TAP CHi KHOA HOC so 13*2016 39 DIEU KIEN DE VANH NITA HOAN CHINH LA VANH CHUOI TONG QUAT Le Thi PhirOTig* Tdm t^t Vdnh R dugc ggi Id nua hoan chinh neu R/J la vdnh niia dem vd mgi lUy ddng ndng dugc t[.]
TAP CHi KHOA HOC so 13*2016 39 DIEU KIEN DE VANH NITA HOAN CHINH LA VANH CHUOI TONG QUAT Le Thi PhirOTig* Tdm t^t Vdnh R dugc ggi Id nua hoan chinh neu R/J la vdnh niia dem vd mgi lUy ddng ndng dugc theo modulo J vdi J=rad(R) Vdnh R dugc gpi Id chuoi long qudt neu R Id tdng true tiep eiia cdc mddun chuoi Trong bdi bdo ndy, chimg tdi dua vi du phdn biet hai lap vdnh vita neu tren vd lam tirdng minh kit qud ve top vdnh nua hoan chinh Id lap vdnh tong qudt cua lap vdnh chuoi cdc tdi lieu [1] vd [5] Han nira, chung tdi lam ro mgt sd dieu hen de vdnh nua hoan chinh Id vdnh chuoi tdng qudt phdi (hogc trdi) Tir khoa: vdnh nua hoan chinh vdnh chuoi tdng qudt Gidi thieu Trong bai bao nay, vanh R da cho la vanh cd don vi va mgi ff-mddun la mddun phai unita De thuan tien, ta se ndi mddun thay cho mddun phai va ki hieu Mthay cho ki hieu MR Khi can thiet, ta se ndi ro M la mddun phai hay trai Cac ket qua ve vanh nira hoan chinh ta cd th8 xem [5] Khi i? la vanh nira hoan chinh thi RR=eiR®e2R@ @e,Li va RR=Rei®Re2® ®Re„ vai e\, , e„ la cac luy dang nguyen thiiy (ddi mgt) true giao Chiing ta dung ky hieu ve vanh nira hoan chinh sudt bai bao Mddun Mdugc ggi la mddun chudi neu cac mddun ciia nd dugc sap tuyen tinh theo quan he bao ham (nghia la, n%uA vaB la hai mddun cua Mlhi A^B hoac B^A) Mddun Mdugc ggi la chudi tdng quit neu Mia mdt tdng true tiep cua cac mddun chu6i Vanh R dugc ggi la vanh chudi (chudi tdng quat) phai neu mddun Rn la chudi (chudi tdng quat) Mdt vanh chudi (chudi tdng quat) trai dugc dinh nghTa tuong tu Vanh R la vanh chudi (chudi tdng quat) neu R la vanh ehudi (chudi tdng quat) trai va phai Mgt vanh chudi tdng quat bit ky ludn la vanh nua hoan chinh Vi du nhu vanh (F R =\ F\ - , , (vdi F la mdt trudng) la vanh chudi tdng quat va tat nhien nd la vanh nua hoan chinh Cae khai niem lien quan d6n phan ma khdng nhac den bai bao chiing ta cd the xem[l] va[4] Ket qua M^nh d l Vanh chudi tdng quat phai (hoac trai) la vanh nua hoan chinh Chung minh Gia sir vanh R la chudi tdng quat phai Khi dd R=e]R@e2R^ ®e„R vdi mdi mddun e,R la chudi va I = T ] e, e, la cae ffiy dang ddi mdt true giao khdng phan tich dugc Vi * CN, Trudng PT Dan tpc npi trii tinh Phu Yen TRUONG DAI HOC PHU YEN e,R la chudi nen ciR cd nhat mdt mddun cue dai Cho nen efi la dia phuang Suy R la vanh niia hoan chinh Nhu vay, vanh chufii tdng quat phai (hoac trai) la vanh nira hoan chinh Tuy nhien diSu ngugc l^i chua c h k dung Vi dy sau cho ta thay diSu dd Vi du vanh nira hoan chinh nhung khdng la vanh chudi tdng quat trai Cho vanh 7?= vdiO va E la cac trudng Khi dd R la vanh niia hoan chinh [O R) nhung khdng phai la vanh chudi tdng quat trai Chung minh Ta cd {eii; 622} vdi 6,1= fl o^i (0 Oj f o O^i ; e^^ - [0 la mgt tap hgp day du cac luy dang nguyen thiiy true giao cua R (Q m (0 0] ChoncnR,=\^ \@\ J = e,,R@e,,R Ta dl dang kiSm tra dugc modun thuc su khac khdng nhat ciia e,,/? = I Ml oJ Do dd euR la mdt mddun chuoi cd dd dai ciia day hgp bang Hon (0 o^ niia, e~.,R = '' ^0 R) la mddun dan nen £22^ la chudi Do dd 7?/( =e,|7?©e22^ '^ ^ ^ h chudi tdng quat phai Artin phai Khi dd theo ([1], Theorem 10.3.5) ta suy R la vanh nira hoan chinh vdi tap cac luy dang nguyen thiiy true fl 0] (0 giao =^^ QJ;^22=[O 0] (0 R ^j^iJ = J(R)=[^^ ^ Gia sii R la vanh chuoi tong quat trai Khido,R = Re,,®Re„=\ ' ® vpi^e,, vafe22 la chuoi " [0 oj 1^0 RJ Gia sir K la truong trung gian giua Q va R Khi Q g ^ £ R Ta dl dang la kigm tra dupe \\i fie22-m6dun trai Suy ta 10 ojKo o]4o SKo 3""™Chpn A: = X : | = Q [ V ] va A: = /i:, = Q^^/3] Khi dp Qg^,£IR va TAP CHi KHOAHOC SO 13 * 2016 41 Q £ ^2 £ ^ la hai day khac Vi thS suy Reii khdng la chudi va dieu la mau thuin Vay RR khdng la chudi tdng quat trai Nhu vay, vanh nira hoan chinh khdng la vanh chudi tdng quat phai (hoac trai) The thi nao dieu xay Cac menh de sau cho ta cac dieu kien de vanh niia hoan chinh la vanh chudi tdng quat phai (hoac trai) Nhu ta biet, mdi mddun xa anh hiiu han sinh tren mdt vanh nira hoan chinh la dang cau vdi mgt tdng true tiep ciia cac mddun xiclic xa anh efi Gia sir R la vanh nira hoan chinh Khi R dugc biSu diln R=e[R® ®eJi=Rei@ ®Re„ Vi thg mdi phin tii ;c e fi ta cd the dugc viet r = S ^; vdi r^ = e,re^ e ^,R^j = -^y • -Do mdi phan tu cua mdt vanh nira hoan chinh dugc bieu dien bdi mdt ma tran vudng cap n: A(r)^{r,f) cho phep nhan vanh tuong ling vdi tich cac ma tran: A(rs)=A(r).A(s) Hem niia, neu R la mgt vanh chudi tdng quat thi R,j dugc ky hieu Rij=efie, la mdt nhdm aben vdi /, j = 1, 2, , n Khi dd R,=R„ la mot vanh va R,j la mgt RrRj - song mddun Mfnh de Cho R la vanh niia hoan chinh Khi R la vanh chudi tdng quat phai neu va chi nSu vdi moi r&R_ , G i?,^ ta cd w e ^^^, v^R^^ cho r = su hoac s = rv Chung minh (=>) Gia sir i? la chudi tdng quat phai va 7", s^R S u y m r & s R hoac s&rR Neur Gsi?thi r = 5/, IGR.VI seRe^, dugc chgn nhu tren Khi r,s&e^R rGRe^nBnsu=settej=stej=rej=rvau=ektej&R^ Neu serR, tuang tu () Gia su i? la vanh chudi tdng quat Ta chiing minh bang phan chiing Gia sir ring resRvassrR cimg xay ra, nghTa la r = ?u va 5- = rvvdi « G R^^ va V e R^^ Vi vay r = rvu va vueRj N6uvu e J(i?^)vdi J(Rj) la can Jacobson cua vanh Rj thi TRUONG DAI HOC PHU YEN e - VH e U ( ^ ) va riej-vu)=0 vdi \J(R) la tap cac phan tii kha nghich R Suy r = Di6u mau thuln vdi gia thifit Vi the vu e \J(Rj) • Han nua, vi R la vanh nira hoan chinh nen ta suy e^R = e^R (xem ([4], Fact 1.20) Dieu la mau thuan (P (i = l,2)va P, Pi, Pi^ieyR e„R} thi tdn tai ddng cku x:P^ >?; cho f\ =fzx hoac ton tai ddng cau y' /^ (iii) Hai f,:P ddng cau khac bat >/^ 0' = ) vaP,PuP2^{Re\, cho yS=x/2hoac tdn tai ddng cau y\Py^ >P^ cho_^=/iy ky cua cac i?-mddun xiclic trai Re„) thi tdn tai ddng cau x\P.^ (phai) >/^sao */^sao eho fi.=yf\- Chirng minh ((i)=>(ii))Gia su vanh R la chudi tdng quat phai Khi dd Im/elm/^hoac Im/^clm/ Trudng h g p h n / c l m / ^ : Vi P, P\, P^ la cac iJ-mddun xiclic va P, P\ PzeidR nen P[ la xa anh Ta ed f^ :P^ fl '• ^2 ^^ la toan cau Vi Pi la xa anh nen ta cd bieu dd sau giao hoan *-Imf2 ejl} >Pla ddng cau Vi Pi va P la cac mddun xiclic nSn ••O, hay tdn tai x:P^ >>/^ saocho/i=/2X TAP CHI KHOA HOC s o 13* 2016 43 Trudng hpp I m / , cz h n / : Tuong tu, ta co P2 la xa anh nen suy bieu sau giao hoan ft y / , p -K), hay tdn tai y : /^ >P^ choj^=/ij' ((ii) => (i)) Gia sir vanh R khdng la chudi tdng quat phai Khi sd i?-mddun xiclic phai P cd cac mddun M\ va A/2 khac va cac phin tiia, G M , \ M^\ aj^&M^\ A/, khac Do dd cd cac luy ding dia phuang e^\e.^&R^ao cho a^e^ eM^\ M , va a^e^ ^ ^ ^ M • ^9 h't'J P>=^iR va / : i^ >P (i - , ) la cac phep ddng cau bien e, ate; Vi R khong la chudi tdng quat phai nen Imyj P2 cho >Pi cho f2=f\y Di6u mau thuin vdi gia thiSt Vay R la vanh chudi tdng quat phai ((ii) (iii)) Dieu kien tuang duang giiia (ii) va (iii) dugc suy tir cac dang cau }\Qxci(eR,fR)=fRe =}iom(Rf,Re) khi/va e la cac liiy dang eiia vanh R He qua Cho R la mdt vanh nua hoan chinh va eho = ej + +e„la mdt su phan tich cua G R mgt tdng cua cac liiy ding dia phuang ddi mdt true giao Vanh R la chudi tong quat phai n^u va chi n6u mdi vanh eRe la chudi tdng quat phai vdi e la mgt tdng ciia khdng qua ba luy dang dia phuang khac tir tap [c], e2, ,e„} R Chimg minh (=>) Gia sit R la mdt vanh chudi tdng quat phai va eeR la mdt IQy dang khac khdng Khi dd vanh eiJe la chudi tdng quat phai Dat = e + / e f f e = iii, eRf=X.fRe = Y,fRf=R2 Gia sir e = ei + +£„ la su phan tich e tdng ciia cac luy dang dia phuang ddi mdt true giao Gia sii mddun e,Re khdng la chudi Khi dd e,Re tdn tai hai eRe-mo&un M\ vi M2sao cho MiOMi^^M, va MinM2:^M2 Bat M, = M^Rva W^ = M^./? Rd rang M^