Luận văn Một vài mở rộng của định lý Jacobson ( về điều kiện giao hoán của vành )

Lu$nvanTht;lcSI Toan:MQlvaimerQn8cuadinhIy Jacobson Trang20 jD~ 2: DINH LY JACOBSON. (v€ di€u ki~ngiaohoan) A '" ? A & MOT VAI MO RaNG. . Trong ph~nnay cua b~liIu~nvan chungta se v~ndlJng chuang trlnht§n congvao illQtvanht6ngquatquacac budcda:trlnhbay trong illlJC cu6i cling cua ph~nireD.D~cbic$tIa d6i vdi di€u kic$ngiaohmin cuaillQtvanhVI tinhch§tnayduQcbaaloanquaphepI§y t6ngtn!ctie-p con.ClJ th€ Ia takh£ngdinhtinhgiaohmincua illQtvanhdlja vao illQt so'di€u kic$nchotrudc. Nhungtrudch€t Ia k€t quacuaJacobson. §1.DJNH LY JACOBSON Trudckhi chungillinhdinh19Jacobsontac~nillQtso'b6 d€: 86 d~(2.1.1):ChoD la m(Jtvanhchiacod(ics6p * 0vaZ la tamcuaD. Gia sit'm(Jtphdn tit'a E D, a ~Z saDcho apn=a wJi m(Jtn ~1 naodo. Khi do t6nt~iphdntit'XED dt choxax.l=ai * a wJi m(Jtsl]'nguyeni naodo. Til b6 d€ ta co th€ chung illinh illQt dinh 19r§t n6i tie-figcua Wedderburn: Menh d~(2.1.2):(Dinh19Wedderburn)MQi vanhchiahilu h~nd~ula truiJng. He qua(2.1.3):ChoD la m(Jtvanhchiacod(ics6'p * 0 vaG c D la m(Jt nhomconnhanhiluh~ncuaD thEG la m(JtnhomAbel (nenla cyclic). GV hu'dn8diln: PG6'>- T6'>I)lii Tu'dn8Tri IIV Thu'chien:Phan Tru'dn8Linh Lu~nvanTh(;lcsI Tolin:MQlvaiIDdrQn8cuadjnhIt Jacobson Trang16 86 d~(2.1.4):ChoD la mQtvanhchiasaDchovdimQiaE D a~ut6ntCJi mQts6nguyen(a) >1al choan(a)=a.Khia6D la mQtrudng. CHUNG MINH: Ta co 2 E D va 2m=2 vdi m> 1DenD co di;lcs6nguyent6p *-o. Ne"uD kh6nggiaohmlnthl t6nt(;limQta E D va a ~Z vdi Z la Him cuaD. GQi P la truongnguyent6 cuaZ. VI an(a)=a Dena d(;lis6 tIeDP. Tn doP(a) la mQtruonghuuh(;lncopkphftnill'vataco apk =a. V~ymQidi~uki~ncuab6d~(2.1.1)d~uduQcthoamand6ivdi a Dent6nt(;limQtphftnill'bED d6 chobab-l=ai *-a. Quanh~naycungvdislfki~navabd~ucoca'phuuh(;lndftnd€n a va b sinhra mQtnhomconnhanhuuh(;lnG trongD. V~ytheoh~qua (2.1.3)thlG giaohmln. Do a, bEG va ab*-bathldi~unayla mallthu~nva b6 d~duQc chungminh.. Baygiotacoth6chungminh: Menh d~(2.1.5):(Dinhly Jacobson)ChoR la mQtvanhsaDchovdimtJi phdntita E R a~utantCJimQtso'nguyen (a)>1,phl;tthuQca,al cho an(a)=a thiR giaohoan. CHUNG MINH: Trudche"tachungminhR la nll'addn. Thlfc v~y,gia sll'a E J(R) thl doan(a)=a, n >1ta suyraa =O. VI khi do a.an(a)-l= a vdi an(a)-lE J(R) va ta bie"tding ne"uux =u, wJi x E J(R) thlphaico u =O. [CM: x E J(R) ~ -x E J(R) Den3x' E R: -x +x' - xx' =0 0 ( , ' ) " " 0]~ =u -x +x - xx =-ux+ ux - uxx =-u+ux - ux=-u~ u = V~yJ(R) = (0). Bay gio,do R la nll'addnDentheom~nhd~(1.5.3)thlR la mQt t6ngtrlfctie"pconcuacacvanhnguyenthuyRa.Ma m6iRala mQtanh d6ngca'ucuaR Denke"thnadi~uki~nan(a)=a.Hdnnua,m6ivanhcon vaanhd6ngca'ucuaRaclingthoadi~uki~ndo. ev hllc'5n8d&n:pes -TSBliiTlldn8Tri IIV ThllC bien:Phan Trlldn8 Linh 1u~nvanTht7csI Toan:MQLvaimdrQn8cuadinh15'Jacobson Trang17 LamQtvanhnguyenthuynen theom::JDn ho~cmQiDm,voi D la mQtvanh chia, dSu la anh d6ng ca'ucua mQt vanhconcuaRa. V~yne'uRakh6ngla mQtvanhchiaD thl t6n t~imQtDkvoi k > 1 ke'thliadiSuki<$ncuagia thie't.DiSu nayhi€n nhiensaiVIph~ntti': 0 1 0 ... 6 0 0 0 ... 0, D=e12 E ka= 0 0 0 ... Q thoa diSu ki l1a diSumallthuftn. V~yRa phai la mQtvanhchianenphai giaohoantheob6 dS (2.1.4). Tli do, VI R la mQtt6ngtn!c tie'pcon cua cac vanh giaohoanRa nendinggiaohoan. " ... ?,,? , §2. MQT VAl MO RQNG CUA DJNH LY JACOBSON Dinhly Jacobsontuychodu'QcmQtdiSuki<$ncuatlnhgiaohoan, nhu'ngclingconmQtnhu'Qcdi€m: Co quait vanhgiaohoanthoagia thie'tcuano.Do la ly domataphaiOmcachmafQngdinhly nay. Sailday,chungtasexetWimQts5mafQngnhu'v~y.Nhu'ngtfu'OC he'tac~n: Dinh nghia:Trongm(}tvanhR tilyy, tagQi: 1)M(}tgiaohoantit (c{{p2)cuahaipht1ntitx,y fa: [x,yJ =xy - yx. 2) M(}tgiaohoantitcapn (n >2) cuanpht1ntitdU:(jcdjnhnghza bangquingp: [Xj,X2,...,XnJ = [[Xj,X2,...,Xn-lJ,XnJ GVhu'dn8di\.n:PGS-TS ])liiTu'C5n8Tri liV Thu'chien:PhanTru'C5n81inh Lu~nvanTh~csTToiw:MQlviiimCirQn8clladinh15'Jacobson Trang18 Nhanxet: 1)Vdi mQix,Y E R thl taco: x glaDhoanveiiy [x,y]=0 2)Vdi mQin ~2 thlgiaohmlntli'ca'pn cuan ph~ntli'co tinhcQng tinhtheotungbien.Tucla taco:Vdi mQii, (1~i ~n) [Xj,...,Xi-j,Xi + Xj,Xi+j,...,Xn]= [x j,...,Xi-j,Xi>Xi+j,...,Xn]+ [x j,.. .,Xi-j,.x:;,Xi+j",.,Xn] 3)Neu A giaohmlnvdi mQiXk,(1~k ~n) thl: [Xj,...,Xi-j,Axi,Xi+j",.,Xn] = A[Xj,...,Xi>...,xn] Tu khaini~mtrentaco: Menh d~(2.2.1):(Dinhly Jacobson-Herstein)ChoR LamQtvanhsaDcho vdimQix,y E R dJu t6nt(IimQts6nguyenn(x,y)=n >1 (phl;lthuQcx vay)d! cho[x,yt(X,y)=[x,y]thiR glaDhoan. Trudchet,tachungminhdinhly naychotruonghqpd~cbi~t: 86 d~(2.2.2):Nlu D LamQtvanhchiathoagid thilt cuam~nhdJ (2.2.1) thi D glaDhoan. CHUNG MINH: Gia sli'D la mQtvanhchiathoagiathie'tmaD khonggiaohmlnthl t6nt(;lia, bED saochoc =[a,b]"* O. Theo giathietthl cm=cvdimQtm>1naodo. Neu A "*0 thuQctamZ cuaD thl ta co AC=A[a,b] =[Aa,b].V~y theogiathie'tdingcomQts6tvnhienn >1d€ cho(Act =AC. Tu do,ne'ud~tq=(m-1)(n-1) +1thltaco(Acf =ACva cq=c. V~y:AC=Aqcq=Aqchay (Aq-A)C=O. Do D la mQtvanhchiava c"*0DentasuyraAq=A. eV hudn8d&n:pes- TS l)lii Tuc'5n8Tri IiVThuchien:PhanTruc'5n8Linh Lu$nvanThc;tcSI Toan:MQlvaimdrQn8cuadinh15'Jacobson Trang19 V~y,voi mQi'A E Z d€u t6ntC;liq > 1 d6 cho'Aq= 'A,DenZ la mQt tru'ongco d~cs6p =j:. O.GQiP la tru'ongnguyent6 cuaz. TakhAngdinhrAngcoth6chQna,bED saochochAngnhung e =[a,b]=j:.0 maconco e ~Z VIneukhongthlmQigiaohoantu(dfp hai)trongD d€u thuQcz. Tli dothle E Zva ae=arab- ba)=arab)- (ab)a=[a,ab] E Znen tasuyraa E Zmau thu§nvoi di€u kit%ne =[a,b] =j:.O. V~ytacoth6giasue=[a,b] ~Z. Theogiathie'thlem=eDenedC;lis6trenP. V~yCpk=c voimQts6 nguyenk >0naodo. Toi daythlmQigiathiefcuab6d€ (2.1.1)d€u du'Qcthoamanvoie Dent6ntaiph~ntuxED d6choxex-l=ei =j:.e, tilc la xc =eix. Noi rieng thl d =ex- xc=[e,x] =j:.0 va theogia thie'tthl t6ntC;lis6 nguyent >1d6 d =d hayd co ca"phuuhC;lntrongnhomnhanD* cuaD. Nhu'ngta lC;lico: de= (xc - ex)e=eixe- e(eix)=ei(xe- ex)= eidhay deal =ei =j:.e (va noi riengthlde=j:.cd). Voi di€u kit%n ayva e,d d€u co ca"phuuhC;lntrongnhomnhanD* ta suyra nhomcon nhansinhbdi e va d trongD la huu hC;lnDengiao hoan theoht%qua (2.1.3).Di€u nay mall thu§n voi di€u kit%nde =j:. cd. V~yb6 d€ dadu'Qchungminh.. Voi b6 d€ naytaco th6chungminhmt%nhd€ (2.2.1): CHUNGMINHM~NHDE(2.2.1): GiasuRIa mQtvanhthoatinhcha"t: '\Ix, Y E R, ::In=n(x,y):[x,yT(X,y)= [x,y] Ta chungminhR la mQtvanhgiaohoan. Ta xetcactru'onghQpsail: a)NeuR la mQtvanhehiathlR giaohoantheob6d€ (2.2.2). b)NeuR Ia mQtvanhnguyenthuythltheomt%nhd€ (1.4.3),ho~cR la mQtvanhchiaD, ho~ccomQtk >1d6Dkla anhd6ngca"ucuamQt vanhconnaodocuaR. GVhu'cn8d~n:DG8 - T8BuiTu'dn8Tri tIVTht{cbien:DhanTru'dn8Linh Lu~nvanTht;lcsI Toan:MQtVatmdrQn8cuadinh15'Jacobson Trang20 Ne'ukhanangthli hai xay ra thl d~tha'yDkclingke'thlia tinhcha't lieU tronggia thie'tchoR VI tinh cha'tnay baa toanquaphepla'yvanh conva anhd6ngca'u. Nhu'ngkhi do,ne'utrongDktaxetcacphftntli': 1 0 ... 0'1 (0 1 0 ... 6 00...01 ,,1000...0=ell va Y=x= =e]2 0 0 ... 0) ,,0 0 0 ... 0, thl ta co: [x,y] =xy - yx=e]2 =y val =0 lien [x,yl =l =0 *-[x,y]. V~ytasuyra [x,yt *-[x,y]vdimQin >1.Mauthuftnvdi di€u ki<$nDk thoagiathie't. V~yR phaila mQtvanhchialienR giaohoan. c) Ne'uRia mQtvimhnrladdnthl theom<$nhd€ (1.5.3),R d~ngca'u vdi mQttangtn!ctie'pconcuacacvanhnguyenthuyRa.Ma m6ivanh nguyenthuyRa, la anhd6ngca'ucua R lien ke'thlia di€u ki<$ncua gia thie't,v~yno phai giao hoantheob). Tli do R cling giao hoan VI tinh giaohoandu'QcbaatoanquamQtd6ngca'uva phepla'yvanhcon. d) Ne'uR la mQtvimhtuyy thl ta xetvanhmladdnR/f(R) cling thoagiathie'tliengiaohoantheoc).Tli do,vdimQix,y E R thlxy-yxE feR). Trongtru'onghQpnaythldoxy- yx=[x,y] E feR) va [x,yt =[x,y] vdin >1,lientasuyra[x,y]=0 [dotinhcha't(ux=u,xEf(R) =>u=0]. V~y,vdi mQix,y E R thltad€u co [x,y]=0 lienR giaohoan.. Bay gio ta xetmQtmarQngcuadinh19naychotru'onghQpgiao hoantli'cuanphftntli'(n>1)trongR. Menh d~(2.2.3):Nfu R la m(}tvanhkh6ngchrianil idealkhdc(0)(ho(icR nrladan)saocho: (1)Co m(}ts6nguyen >1naodomaWJimQiXj,"',XnE R thitan tc;zim(}tm >1 (ph{lthu(}cxj,...,xn)di cho[xj,...,xn]m=[xj,...,xn]. Khi doR la m(}tvanhgiaohodn. GV hu'dn8d&n:PG8 -T8 BuiTu'dn8Tri liVThu'chien:PhanTru'dn8Linh Lu~nvanTh~cSf Toan:MQtvalillC'5rQn8cuadjnhIf Jacobson Trang26 DS chungminhm~nhd~naytacffn: B6 d~(2.2.4):Ne'uR ia m()tvimhchiathoaddu ki~n(1) thztaco: (2) [xj,...,xnl=0 wJi mQiXj,...'XnE R (TrangtrlliJnghC;pnaytanoi: [xj,...,xnliam()td6ngnhdtthactrenR) CHUNG MINH: Gia sii't6n t(;liXj,".,XnE R saocho a = [xj,...,xnl*-O.Khi d6, theo giathie't,t6nt(;limQtm> 1dSam=a. GQi Z la Himcua R thl VA.E Z ta c6 A.a=A.[xi,...,xnl=[Axi,...,xnl clinglamQtgiaohmintii'ca"pn trongR nen3k >1 dScho (A.al =A.a. V~yne'udi;itq =(m-l)(k -1) + 1 thl ta suyra aq=a va (A.a)q=A.a. Tli d6 ta du'QcA.a=(A.a)q=A.qaq=A.qa.V~y (A.q- A.)a=O.Ma RIa mQt vanhchiava a*-O nentadu'QcA.q- A. =0,VA.E Z. V~y,voi 2 E Z thl 3q > 1 dS cho2q=2 nen tru'ongZ (hoi;icvanh chiaR) c6 di;ics6p *-O.GQiP la tru'ongnguyent6 cuaz. Ngoaita,taconc6thSchQnXj,.",XnE R saochoa =[xj,...,xnl~Z VIne'ukh6ngthlmQigiaohmintii'ca"pn trongR d~uthuQcZ. Khi d6ta xeta =[xj,...,xnl*-0 va ne'udi;itb =[xj",.,xn-ilthl tac6 [b, bxnl=b(bxn) - (bxn)b=b(bxn- xnb)=b[b,xnlvoi [b,bxnl=[xj,...,xn-i,bxnlva [b,xnl =ad~ula giaohoantii'ca"pn trongR nenthuQcZ.V~y [b,bxnl=baE Z voia E Z vaa*-Onentadu'Qcb E Z (vIR la mQtvanhchia).Tli d6ta suy ra a =[b, xnl = 0, mall thu§:nvoi gia thie'ta*- O. V~yta c6 thSgia sii'3xj,...,xnE R saochoa =[xj,...,xnl~z.Mata l(;lic6 am=a,m>1nena d(;lis6trenP. Dod6praYla mQtmarQngd(;li s6,nen cling la mQtma rQnghii'llh(;ln,cuaP. Tli d6 P(a) c6 ca"pptvoi mQtt ~ 1 nao d6. N6i cachkhac,t6n t(;lit ~1 dS cho apt =a. Tli d6,a thoacacdi~uki~ncuab6d~(2.1.1)nent6nt(;lix E R dSchoxax-i=ai *- ahaytu'dngdu'dngxa=aix*-ax.N6iriengthlc =[a,xl =ax-xa*-O. Nhu'ngvoi di~uki~nnay ta l(;lic6 ca =(ax - xa)a=a(xa)- (xa)a= a(aix)- (aix)a=airax-xa)=aicnentasuyracac-i=ai *-a,n6iriengthl ca*-ac. ev hudn8d&n:pes -TSBidTudn8Tri IIVThuchien:PhanTrudn8Linh Lu~nvan Thl.1csTToan:MQLVal ffidrQn8cua djnhIt Jacobson Trang27 M~tkhac,ta conc6 a =[Xj"",xnlvac =[a,xl =[xj,...,xmxl= [[xj,x2l,x],...,xmxld~ula giaohoantuca"pn trongR nen c6 ca"phUll h(;tn trongnh6mnhanR* cuaR. Vdi cacdi~uki~ntrenthlnh6mconsinhbdia va c trongR* cling c6 ca"phUllh(;tn entheoh~qua(2.1.3)la nh6mAbel,mallthuiinvdi tinhcha"tca=I:-ac.B6d~da:du'Qcchungminh. B6 d~(2.2.5):Khdng dtnh cila b6 d~(2.2.4)Gangdungdoi WYimf)tvimhR tuyy. CHUNG MINH: Ta xetcactru'onghQp: a)Ne'uR la mQtvanhchiathlb6 d~(2.2.4)da:du'Qchungminh. b) Ne'uR la mQtvanhnguyenthilyva thoa (1). Khi d6, ho~cR la mQtvanhchia D, nenkh~ngdinh(2) dungchoR, ho~c~k> 1 d6 Dk la anhd6ngca"ucuamQtvanhconnaod6cuaR. Ne'ukha nangthuhai xay ra thl do tinhcha"t(1) du'Qcbaatoanqua phepla"yvanhconva anhd6ngca"unen(1)clingdungchoDk' Nhu'ngkhi d6,ne'utrongDktaxetcacphftntu: 0 0 0 ... 6 1 0 0 ... 0 Xl = 1 0 ... 0 ... I 0 0 ... 0=e2l va X2=...=Xn= I =ell 0 0 0 ... 0) ,,0 0 ... Q thl ta c6 [Xj,...,xnl=Xl =I:-0 va [Xj"",xnf = 0 nen [xj,...,xnlm=I:-[xj,...,xnl, vdi mQim> 1,mallthuiinvdi di~uki~nDkthoatinhcha"t(1). V~yR phaila mQtvanhchianen(2)clingdungchoR. c) N€u R la mla donthl R d~ngca"uvdi mQtt6ngtn!ctie'pconcua cacvanhnguyenthuyRa.Ma theophftnchungminhtrenthlkh~ngdinh (2) da:dungcho m6i Ra.Hdn nua,tinh cha"t(2) baa toanquaphepla"y t6ngtn!ctie'p(vI cacpheptoancQngva nhanthlfchi~ntrentungthanh phftn),phepla"yvanhconva anhd6ngca"u.Do d6 (2)clingdungchoR. ev hudn8d&n:Des - TS 511iTuon8 Tri liV Thuc hien:DhEinTruon8 Linh Lu~nvanThl,icSI Toan:MQLvalllldrQn8cuadjnh15'Jacobson Trang28 d)NeuR la vimhtilyy thlR/i(R) la ml'addnlienkh~ngdinhdung choR/i(R).V~y,vdimQiXj,"',XnE R taco[xj,...,xnlE feR).Ma theogia thietthi [xj,...,xnlm= [xj,...,xnl,m > 1. Tli do ta suyra [xj,...,xnl= O. [Theotinhcha't:ux=u,X E feR) ~ u=0]. V~y ta da:chungminhkh~ngdinh trongb6 d~(2.2.4)cling dung chomOtvanhR tily y.. Tli cacb6d~naytacoth~chungminhm~nhd~(2.2.3) CHlJNGMINHM~NHDE(2.2.3): Tru'dchet ta xet tru'onghqpR khongchuanil ideal khac (0).Theo b6 d~(2.2.5)thl tada:co [xj"",xnl la mOtd6ngnha'tthuctrenR. Do R khongchuanil ideal khac (0) thl theom~nhd~(1.5.3),R la mOtt6ngtn!c tiep con cua cac vanh nguyento'Roc.Do tinh giao hoan baaloanquaphepla'yt6ngtn!ctiepvavanhconlientachic~nchung minhRocgiaohoan,vdimQioc. Noi cachkhac,ta co th~gia sii'R la mOtvanhnguyento'va [Xj,...,xnlla mOtd6ngnha'thuctrenR vdi n > 1.Neu n >2 thl ta se chungminh[xj"",xn-llclinglamOtd6ngnha'thuctrenR. Tht!cv~y,choXj"",Xn-lER tilyy, do[xj,...,xnl=[[xj"",xn-ll,xnl=0, '\IxnE R lientaco[xj"",xn-llE Z =Z(R).V~ymQigiaohoantii'ca'pn - 1 trongR d~uthuOcZ. Bay giOgia sii't6n t(;liXj,"',Xn-lE R saocho a =[xj"",xn-ll =I:0va ne'ud~tb =[xj"",xn-21thltacoc =[b,bxn-ll=b[b,xn-d=b[Xl"",Xn-l1= ba.Nhu'ngVI a = [xj"",xn-ll =I:0 va c = [b, bxn-ll= [Xj,...,Xn-2,bxn-lld~ula giaohoantii'ca'pn - 1trongR lienthuOcZ.V~ytacoc =bavdi a, c E Z, a=l:O. Nhu'ngvdi d~ngthucnaythl tasuyraa(bXn-l)= (ab)Xn-l= (ba)Xn-l= CXn-l=Xn-lC=xn-lba) = (xn-lb)a= a(xn-lb)haytu'dngdu'dnga(bXn-l- Xn- Ib) =a[b,xn-ll= O. M~tkhac,VI a =I:0 va R la mOtvanhnguyento'lien khongco ph~n tii'nao thuOctamcuaR la u'dctht!cst!cua O.V~y tli d~ngthuctrenta phaico[b,xn-ll=[xj,...,xn-d= 0,mallthuftnvdidi~uki~na=I:O. V~y ta da:chungminhdu'qcr~ngneu [xj"",xnl la mOtd6ngnha't thucchoR thl [xj"",xn-ll clingla mOtd6ngnha'thucchoR. CV hu'dn8dttn:PC&>- T&>l)lii Tu'dn8Tri flVThu'chien:PhanTru'dn8Linh Lu~nvanThl;icsTToan:MQLvt'iilid rQn8elmdinhI)'Jacobson Trang29 Tie"pWc qua trlnhtren thl cuO'icling ta du'Qc[x], X2]phai 1aIDQt dang nha"thuc cho R hay 1a[x], X2]=0, '\Ix],X2E R nen R giao hoan. Bay giO,ne"uRIa IDQtvanhmladdnthlR khongchuanil idealkhac (0) [vI IDQinil ideal cua R d€u chuatrongJ(R) =(0)]nentheoph~n chungIDinhtren,R 1avanhgiaohoan.. Tru'dckhi chuy€n sang xet IDQtsO'IDd rQng khac cua dinh 1y Jacobsonta c~nIDQtvai dinhnghlava IDQtsO'b6 d€ lien quailde"ncac IDdrQngtru'ongsailday: Dinh nghia:ChoK la mQtmdrQngdgj sf)'cila mQttn(CJngF. Phdn tita E K dU:(jcgqi la tachdU:(jctrenF ntu da thacto'ititu cila no trenF khongco nghi~mbQi. Nhanxet: VI IDQtdathucp(x)co nghi~IDbQikhi va chikhip(x)va dp(x)/dx coIDQtnhantti'chung.Dodo,IDQtdathucba"tkhaquiconghi~IDbQithl phaicodp(x)/dx1adathucO. Tli dotaco: 1)Ne"uF co d~csO'0 thl di€u nay suyrap(x) 1aIDQtda thuch~ng. Trongtru'onghQpnaytaco IDQiph~ntti'trongK d€u tachdu'QCtrenZ. 2)Ne"uF cod~csO'p"*0 thldp(x)/dx=0 suyrap(x) =g(:?) vdi g 1a IDQtda thucnao do. Khi do, ne"ua E K thl tan t(;liIDQtsO'nguyenk sao cho apk tachdu'QCtrenF. Tuynhien,trongtru'onghQpnayhoantoanco khanang1aclingco apk E F. Khidotaco: Dinh nghia: Cho K la mQtmd rQngdC;lisO'cila mQttrU:CJngF. Gid sit mQt phdntita E K saDchot6ntc;limQtso'nguyenk ~0 dt apk E F thitanoia la hoantoankhongtachdU:(jctrenF. GV htfdn8d&n:PG6 -T65ui Ttfdn8Tri lIVThtfchien:PhanTrtfdn8Linh Lu~nvanTh/;lcsI Toan:MQtval merQn8CUBdinh ly Jacobson Trang30 Mf)t m(jrf)ngdt;lis6 K cila F dzt(lcgQifa m(jrf)ngtachdzt(lc(tztdng ling: hoanroankhongtachdzt(lc)trenF ne'umQiphdnta cila no d~utach dzt(lc(tztdngling:hoanroankhongtachdzt(lc)trenF. Nhiin Kef: Nguoi tada:chungminhduQc: Tijp cacphdnta trongK hoanroankhongtachdzt(lctrenF lijp thanhmf)ttrztiJngconcilaK. Vake'tquatztdngtT;tGangdungchotijpcac phdntatachdzt(lctrenF. B«J(t~(2.2.6):ChoK fa mf)ttrztiJngm(jrf)ngcila trztiJngF, K '*F va giGsa VdimQia E K d~utan tt;limf)ts6 nguyenn(a) > 0 di choan(a)E F. Khi do: 1)Rogc K fa hoanroankhongtachdzt(lCtrenF. 2) Rogc K co dgcs6 nguyento'va fa dt;lis6 trentrztiJngnguyento'P ? ,cuano. CHUNG MINH: N€u K la hoantoankhongtachduQctrenF thl khongco gl d€ chungminh. Giii suK khongla hoantoankhongtachduQctrenF thltant~imQt phftntua E K, a ~F la tachduQctrenF. Do anE F nena la d~is6va tachduQctrenF nentruongF(a) nhungduQcvaomQtmarQngchuffn tilchuuh~nL cuaF. TlnhchuffntilccuaL chotamQttlfd~ngcffu$cua L c6dinhF saochob =$(a) '* a. Tal~icobn=$(at =$(an)=anVIanEF. Tudob=vavdiV'* 1E L la mQtdin b~cncuaddnvi. Tudngtlf,VI $(a + 1) =b +1va (a +1r E F nentant~imQtphftn t-«\-tE L sac cho \lm=1vab+1=t!(a+1)hayva+1=~(a+1). Ta l~ico fl '*v VI n€u khongthl b + 1 =v(a +1)=va +v =b +v mallthuffnvdi di~uki~nv '* 1. Giiii l~itheoa taduQca =(1- fl)/(V- fl). Do fl vav d~ula din cuaddnvi nend~ud~is6trentruongnguyen t6P vadodoad~is6trenP. GV htfC:Jfi8clan: PG6 - T6 Bui Ttf<':Jn8Tri lIV Thtfchien:Phan Trtf<':Jn8Linh Lu~nvanTh~csI Toan:M(>lvaimer(>n8cuadinh15'Jacobson Trang31 Tac~nchungminhP cod~cs6p *- O. D~tLola marQngchu~nt~chuuh(;lncuaP chuaa.Cacly lu~ntren choa clingapd1.JngduQcchoa + i, vdimQis6nguyeni (vI ne'ua tach duQCtrenF thla+i clingv~y).Tll dotacoa +i =(i - f.li)/(Vi- f.li)trong doVi, f.lid~ula din cuadonvi vathuQCLo.Dodo,ne'uP cod~cs60thl cacph~ntii'a + i la phanbi<$tvakhidoLola mQtmarQnghuuh(;lncua truonghuud l(;licomQts6vo h(;lncacdin cuadonvi phanbi<$tla di~u phily.V~yP phaicod~cs6p *- O. Bay gio, ne'uf E F thl a +f clingtachduQCtrenF nena +f la d(;li s6 trenP. Nhunga la d(;lis6 trenP nen ta suyraf cling d(;lis6 trenP. V~yF d(;lis6trenP. NhungKia d(;lis6trenF nentll do K d(;lis6trenP. B6 d~daduQcchungminh.. 86 d~(2.2.7):(EJjnh If lacobson-Noether)Ntu D la m(}tdc;zis{/chiakh6ng giaohoanva la dc;zis{/trentamZ cuano thi t6ntc;zim(}tphclntit thu(}cD, kh6ngthu(}cZ la tachdu:c;ctrenZ. CHUNG MINH: Ne'uD co d~cs60 thlkhongco gl d€ chungminhVImQiph~ntii' cuaD d~utachduQCtrenZ. V~ytahayxetmQtvanhchiaD cod~cs6p *- O. Ne'ukhAngdinh cua b6 d~la sai thl D la hoan loan khong tach duQCtrenZ, tuc la, vdi mQixED thl xpn(x) E Z vdimQtn(x)~0 naodo. V~ythl t6nt(;limQta E D, a ~Z saochoaPE Z. GQi0 la anhX(;ltrenD xacdinhbai xO=xa - axthldoD cod~cs6 p*-OnentacoxoP=xaP- aPx=0VI aP E Z. L(;lico a !2:Z nen0*-O.V~yne'uyO*-0 thl t6nt(;limQtk > 1 saocho yok=0 nhung yok-1*- O. D~tx =yok-1thl dok >1tacox =wO=wa - aw. L(;litll xO=0 tasuyraxa - ax=O.Hon ml'a,doD la mQtvanhchia nentavie'tduQcx =auvax giaohoanvdia nenuclingv~y. Do doau=wa- awvatasuyraa =(wa - aw)u-1=(wu-1)a- a(wu-1) ". -1 T ' d" d 1 -1=ca - acVOlc =wu. u ay ta uQcc = +aca . e VhUdn8d&n:pes - TS I':>liiTucn8 Tri lIVThuchien:PhanTrucn8Linh . Lu~nvanTh~cSf Toan:MQtvaimdrQn8cuadjnh15'Jacobson Trang32 Nhu'ngvdi mQtt naodo ta1C;lico cpt E Z nencpt =(1+aca-iy' = l+(aca-iy' =l+acp'a-i =l+cP' VI cpt E Z. Tli day dftnd€n di€u mall thu§nIa 1=O.B6 d6 dffdu'Qchungminh.. Tli haik€t quatrenthlI. N. Hersteindffchungminhdu'QcmQtdinh 1:9marQngkhacchodinh1:9Jacobson[m~nhd€ (2.1.5)]. Menh d~(2.2.8):ChoR la meltvanhcotamZ vantuvdimQia E R thitan t(Iimeltso'nguyenn(a) >0 dl choan(a)E Z. Khi do ntuR khongconil idealthinogiaohoan. [Hay tzt(jJ1gdztang:idealcaegiaohoantitcuaR phdi la nil]. CHUNG MINH: Tru'dch€t tachungminhk€t quachomQtvanhchiao N€u R 1amQtvanhchia thl doan(a)E Z vdi mQia E R nenR dC;lis6 trenZ. Theo b6 d€ (2.2.7)thl,ho~cR giaohoan,ho~ct6ntC;limQtphffn ttl'a ~ Z tach du'QCtren Z. N€u kha nang sail xay ra thl tru'ongZ(a) kh6ng1ahoanloankh6ngtachdu'QCtrenZ va thoacacgia thi€t cuab6 d€ (2.2.6)nentasuyra Z(a), va dodoZ, 1adC;lis6trentru'ongnguyent6 p vdi d~cs6p *-O. V~y n€u x E R thl VI x dC;lis6 trenZ nen cling dC;lis6 trenP. Di€u nay cho thffyP(x) 1amQttru'onghUllhC;ln.Tli do xm(x)= x vdi m(x)> 1 naodo.V~ytheodinh1:9Jacobson[m~nhd€ (2.1.5)]thlR giaohoan. BaygiOtaxettru'onghQpR Ia mQtvanhnguyenthuy.Khi do,ho~c R 1amQtvanhchiaD, ho~cvdimQtk >1naodothlDk1aanhd6ngcffu cuamQtvanhconcuaR. Nhu'ngvdikhanangsailmatrongDktaxetphffnttl' 1 0 ... 6 0 0 ... 0x= =ell 0 0 ... 0, thltacoxm=x vdimQimvdix kh6ngthuQctamcuaDk,mauthu§nvdi di€u ki~nDkk€ thliatinhchfftneutronggiathi€t choR. V~yR phaiIa mQtvanhchianenno giaohoantheochungminh tren. CV hudn8clan:PCS -TSI'>liiTudn8Tri liV Thuchien:PhanTrudn8Linh Lu~nvanThf;lCSI Toan:MQtval ffidrQn8cua djnh It Jacobson Trang33 Trong ph~ncon l(;licua phep chungminh thl, theonhu'sd d6 dff nell, Ie ra ta chI con phai chungminh cho tru'onghQpR Ia mla ddn. Nhu'ngnham d(;ltdu'QcmQtke'tqua sau hdn ta se chuy~nsangmQt hu'ongkhac. Bay giOgia si'tR la mQtvanhkhongco nil idealkhdc (0) va thoa di~ukit%nvoi mQia E R thl3n(a) >0 d~an(a)E Z. Theo mt%nhd~(1.5.3)va chuthichngaysaudo thl ta co th~bi~u di~nR thanhmQtt6ngtn!c tie"pcon cua cac vanhnguyento'Ra va co tinhcha't:Voi m6iRad~ut6nt(;limQtph~nti'tkhongliiy linhXaE RasaD chovoi mQiidealkhackhongUac RathlXam(U)E U voi m(U»O naodo. VI Ia anh d6ngca'ucua R DenRa thoa gia thie"tan(a)E Z. V~y d~ chungminhmt%nhd~tachIc~nchungminhchoRa. Noi cachkhac,taco th~gia si'tR Ia mQtvanhnguyento"thoadi~u kit%nan(a)E Z va co themtinhcha'tla t6nt(;limQtph~nti'tkhongliiy linh b E R saDchovoi mQiideal U khac(0)trongR thl bm(U)E U. Do bn(b)= C E Z cling khong liiy linh va cac liiy thua cua no di chuy~nquanhcac ideal khac (0) cua R Den ta co th~gia thie"tngay chinhb E Z. M~itkhac,VI R nguyento"Denkhongco ph~nti'tnaothuQc Z la u'occua0 trongR. D~t .!J1t={(r, z) IrE R, z "*0 E Z} va trong.!J1tta dinh nghIa quail ht% xac dinh bdi: (rj, Zl) ~ (r2, Z2) ne"u rlZ2 =r2Z1 thl day la mQtquailht% tu'dngdu'dng.D~tR* la t~pcaclOptu'dngdu'dngvaki hit%u[r, z) la lOp tu'dngdu'dngcua(r,z).TaclingdinhnghIacacpheploan: [rj, Zl)+[r2,Z2)=[rlZ2+ r2Z1, ZlZ2) va [rj, ZlJ[ r2,Z2)=[rlr2, ZlZ2]. Do cacph~nti'tcuaZ khongla u'occua0 trongR Dencacpheploan naydu'QCdinhnghIato't vaR* la mQtvanh.Hdnmla,anhX(;lr~[ rz,z) la mQtphepnhungR vaoR*. Saucung,tamcuaR* la Z*={[r, z)1rE Z} va suyrangayZ* la mQttru'ong. Ngoai fa, ne"u[r, z) E R* thl [r, zt(r) = [rn(r),zn(r))E Z* DenR* cling ke"thuatinh cha'ttronggia thie"tchoR. Tuy nhien,R* l(;lila mQtvanh ddnVI ne"uU*"* (0) la mQtideal cuaR* va d~tU={rER/ [r, Z)EU* veYi m(JtZ E Z newdo}thl U"* (0) la mQtidealcuaR Denbm(U)E U. CV hudn8 diln: PC- TBui Tudn8 Tri liV Thuc hien:Phan Trudn8Linh Lu~nvanTh~csTToan: MQL vitimdrQn8cuadjnh15'Jacobson Trang34 Ma0* bm(U)E Z (vIbkhonglliy linhva b E Z)Dentil dotasuyra U* chuamQtph~ntakhac0cuaZ*.M~tkhac,VIZ* la mQttruongDen ta duQCU* =R*. V~y ta da chungminh R* la mQtvanh ddn. La mQtvanhddnvacoddnvi DenR* la nguyenthuyvadodophai giaohoantheochungminhtren.Mc$nhd~daduQcchungminh.. Nhanxet: Mc$nhd~naythtfcstfla mQtmdrQngcuadinhly Jacobson[mc$nh d~(2.1.5)]VIneuR la mQtvanhthoaxn(x)=x,n(x)>1thlR khongco chuaph~ntaliiy linhkhac0Denclingkhongconil idealkhac(0). M~tkhac,neue la mQtliiy dAngtrongR thlvdi mQix E R, b~ng pheptinhddngiantaco: (xe- exe/ =0 =(ex- exe/vasuyraxe- exe= ex- exe=0 (vIR khongchuaph~ntaliiy linhkhac0).Til dothlxe=ex. V~ymQilliy dAngtrongR d~uthuQctam. Do do, neuan(a)=a, n(a) > 1 thl e =an(a)-llamQtliiy dAngva theo chungminhtrentasuyraan(a)-lE Z. V~y cac gia thief cua mc$nhd~(2.2.8)d~uduQcthoa manDenR giaohoan. Trongph~nconl<;licualu~nvannaytaxetthemmQttruonghQp mdrQnghdnmlacuamc$nhd~vilar6i. Menhd~(2.2.9):ChoR fam()tvanhmladanthoadduki~n: (1) '\Ix,Y E R, 3m=m(x,y), 3n=n(x,y) (m,n >0)dt cho[xm,ynJ=O. thiR giaohodn. Trudckhi chungminhmc$nhd~tanh~cl<;lir~ng: * M()tph6ntrlx E R, wJi R fa vanhtityy, dzt(lcgQifa t1!achfnhqui phdi nfu tan t(li m()tph6n trl x' E R saDcho x + x' + xx' =O.Khi do x' dzt(lcgQifa t1!angh;chadophdi cuax. GV hudn8d~n:PGS - TS BliiTudn8Tri liVThuchibi:PhanTrudn8Linh Lu~nvanThf;lCsT Toan:MQtvitimdr(>n8cua djnh ly Jacobson Trang35 * Hdn mIa,ne'uR co ddnvi 1 thix E R la t1;lachinhquiphai khi va chi khi 1 +x khanghichphai trongR. [vI khi do taco (1 +x)(1 +x') =1 vadodox' =(1 +xrl - 1] * Dinh nghlatl1dngtt;!chophffntil't1;lachinhqui trdi va t1;langhich daotrdi. * Ta cling chuyding ne'ua d6ngthili Ia tt;!achinhqui tnli va phai thlcaephffntil'tt;!anghichdaotraivaphaicuaa la trungnhau.Khi dota gQitAta Ia t1;lachinhquivaa' la t1;langhichdaocuaa. Khi do,ne'uR coddnvi 1thla la t1;lachinhquikhivachikhi1+a la khanghich(haiphia)trongR. Ngoaifa,taclingchungminhtrl1ocaeb6d~sailday: ~~ (2.2.10):Ne'uR lamiltvanhcoddnvi1vathoa(1).Khidone'ux,y E R saDchox va(1+x')ydiu la t1;lachinhquithit6nt(limiltk =k(x,y) >0dt cho: (2) [x, /][(1 +x')y]'(1 +x')[x,yk]=0 (trongdo [x,yk]la giaohodnta(caphai)cuahaiph8ntax va/) CHUNG MINH: Theogia thie'thl vanhR co ddnvi 1va x, (1 +x')y d~ula tt;!a chinhquiDen1 +x va 1 +(1 +x')yd~ukhanghich.Do do tichcua chungla: (3) (1 + x)[1 + (1 +x')y] =1 +x +y clingkhanghich. [trongdotachuyding (1 +x)(1 +x') =1] Ne'ud~t: Z =(1 +x +yrly( 1 +x +y) t=(1 +xrly(1 +x) Thltaco: (1+x +y)l =/(1 +x +y) (1 +x)t =yk(1+x) Tru ve'theove'tasuyra (1 +x)(l- tk)=/y - yl GV hudn8d&n:PG8 - T8 BuiTudn8Tri tIV Thucbien:PhanTrudn8Linh 1u~nvanThl;lcsI Toan:MQtvalmdrQn8elmdinhIy Jacobson Trang36 Tlido: (4) x(l-l) =yky- (y + l)l + l M~tkhac,doR thaa(1)Den[ym,Zn]=0va[l, t5]=0 N6ula"yk la bQichungnhanha"tcuam,n, r, Sthltaclingco[l, l] =[l, l] =0 tuclaykgiaohoanvoical va l Dentli (4)tasuyfaykgiao h " /. ( k k )oan VOlx Z - t . V~y[x(l-l), yk]=0vatli dotaco: (5) [x,l](l-l) =0 [vI(l-l) clinggiaohoanvoil] B" .'- " h ( k k )ay glo tatm Z - t . £)~tv=[(1 +x')y]' thldo[1 +(1 +x')y](1 +v) =1Dentli (3)taco 1+x=(l +x+y)(l +v) Dodo: (6) (1+x +yrl =(1+v)(1+xrl Va: 1+x =1 +x +y +(1 +x +y)v Hay tudngdudng: (7) v =-(1+x +yFly Tli cacd~ngthucnaythl: l =(1+x+yrile 1+x +y) =(1+x +yrlyk(1+x) +(1+x +yrlly =(1 +v)(1 +xrlyk(1+x) - vl [do(6)va(7)] =(1 +v)l -vl k ( k k )=t+vt-y V~y: k k ( k k )z-t=vt-y =v[(1+xrlyk(1+x) -yk) =v(1+xrl[yk(1+x) - (1+X)yk] =v(1+xrl(lx -xyk) =v(l +xrl[l, x] =- v(l +xrl[x, yk] =-[(1 +x')y]'(1 +x')[x,l] Thayvao(5)taduQC(2)vab6d~duQchungminh. CV hl1c'Jn8d&n: PC- TBili TI1dn8 Tri tIV Thl1chien:PhanTrl1dn81inh Lu~nvanTh!;tcsI Toan:MQtviiilid rQn8cuadinhI)'Jacobson Trang37 Be}d~(2.2.11):Ntu D Ia mQtvanhchia thoa(1) thi wJi mQix, Y E D a~u t6n tc;zimQts{fnguyenk =k(x,y)>0al cho[x,lJ =O. CHUNG MINH: N€u y =0 thlhi~nnhienkh~ngdinhcuab6d~Ia dung. N€u y "*0 E D vagill sax va(1+x')yd~uIa t1,1'achinhquitrongD thltheob6d~(2.2.10),t6nt~imQtk=k(x,y)>0d~cho: [x,l][(1 +x')y]'(1+x')[x,ykJ=O. Til do,n€u [x,lJ"* 0 trongvanhchiaD thltaphaico[(1+x')y]'=O hay1+x' =O. Nhu'ngtaI~ico 1 +x' Ia khanghichtrongD DenphaikhacO.Va theodi~uki<$ndangxetthly "*0 Denta clingco (1 +x')y "*O.Til do ta suyra[(1 +x')y]'clingkhac0 Iadi~umallthu§:n. V~ytadachungminhdingn€u x va(1 +x')yd~uIa t1,1'achinhqui trongD thlIuan t6nt~imQtk =k(x,y) >0d~cho[x,lJ =o. Ngoaira ta coXED khangIa t1,1'achinhquikhi va chIkhi 1 +x khangkhanghichtrongD tucIa taphaico 1 +x =0 hayx =-1.Trong tru'onghQpnaythltaIuanco [x,yJ =[-1,yJ =0Denb6d~du'dcnghi<$m dung. Tu'dngt1,1':(1+x')ykhangIa t1,1'achinhquitrongD khivachIkhi 1 +(1 +x')y =0 haytu'dngdu'dng1 +(1 +xt1y=0,tucIay =-(1 +x). Trongtru'onghQpnay taclingco [x,yJ =[x, -1 - xJ =0 Denb6 d~du'dc nghi<$mdung. V~ytrongmQitru'onghQpthlb6 d~Iuandu'Qcnghi<$mdungIa di~u phaichungminh. Voi cacb6 d~vila du'Qcthi€t I~ptadaco th~chungminhm<$nhd~ (2.2.9): CHUNGMINHM~NHBE (2.2.9): Nhu'thu'ong1<$ta xet cac tru'onghQp: a)N€u R Ia mQtvanhchiathoa(1).Khi do, gill saa, b E R tuyyta xetvanhconRl sinhbdiavabtrongR. Theob6d~(2.2.11)thlvoi mQix E Rl (c R Ia mQtvanhchia)d~u t6nt~icacs6nguyenm,n >0d~cho[a,xmJ=0va[b,xnJ=O. GV huCJngclan:PG6 -T6 Blii Tudn8Tri liV Thucbien:PhanTrudn8Linh Lu~nvanThlfcsI Toan:MQl ValmerQn8cua dinh 15'Jacobson Trang38 NSu H(yk la bQichungnhanha'tcuamva n thlk >0 vaXkgiao hmlnduQcvdicaavab.V~yXkE Z(R]). M~tkhac,doR la vanhchiaDenR]khongcoph~ntiIlfiy linhkhac 0vadodoclingkhongchuanil idealkhac(0). V~ycacdi~uki~ncuam~nhd~(2.2.8)d~uduQcthaamanchoR] DenR]giaohoan.Tucla tacoab=ba.Tli daytasuyraR giaohoan. b)NSuR la mQtvanhnguyenthaythltheom~nhd~(1.4.3),ho~cR la mQtvanhchiaD, ho~ccomQtk >1d€ Dkla anhd6ngca'ucuamQt vanhconnaodocuaR. NSukhaDangthuhaixayrathld~tha'yDkclingkS thliatinhcha't lieUtronggia thiStchoR VI tinhcha'tnaybaaloanquaphepla'yvanh convaanhd6ngca'u. Nhungkhido,nSutrongDktaxetcacph~ntiI: 1 0 ... 01 r l 1 0 ... 6 0 0 ... 0I , 0 0 0 ... 0=ell va Y=x= =ell +en 0 0 ... 0 0 0 0 ... 0 h' ".. 0 " m n A [ mn J [ ] 0t I VOl mQI m, n > taco:x =x,y =y lien x ,y = x,y =en *- . Mauthuffnvdidi~uki~nDkthaagiathiSt. V~yR phaila mQtvanhchiaDenR giaohoan. c)NSuR la mQtvanhmladdnthltheom~nhd~(1.5.3),R d~ngca'u vdi mQtt6ngtn!cliSp concuacacvanhnguyenthuyRa.Ma m6ivanh nguyenthuyRa,la anhd6ngca'ucuaR DenkS thliadi~uki~ncuagia thiSt,v~ynophaigiaohoantheob).Tli dotasuyra R clinggiaohoan vam~nhd~daduQcchungminh.. CV hudn8d&n:PCS - TS BuiTudn8Tri tIV Thucbien:PhanTrudn8Linh