a ic-
r e p H C T H K H ? * 2 . T o r n a ^ a H T H K O M M y T a -
T H B H a.
flOKASATE/IbCTBO. PBCCMOTPHM nHHefiaoe nonnpocrpaHCTBO -^ ,
nopoacasHHoe KBanparaMH ajieweHTOB anre6pbi CK- . B cnny cnencTBaa
neMMbi 1; ^7 _ Hflean anre6pbi (X. . fljia noKaaarenfaCTBa reopeMbi flo-6
CTBTOHHO noxaaarfa, HTO ^7. ffl, , TBK KBK rorna B cnjiy npocroTw ajire6-6
pw OL J, = O H Bee aonasano. PBCCMOTPHM npHcoeaHHgHHyio anre6py
aEOHcupoaaH BnoxoM B H 73 .
O Hexoxophix Knaccax HeKOMMy.xaxHBHhix ftopaaHOBbJX KQJieii 411
,w . „„, nonyneHHyro sseaeHHeM Ha jmneftHOM npocTpaHcrse c/f HOBOPO
\ u "* J~(&. b + od^) , rne etc? - yMHo*eHHe B anreSpe
(% ", /lerico npoaepHXb., KTO CK. - ftopaanoaa HHniy-anre6pa. Flo reope-
~\rfr^ /-rt/^\ =L W
MexHM xenept., H.TQ u£ — Ct , %= (& ) KBK MHO^KecTBa.
^ = Q .
TeopeMa
OnPEflE/IEHHE 4. HeKOMMyraTHBHoe flopnaHOBO KOHbUO
saercH craHaapTHbiM, eonn B •*(% BbinonueHbi
B qacTHOCTH, KBK accounaxMBHtje, xax H" ftopaaHOBbi Kons.ua HBJIHIOT-
OOPEAEAEHHE 5. KOJIBUO CK naabiBaexcH npaBoanfaxepaaxHBHbiM,
ecnH B Ot BbinonneHO ro>KnecTBO
— O. (8)
OnPEflE/lEHHE 6. 3/iacTHHHoe np%Boani>TepHaTHBHoe
Baexca a/ibxepHaTHBHtiM.
OnPEflEHEHHE 7. HeKOMMyrarHBHoe ftopnanoBo Koni>uo (X,
erca o6o6meHHo— craHnapTHbiM, ecan B i^ BbinonneHbi
, 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
Bee ocHOBHue reopeMbi, aoxasauHbie B H13H IUa4>epoM ana KOHBH -
o6o6meHHo-CTaHflapTHhix anre6p, Sbrna aHOHcnposaHbi B C8U
BJIOKOM ana KOHeiHOMepHbix HeKOMMyxaTHSHfaix ftopnaHOBbix anre6p yno-
ToacaecTBaM (10) a (11).
3aM6THM, HTO BcflKoe ftopaBHOso HJiH anbrepHaTHBHoe Konfauo yao -
TOHcaecTBaM (10) H (11). Eojiee roro, nerxo npoaepHTb,
H BCHKOS craHnapTHoe Konbuo yaoBneTBOpaer STHM TO«necTBaM.
JK8M renepfa, qTO H B nro6oM oSoSmeHHO-craaaapTHOM KOJIMIS
TOKnecTBa (10) H (11).
nPE/UIOKEHHE 1 . H e K o ' M M y x a T H B H o e . f i o p a a -
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 -
a e c T B y ( 9 ) , y n o s n e T B O p a e T H T o a c a e c r -
B a M (10), (11).
AOKA3ATE;ibCTBO. MeMa B (8) y H u) MecraMii H
Baa nonyHSHHoe TOJKflecTBO c (9), nonynHM B cnny (1 )
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HrtH
6r,y, zuJ)+(u>,y,xz)+(Z,t/,x-u))-)-(x,u),z y)-t-( z/,u>,xz)+ (z, u>,scy)=O ( 12)
Menaa anecfa JC H Z MecraMH H BbiHHTaa nonyneHHoe xownecTBO HS
(12), nonynaaM
/( to, y , Lsc, zl) -I- (y, w, f^r, zip = O ( 1 0
oTKyaa cneayer (10).
flanee, H3 (12) nojiynaeM BBH^y (9)
• O = \^C^,
C &, (z, u),y)] + fjr, (z, y, uJf\ o ,
HJIH npH ii)=
flopaaHoso xo>KnecTBO (7) ,
O HenoTOpbix KJiaccax HeKOMMyxaTHEHbix fiopgaHOBbix icojieu 413
u),z/,JC') = O,
orxyna iipn z/ = re , u)=Z HMBCM B cHjiy ( l')
= £7,
HJIH
C,xz) = (z, (C ,
renepb B (9) U= 3C , U) — Z • riojiyiHM B CHny ( lj H (13)
STO TOjKflecrBO c npenwayrnHM, nonynaeM
B CHny ( 10 )
STO xo>KflecTBO, nonyiHM
flanee, HMBOM BBH«y (13'), (!') H (9)
[&,(y, z, u))} + £(sr, y,x),uf\ - C^r, Cy, u>, zj] + L i^, (i/. sc, z
»y> EX " ) — r u), Z, jcy> - C^c, x, z/W) - (y, x,
STO B (2) , nonynaeM BBHfly (1^) H
B cany (14) H ( r ) nonynaeM
TO ecn. (11) HOKaaaHO, HTO oaHoapeMeHHo noKaabiBaeT npeanoxceHMe.
TaKHM oSpaaoM^KJiacc Konen, BBeaeHHbift EJIOKOM, coaepxiMT see
o6o6ineHHO-cTaHnapTHwe nonbua.
n.3. B H 143 A.TanH onpeaennn Knacc W -nonycTHMbix a^re6p cne
o6paaoM:
414
OnPEflE-TIEHHE 8. HeKOMMyraTHBHas flopnaHosa anre6pa C%, Haa
noneM £" xapaKTepHCTHKH ?* 2 HasbisaeTca W -AonycTHMoft, ecnH ana
n^ Qt IL € £~ > TBKHX, HTO nn5o •?+ 2p 4- <5<5 ~ •*£" 7 - ^ ,
p -\- 4- 'C ^ O , B £?Jf Bbinojikenbi ro>KflecTBa :
0. ( is)
noxasaji, HTO Kaacaas: o6o6meHHO-CTaHflapTHaa anreSpa aajiaeT-
CH W -nonycTHMoit c napaMexpaMH n = — 2 , 0 = £~ = O . C apyroft
CTOpoHHi, OH nocTponji npHMep W -flonycTHMoft ajire6pbi, B KOTOpofl He
BbinonuaHO TO>KnecTBO (10). TBKHM o6pa3OM, K/iacc W -nonycTHMbix .an-
re6p conepwHT ace o6o6meHHO-CTaHflapTHbie anreSpw H He conepxcHTCH B
Knacce Koneu, BBenenHOM BHOKOM.
flanee, TSHH noxaaaji, HTO ocHOBHbie xeopeMbi , noKaaaaHwe LUa<t>e-
POM ana KoneHHOMepHbix o6o6meHHO-CTaHnapTHbix anreSp, cnpasefljiHEbi H
njia KoneHHOMepHbix W -flonycTHMbix anreSp. B HBCTHOCTH, B Knacce
W -nonycTHMwx anre6p cnpafiennHBa crpyKTypHaa TeopeMa BenaepSop -
Ha 06 oxmenneHHH pannxa/ia (STOT 4>aKT oTpawen B HaaBaHHH nanHoro
KJiacca anreSp) .
n.4. SaMeTHM, HTO xax xnacc VV -nonycTHMbix anreSp, TBK H Knacc
Koneu, BeeaeHHbifi BHOKOM, aanniOTca noziKJiaccaMH miacca
HHX ftppaanoBbix Konea, onpenenaeMoro TownecTBOM (15). HaynHM
ropbie cBoftcTBa sToro itnacca moneu, Hcnojifaays MeTOflbi pa6oTbi
FlyCTI. OL - npOHSBOnfaHOe KOJIfaUO. B KOJIblie BCeX 3HAOMOp(})H3MOB
anaHTHBHoft rpynnw xonbua Ot- MHoxcecTBo 3HaOMop4>H3MOB BHna 7 . ,
rae CL^.C/i- , nopo>KAaeT noflKonfauo, KOTOpoe HaabiBaeTca KOJII.UOM yM -
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meft narHHGKOH 6yKBofl Jl ,
OIIPEflEJlEHHE 9. Konfauo Ot MM 6yaeM nasbmarb nonycTHMHM,
ecnH B C)i BbinonueHbi
= 0,
O HeKOTopux maccax HenoMMyxaTHBHbix flopgaHOBbix Konen 415
$lx,yl,y,y) = o. (is)
PaccMorpHM cBo6oflHoe aonycTHMoe KOHUIO «i> or MHoacecTBa CBO-
DoaHbix oSpaayroinHX X — [cT-\ KO.nbno ero yMHOiKennfl £ . FlycTb
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TO ecTb ecnH Ct=£LCZ, • in Ct = GL Cff ,*<>&>& B TOM^T £ f £
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nnHHbi ynopflflOHeHHOCTb neKCHKorpa4)H«iecKaa cnpasa Haneso.
OOPEAE/IEHME 10. CJIOBO w=7" 7" ...7L, H3_^/*6ynaM aa -"* "
3biBaTb -HopManfaHbiM cnoBOM, ecnH ana xawnoro ,— 3 , r . . . t
. ,£,-/ Z-
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n H H e f t H O f t K O M S H H f l U H H ~J — H O p M a n b H b l X
c n o s x o r o «e c o c T a s a O T H o c H T e n b H O
L/ , H T o H VV .
flOKASATE/lbCTBO. Ha jiHHeapHSOBaHHbix TOHcnecTB sJiacTHHHOCTH
( I ) H iiopaaHOBOCTH (7 ) nonynaeM c/ieayiomHe ToacnecTaa B so/ibue
416 H.n.lllecTaKOB
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=i Z^ . rae
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HTO v^ J ' */ HMeror T<TT >Ke cocxaB, HTO H W- , z/. , HOj
Ha TO»cnecTB (17) - (19) nonynaevi
(20)
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. Pas6epeM xawabift HS HecKonbKRx BOSMOWHUX cnynaeB.
Ha (21) H (22)
(23)
3aM9THM renepb, AITO Knacc nonycTHMtix xoneu aaMKHyT OTHOCHrenfa-
KO aHTRH3OMOp4)R3Ma. CjieflOBareTibHO, ecnH B & cnpaseonHBO HeKOTOpoe
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HS (23)
Tenepb B cany (20) H3 (23), (24)
(24)
(23)
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*g*c -t-gficfij—jtS^Lf-m^ ). (26)
Haxoneu, BBHny (16)
(25)
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noncraBHB renepi, B W BMSCTO Ts 77 TV/ cooTBercrByiomyio npa -
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2. /=<3/, C<d.
Ha (21) nonynaeM
HHBepcHoe
S L L = °- (30)
B W BM6CTO ^V /" 7"/ COOTBBTCTBy KDIHyK) npaBylO HaCTb H3
(23) - (28), c yifeTOM (29) H (30) BHOBI> nonyiHM W=2cx-W-z ^ lj
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H HHBepcHoe
VgLj^ct - O. (32)
flanee, na (16) cnenyeT
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(33) nonynaeM, HTO \V=
TOT >K9 COCT8B, HTO H W
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418
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TO \V=2Jct . W- , rne see \V- < W H HMBKDT TOT we cocTaa,
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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.
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