Luận án: Mở rộng module và các vành liên quan trong Lý thuyết vành

[HCHỦ TR NG IHCS PHM O TH TRANG MTS MRNGCAM UNNIXV C CV NHLIN QUAN Chuynngnh: is v | thuyts M s: 9460104 LUN NTINS TONHC Ngihng dnkhoahc 1: PGS. TR NGC NG QU NH Ngih ng dn khoa hc2: GS. LV N THUY T HU ,N M 2022 Licam oan Tixincam oan,cckt qu cnghincuv trnhbytronglun nny l caringtihocca ngtcgiv cccngtc gi cho php tỉ avolunn. O TH TRANG Licmn Li utin,tixinbyt Ing bit nsusc nhaiThyh ngdn, 1: PGS.

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s5 +cszszvrrtexererereeveserere 25 2M UNB TBIN NGCUV CCVNHLIN QUAN 29 2.1 nhnghav cctnhchtcam unbtbin ngcu. 29 22 Vnhcct ngcucam unbtbin ngcu.3 Munbtbin ngcutrn vnh Goldie.4 VnhtaFrobenusv vnhbtbin ngcu. 48 3 VNHMMIIANPHIHUHNSINHB TBIN NGCU 50 3.2 /a-v nh phi v i chi u Goldie h u hT.3 fa-v nh ph ¡ kh ng suy bỉ I. cv ng ng nếp 75 Ktlunv kinnngh.----scsxscsseeserereersrsesrxre 77 Danh m ccccng tr nh c a fC gÌL.--ccccsreereerrree 79 Danh mccckhiu K hiu Nghacakhiu Tiliuth i mc"Tiliutham kho" Baonix cam unm Vnhcct ngcucar-m unM nhca ngcuf Htnhnca ngcuf Vnh ma trnvungcpnlycchttrnvnhkR MIR-m un phi M]R-m un tri NÌm unconcaM NÌm unconctyucaM Nl m uncon ictyucamMm Nl hngt trctipcam unm Munn ngcuvim unm Tngtrctipcam unNvm unm Cncam unm camunM Danh m c cc thu tng Thutng ting Vit m unbtbin ngcu xyclic unconctyu uncon ictyu unm rng unt do unnix untanix BBBSBEBEBEB rngctyu B un gi nix N-nix phnb suy bin kh ng suy bin m unlintc m untalintc tr c giao naArtin nan vnh di truy n vnh ta Frobenius vnht nix tnhchttrao i trctiphuhn Thutng tỉ ng Anh automorphism invariant module cyclic essential submodule superfluous submodule extending module free module injective module quasi-injective module essential extension pseudo injective module N-injective complement singular nonsingular continuous module quasi-continuous module orthogonal semi Artinian semisimple hereditary ring quasi Frobenius ring self-injective ring exchange property directly-finite L.1mu L thuy tvnhkthp kh ng giao honnichung ra ihn100nm.Hin nay, Ì thuy tn y vn kh ng nợ ng cccnh tonhc quan t m, nghỉ n c u.

Trong,h nợ nghincucutrccavnhdatrnvicnghincucctnh chtcam untrnchngthu cnhiuktqu quan tr ng. Trong l thuy ft m un,lpm unnix c mtv tr trungtm cbitmt ccnhton hcluntmcchm rng theo nhuh ngkhcnhauv crtnhiulp m unm rngcam ra iNm1940,Baer giithiuv Ìp R-m unM lI hnợøt trctipcamimunchan.Baer ch ra iukinnyt ng ngviiukinmir- ngcugt mtianibtk caRvomu tntiphnt m€Msao cho o(x)=mx vi mix €! ([10]). Nm 1956, Cartan v Eilenberg dngngnngca is ngiu din tlpmunny v dngthutng "nix" ch chng([14]).C thhmtm unm cgi l nix numi ngcut munconAcam unBVOM uc thm rnợg nmt ngcutBVOM.Nm 1961,Johnsonv Wong giithiu mtm rngcam unnix 1m untanix ([33]).Mtm un cgiltanix numi ngcut mtm unconcan von m rng nt ngcucan.Cc ctrngcam untanix cmtstc gi nghin cu trong [7], [21], [23], [46]. Mtm rng quan trng calp muntanix 1 ginix,n cJainv Singh giithiu trong [32].M tm un cgil gi nix numi ncutm unconcan vonecth m rng nmtt ngcucan.Sau ,nhiutcgicng tiptcnghin culpm unnynh M.

V ¡ ph ng ph p mrnglpmunnixnhtrn, crtnhiulpmun cgiithiuv nghincu. Chng hn, Alahmadi, Er v Jain trong [4] aralpm ungi ni x ctyu, M. Harada ([24]) aralpm unca-nix,C. Smith ({15]) aralpm untac-nix,.Ngoira,mts tcgicn 8 m rnøglpm unnix da theo Tỉ u chu n Baer ([10, Theorem 1]); cth k nnhl lpm unp-nix ([44]),m untap-nix ([50]).

C th nhnnhnkhacnhtnhnix caccm unquaquan imbtbin, trong [33] Johnsonv Wong ch ralpccm unbtbind ittc cct ngcucabaonixcantrngvilpccm untanix. yl mt trong nh ng kt qu quan trngcalpmuntanix,n chothytnhta nixcamuncth cdin tthngquacctnhchtniticam un .Nm 1969, Dicksonv Fuller nghinculpm unbtbind icct ngcucabaonix can chotr nghp is huhnchiutrnmt trnøg,cecmunnhvy cgingngnl munbtbin ng cu ([16]). Nm 2013, Lee v Zhou trong [40] nhngham unbtbin ngcu cho vohv munbtk;theo ,mtr-munm ce gil btbin ngcu nuMbtbinquattc cct ngcucabaonix caM,tcÌo(M)< Mvimit ngcu@gcabaonix cam.Haitc gi Leev Zhou ch ra lpm ungi nix 1 btbin ngcu.Tuynhin,ktqu_ tph trong ch ny thu c v Er, Singh, Srivastava v Asensio. Trong [22] Er, Singh v Srivastava phntchmtm unbtbin ngcuthnhtngtrctipcamtm un tanix v mtm unkhngchnh ph ng.

H ch ralpm un bt bi n ngcuchnh! lpm ungi nix.Sau_, Asensio v Srivastava trong [6] chng minhrnglpm unbtbin ngcuthamntnhchttrao iv aracc ctrngcavnhv m unclean. mth ngtipcn kh c, khi xem v nh R nh l im unphiv mii an phinh 1 mtrm uncon. Nm 1969, Jainv Singh nghinculpvnh m miianphil tanix,lpvnhny c gil qvnhphiv h ch ramts _ c trng quan trng cho lp vnh ny (([30]). Sau, Ivanov tng qutlpq-vnhphi,gil fq-vnhphi, 1 lpvnhm mii anphi huhnsinhl tanix.Tcgi Ivanov nghincufq-vnhlinktvicc khinimly ngnguynthytr mtv ly ngkhngsuybin,t tec gi thu cmts ktqu th v ({29]).M rngcclpvnhnitrn thohngt tnhtanix ntnhbtbin ngcu,cctcgïi Ko san, Qunhv Srivastava giithiulpvnhm mii anphil btbin ngc ulpvnhny cgilavnhphiv h thu cnhiuktqu pve utrecalpvnhn y.Chnghn,mta-v nh philtngtrctipcavnhn anchnhphng y v vnhkhngchnhph ngphi.Cctcgi cng thu 10 c nhl v cutrechomta-vnhphikhngphntch ¢, Artin phi, khng 11 suy binphi cbiudinnh 1 mtvnhcc ma trn tam gi c khi ({35]).

Tiptcnghincutheoh ngny, chngti aralpvnhm mii anphi huhnsinhl btbin ngcu,chngtigi 1 lpfa-vnhphi.Cckt qu lin quan n fa-v nh cnghincu trong [48], [53]. Vnhcct ngcuEndM)cam unbtbin ngcuMcngc nhiu tnh chtt ngt vivnhcct ngcucam untanix. Trong [6], Guil Asensiovw Srivastava ch ra r ng End(M)⁄(End(M)) Ì v nh ch nh quy von Ñeumann viccly ngnng c modulo /(EndM)) v cn Jacobson J(End(M)) gmftc cctft nøgcucaM€C nhnctyu.Khng nhny tngt viccktqu c ch ng minh bi Faith v Utumi trong [23] cho tr nghpcecem untanix vl mrngktqu tre cachnhUtumi cho mun nix([55]).Nghincutheocch ngny,chngtithu cccktqu tngtcholpmunbtbinngcuv vnhcct ngcucan ([53)). Qua s l cv cchngnghincuv ccktqumnhiutcgithu c nh trn,chngtithyrng nghincuv lpmunbtbin ngcu vvnhcctngcucan cnợønh cclpvnh khclin quan nh lpí#a- vnh,vnhbtbin ng cu, vnh ta Frobenius,.

Ch ngticnphihiurv Ipm unnixv cemrngcan. 11 dochngtichn ti:"Mts mrngcam unnix v ccvnhlinquan" cholun nny. Trong lun nny, ch ng t¡ chia th nh ba ch ng, trong : Ch ng 1,chngtitrnhbycc nhngha,k hiuv mts tnhchtc bncanhngkhinim cs dngnhiulntronglun n. ngthi,chng ticnglitk ccktqu ctrchdntronglunn ngi cthun tintm kim, theo di.

Ch ng 2,ch ng ti nghincumts tnhchtcam unbtbin ngcu. Btutvnhcctngcucamunbtbin ngcuvchngtithu cktqu chnhsau jy: 12 nhl 2.7;Num1m unbtbin ngcuvichiu Goldiehuhn thi antriccicavnhEndM)cdngA,viulphntunoca M. Hnna,End(M) Ì vnh na hon ch nh. Ch ng ti bit r ng vnh Rickart khngc tnh ixng,Chase ch ramt v d v vnh Rickart tri nh ng khng Rickart ph ¡.

Tuy nhi n, ch ng ti khng nh crngnuvnhbtbin ngcuphiv Rikartphith n l vnh Rickart hai ph a.211:Chol vnhbtbin ngcuphi. Khi (1)NÑuaRlx nhvia€CRth Ral x nhnh l mti antricaR. (2)ÑuRlvnh Rickart phith ml vnh Rickart tr i. Nh chngtigiithiu trn,lpm unbtbin ngcul mrngca lIpmuntanix,lpmuntanixlmrngcalpm unnix.

Nhiunh tonhe aramts iukinchovnhe s hocchom un Ipmun "In" trngvilpmun "b ".Chngti aramtstrnghp Ipmunbtbin ngcutrngvilpm untanix hoclpm un nix.Mttrongnhngkt qu cchngtikhngnh,1: nhl 2.7: Cho rl vnh Goldie phinguynt v Ml rm unthamn udim(M/z(M )) > 1. Khi ,M1 m unbtbin ngcukhiv ch khim 1 munnix. Nm 1939, Nakayama arakhinimvnhta Frobenius, 1 vnht nix haiphav Artinhaipha.Nhiu ctrngcholpvnhny cnghin cu,trong Faith ctrngcholpvnhnyl t nix mtphav tha mn iukinACCmtphatrncclinhhat.Chngti Imyuiukin caFaithttnhtnix sangtnhbtbinngcuv thu cccktqu: nhl 2.2: Mt vnh1 ta Frobenius nuv ch nun l vnhbtbin ngcuphiv thamn iukinACCtrncclinhha t phi sao cho mi ian phititiucanlmtlinh hat phi. nhl 244Cciukinsau yltng ng ivivnhR cho: (1)Vnhl ta Frobenius.

(2)1 vnhbtbin ngcuphiv thamn iukin ACC trncclinh hat phisaochomii anphititiucarctyutrongmthngt tre tipc a Ra.5: Mtvnhl ta Frobeniusnuv ch nun l vnhbtbin nợ cuphi,eƒ-m rngphiv thamn iukinACCtrncclinhhat phi. Ch ng 3, chngtitrnh bycc kt qunghincuv vnhm mii an phihuhnsinhbtbin ngcu.Mtktqu nitingca Wedderburn-Artin ch rarngmtvnhnan ngcuvitchtretiphuhnccvnh ma trntrnccvnh chia. Trong [35] Ko san, Qunhv Srivastava aram t nhl cutrecholpa-vnh, 1:mta-vnhphil tngtrctipcavnh na nchnhphng y v vnhkhngchnhph ng.T cc t ngtrn, chngtiaracutrccamt fa-vnh phi,1: nhì 3.9:Mt#-vnhphi ngcuvivnh ma trn tam gi c hnh thc s0 c dng visl vnh chnh quy von Neumann t nix phi chnh mM T phng y ,rl vnhkhngchnhph ngphiv MÌ T-ssongm un. Trong [35],cctcgi ch ramtvnhl avnhphinuv chnum ii anphictyucan lbtbin ngcunuv chnun Ìl vnhbtbi n ngcuphiv miianphictyucan ] r-m untrivirlvnhcon can csinhbiccphnt kh nghchcan.Chngti thu ckt qu t ngt cho fa-vnh vichiu Goldie phihuhn(Mnh © 3.

Trong [30] v [35],cctc gi ch ravin >1] s nguy n,vnh ma trn M,(R) la- vnh phi khi v ch khi M,(R) a-v nh ph ¡ khi v ch khi Rl vnhna n.Ch ngti chng minh crng: Mnh_ 3.Cc iukinsau yltng ng ivivnh khng suy bin phir: (1)rk 1 vnh chnh quy von Neumannt nix phi. (2)M,(R) | fa-v nh phi. (3)M,(R) bt bin ng cu phi. Khini mƒa-v nh khngc tnh ¡ixng.

Tuy nhin, ivivnhnaArtin thchngti chratnhphi,tric am tía-v nh kh ng suy binl nh nhau.17:Mt vnh 1 faevnh phi na Artin phi khng suy bin phi 14 nuv ch nunl fa-vnh trina Artin tri khng suy bintri. 15 Ch ng1 KI N TH C CHU N B Trong ch ngny,chngtigiithiucckhinime bn,k hiuv mts tnhcht cs dngnhiulntronglun n.Nhngkhinim,k hiuv tnh chtm chngtikhngtrnhby y, cgic thtmthytrongcctiliu [3], [18], [26], [34], [38], [39].1 Mts khinimv tnhcht Vnher cholun c gi thitl vnhkthpe phnt nvv mir- mun cxtlunita,tclx1,=xvimixem. Khinir-m unm, ch ngtiquy c 1R-munphim;khikhngs nhmIn,r-mun™ ta vitgnlmunM.Tak hiuA<M chAlm unconcaM;f:M >N1R- ngcum unt MVON;End(M)]l tpttc cct ngøcuc aM. M un conN caR-m unM cgil ctyutrongM nu mim un con khckhngrtcaMthNnL 0,k hiulN<°M.Khi ,tacngniM Imrngctyuca mun con N.

D dng suy rat nhnghacam un con ctyu,tac du hiunhn bit sau y. M un conN1 ctyu trong M khi v ch khivimiphnt xkhckhng thucM th tntir€RsaochoxrEN v xr z0. 16 ingu,tac khinimm uncon ictyu.M unconkKcaR-m un 17 M cgil ictyutrongmMnuvimim uncontcamthamn K+L=Mth L=M,k hiuk«m.Linquan ntnhctyuv ictyu caccm uncon,tac khinim ncuctyuv toncu ictyu.Mttoncug:M>N cgil ¡ictyunu Ker(g)«M. ChoNn Ï munconcaM,m unconLrcaM cgilphnb caN trongM nuLlmunconti ivitnhchtNnL=0.TheoB Zorn, m un con L nh th luntnti.Mun con ph n b ng vai tr quan tr ng trong vicnghincucutrccamtslpvnhvmun.T khinimphn bcamtmuncon,tactnhchtsau y.

ChoN v LÌ ccm unconcaM.Nu L]phnb cam un con N trong M th N@L<M v (N®L)/L <° M/L.2tathymim unconcamlhngt trctipcamt m unconctyutrongmM.Mtm unconNcam~ cgil ng trong MnuN khngc m rngctyuthcs trongM,tc]l nuN <°kK =M thN=k.Cth chngminh crng,haikhinimm unconphnb v mun con nø trong M Ï tr ng nhau. Cnv lhaicngc khostcc ctrngcavnhv m un.Tip theo, chng titrnh by hai khinimnyv mts tnhcht lin quan. Cho M1 R-m un phi,giaocatte ccm unconti icam cgilcn caM,k hiulrad(m). Nhv y rad(M ) = N pe Za B=M AM trong øl ccm unconti icaM.Quy c,nuR-m unM khngec m un con tỉ ith rad(M) =M.

i1vivnhR choth chngta lun rad(R,) = rad(,R) V cgil cnJacobson c a v nh R,khiu]l 2=(R). Choml rm unphitngcattc ccm uncon ncamM_ cgi 1 ca munM, k hiul soc(m). Nyy y N soc(M) = B= A A= B 18 trong ø]l ccm uncon ncaM.NuR-m unm khngc m uncon nth quy c€Ồsoc(M)=0. Mtphnt ecavnhr cgøil ly ngnue=e.Phnt ly ng eCR cgil natmtrinuRe=eRe hay (1- e)Re=0.D d ng th y r ng, nue€Rl phnt ly ngth End(eR) “=eRev eRe l vnhec phnt n v le.

Sau yltnh chtlin quan nphntly ng. ChoRl vnhv el phnt ly ngca R.CcC iukinsau yltng ng: (1) eR(1 - e) = 0. (3) Rel mti ancavnh R.