`ean 1 mibeg 1.1 zeix`pia zelert izy mr R,+, dveaw idef :beg edn mixkef mleky gipn ip` -y jk,(dn`zda ltke xeaig odl `xwpy).(0,ilxhiip xai` mr) zitelig dxeag `ed R, +.1.(xeaigl qgia) ziaiheaixhqice ziaih`iveq` dlert `ed ltkd.2 :mibegl ze`nbec.dxeag deedn epi` xeaigd ik,beg `l,oaenk,md miirahd. Z, +, :minlyd.1.(irah m lkl) m elecen ltke xeaig md m-e + m xy`k, Z/mZ, + m, m.2.(zncewd dnbeca mb millkp mwlgy,miiteq zecy llek) dcy lk.3 ly xehwe mb zeidl lekiy) x dpzyna minepiletd beg,f [x] f` dcy F m`.4 minepiletd sqe` - R[x] f`,beg R m`,xzei illk ote`a.beg `ed (mipzyn.minepilet ly xeaigle ltkl qgia beg R-a mincwn mr,x dpzyna M n n (R) f` beg R m` :oezp beg lrn zevixhn ibeg xicbdl ozip dnec ote`a.5 xeaigl qgia beg mpn` edfy xexae) R-a mincwn mr n n zevixhnd beg `ed.(zevixhn ly ltkle.(mizcewp ltkle xeaigl qgia) meqg jnez mr f : R R zeivwpetd beg.6 l`nyn ltkl qgia ilxhiip xai` `"f) dcigi yi begay gipp cinz igkepd qxewa miyxec `l m`.(?recn) ilxhiip xai` oi` dpecg`d dnbeca,lynl.(1 epnqpy,oinine xy = 0 xicbpy i"r beg,oiiprn k"k `l ote`a,`id zitelig dxeag lk,bega dcigi meiw jxca,dxwnd epi`y dn,zitelig ltkd zlert m` itelig `xwi beg.dxeaga x, y lkl dnbecde,itelig R m` wxe m` itelig lawznd beg,ziriaxd dnbeca,lynl) llk `id miitelig mibeg ly dxezd.(edylk dcy lrn xak ziaihhenew dpi` ziyingd blfnd dvw lr wx da rbp ep`e,(ziaihhenew dxabl`" qxewa zcnlpe),c`n dxiyr weqrp mda xzeia miaeygd milkd cg` od - zexabl` ly - zeixefph zeltkn,lynl).(miitelig mibeg lrn wx illk ote`a zexcben el` j`,qxewa meyn) R \ {0} lr dxeag dxicbn dpi`,k"ca - dcigi mr mibega mb - ltkd zlert lka avnd edf - dyrnl - (l`nyne oinin) ikted yi bega xai` lkljxkda `ly dxeag deedn R bega mikitdd mixai`d sqe`.(dcy zeedn opi`y epzpy ze`nbec ibeg mi`xwp iktd yi xai` lkl mda dcigi mr mibeg.r -a zpneqn xy` (ziltk) mizirl mi`xwp (dcy mpi`y) welig ibeg.dcy `ed iaihhenew welig bege,welig miwegxe mikaeqn mihwiiae` md mnvrlyk,welig ibeg.miaihhenew `l zecy mb ik m`) mnvrlyk welig ibeg xewgp `l df qxewa.zwtqn dnxa mipaen zeidln lk jxe`l epze` deelzy daeyg dnbec epl didz oehlind ly mipeipxheewd zxabl` dpand ihtyna xzeia aeygd ilkd eidi welig ibeg lrn zevixhn ibeg la` (qxewd.qxewa cnlpy miixwird xeaigde ltkd zelert z` zcakny f : R S dwzrd ef mibeg ly mfitxenened dpi` dpexg`d dpekzd welig beg epi` begd m`y al eniy) f(1 R ) = 1 S zniiwne 1
mfitxenened,xnelk) R = S xy`k `ed aeyg ihxt dxwn.(dipyd on gxkda zraep ote`a,yi oezp R beg lrn minfitxnecp`d lk sqe`l.(mfitxenecp` - envrl begd on didz ef dnbec mb.end(r) epnqpy,(zedfd zwzrd) dcigi mr beg ly dpan,irah minfitxenecp`d beg :xzei lecb beg-zz edf,dyrnl.qxewa zifkxn zeaiyg lra sqe` A, + zitelig dxeag lkl,xac ly ezin`le) R, + zixeaigd dxeagd ly.dakxdle izcewp xeaigl qgia beg deedn A ly minfitxenecp`d ri I m`,r ly (ipni) il`ny l`ici` `xwz I R zixeaig dxeag-zz `xwi ipni l`ici` mbe il`ny l`ici` mb `edy l`ici`.r R lkl (Ir R) oiievi m` `l`,il`ny l`ici` oiivi cinz l`ici` qxewd jldna.iccv-ec l`ici` :mil`ci`l ze`nbec.ynn l`ici` `xwp I R l`ici`.zxg` l`ici`,cinz,edf.il`iaixhd l`ici`d `xwpd,l`ici` `ed {0} beg lka.1.iccv-ec r R lkl,r beg lka xzei illk ote`a.l`ici` `ed mz minlyd bega.2 m` wxe m` ynn l`ici` edfe,l`ici` `id rr := {rs : s R} dveawd.welig beg `ed m` wxe m` ynn mil`ici` oi` begl,okl.r / R R,beg lkl,illk ote`a.iccv-ec ihnehe` ote`a,`ed li`ci` lk itelig bega.3 m`,r z` liknd xzeia ohwd) iccv-ec l`ici` `id RrR dveawd r R lkle.(dcigi yi bega yi bega m`e,(iccv-ec) l`ici` `ed (miccv-ec) mil`ici` ly dler cegi`.4.ynn l`ici` `ed ynn mil`ici` ly dler cegi` f` dcigi R ly (iccv ec) l`ici` `ed ker(f) f` mibeg ly mfitxenened f : R S m`.5.(l`ici` `weec epi`y S ly beg-zz `id f ly dpenzde) xzeia ohwd beg-zzd,xnelk - X i"r xvepd begd z` X -a onqp X R m` zveaw zniiw R-l m`e,r-l mixvei zveaw X-y xn`p, X = R m`.x z` liknd dnld zxfra) raep lirl ziyilyd dnbecd on.ziteq xvep R-y xn`p,ziteq mixvei.(dlkdl qgia) miaxin (miccv-ec) mil`ici` miniiw dcigi mr beg lkay (oxev ly.miiaxin (miccv ec) mil`ici` yi ziteq xvep beg lkay mb d`xi weica oerih eze`,epi`xy itk) miiaxin mil`ici` meiw zghadl daeyg `id dcigi yi begay dyixcd ilniqwn l`ici` meiwe,dcigi `ll beg,il`iaixh oxe`a,`id zitelig dxeag lk zexeag zz oi` dwilg zitelig dxeaga la`.zilniqwn dxeag-zz meiwl lewy.(zeiaxin ozipe, R, + ly zixeaig dpn zxeag `id R/I,iccv ec l`ici` I R m` epilr daeh efd dxcbddy gihadl ick) (r + I)(s + I) = rs + I dltkn xicbdl milawn epgp` efd ltkd zlert mry `ceel lwe,(iccv-ec `ed l`ici`dy yexcl itk,xzei e` zegt,mibdpzn dpn ibeg.dpnd beg `xwpy - R/I lr beg ly dpan :lynl,mitvn epiidy.s = R/ ker(f) f` lr mfitxenened f : R S m`.1 oial I z` milikny R ly mil`ici` oia dlkd zxeneye lr,r"gg dn`zd yi.2.r/i ly mil`ici` 2
.mibeg ly dxyid dltknde xiyid mekqd md jynda zeax eriteiy miaeyg mibyen {(r i ) i I : zexcqd sqe` df i I R i dxyid dltknd mibeg ly {R i } i I dgtyn ozpda ly beg-zzd edf i I R i xyid mekqd,dnec ote`a.mizecwp xeaige ltk mr r i Ri } -nay oaenk.0 md,iteq xtqnl ile` hxt,mixai`d lk oday zexcqd ly i I R i dxwnd `ed aeyg ihxt dxwn.zecklzn zexcbdd izy ziteq dveaw I eay dxw dxyid dltknd xear R n aezkp ( I = n m`) df dxwna.i I lkl R i = R eay.mibegd ly milecen 1.2 lrn ixehwe agxn ly byend ly zicind dllkdd `ed beg lrn `ed lecen ly byend zibel dpigane,micklzn mibyend ipy dcy,dxwna,`ed begd m`y,df oaena) dcy dlina "dcy" dlind ly zllek dtlgd i"r ihxtd byend on lawzn illkd byend zakxene dxiyr `id mibeg lrn milecen ly dixebhwd,mixacd rahn,j` - ("beg",xzei dxiyi dxcbd,ziy`x,`iap.mixehwe miagxn ly dixebhwd on xzei daxd beg ozpda.ixehwed agxnd byen ly ziciind dllkdd ok` idef recn d`xp jk xg`e ρ : R mibeg ly mfitxenenede zitelig dxeag ef R lrn M ( 1 il`ny) lecen,r.(zixeaig dxeagk M ly minfitxenecp`d beg `ed End(M),libxk,xy`k) End(M) ρ(r) : M M divwpet milawn epgp` r R lkl :hexit xzia dxcbdd z` ogap :okl.rm,heyt,e` ρ(r)(m) = r m k"ca onqpe.r(m 1 + m 2 ) = r(m 1 ) + r(m 2 )-y `vei ρ(r) End(M)=y oeeikn.(r + s)(m) = rm + sm,mibeg ly mfitxenened `id ρ-y oeeik,(dakxd i"r oezp ltkd,xnelk) zeivwpet beg `ed End(M),sqepay oeeik,seqal (rs)m = (ρ(r) ρ(s))(m) = ρ(r)(ρ(s)(m)) = ρ(r)(sm) = r(sm) mb lawp.1m = m-e z` aygl mircei ep` m M lkle r R lkl m` :oekp ipyd oeeikd mby xexa ρ : f` (M ziteligd dxeagd ly) mfitxenecp` `id ρ(r)(m) = rm divwpetde rm xzei gp k"ca.l"pd zepekzd yely zeniiwzn m` wxe m` lecen `id R End(M) mixywda la`,ρ dbvdd i"r `le l"pd zepekzd yely i"r yxetn ote`a lecen x`zl mfitxenened `id f : R S m`y xexa,lynl.xzei dgep ef dyib mizirl mihyten dakxdd i"r lecen-r mb `ed M f`,ρ dbvdd i"r oezpd lecen S `ed M -e mibeg ly :milecenl ze`nbec.ρ f.lecen `ed dcy lrn ixehwe agxn lk.1.r lrn lecen `ed (ynn l`ici` `weec `l) I R il`ny l`ici` ozpda.2 ote`a.(ng = g n i"r) Z lrn lecen,ihnehe` ote`a,`id zitelig dxeag lk.3.q lrn lecen `id dwilg zitelig dxeag lk,dnec,oaenk,edf.beg `ed R lrn R[x] minepiletd sqe` R beg ozpday xak epi`x.4 mb,dnec ote`a.r lrn lecen mb j`,(lirl 2 dnbec itl) envr lrn lecen ddfp m` jk,envr lrn lecen okle,beg `ed R lrn M n (R) zevixhnd sqe` mb,dyrnl,edfy lawp r ri n oekiyd zgz M n (R)-a ezpenz mr R z`.r lrn lecen el wwcfp ynn `l la`,ipni lecen ly byen mb yi 1 3
edf) lecen-r ly dpan yi S lr f : R S mibeg ly mfitxenened ozpda.5.(envr lrn lecen lrk S lr miayeg m`,lirl epi`x xaky dn ly ihxt dxwn R, + ly minfitxenecp`d sqe` `ed End(R)-e beg `ed R m`y xak epi`x.6.σ rσ zervn`a lecen-r `ed End(R) xac ly ezin`l.beg `ed End(R) if` mey dyrnl,oi` R, + `id eplgzd dpnn ziaihhenewd dxeagdy dcaerl lecen-r lkl lecen-r `ed End R (M)-y d`xz weica dgkedd dze`e,zeaiyg wx `le,lecen-r-k M ly minfitxenenedd sqe`l wx minvnhvn m`,m.(oldl e`x) zitelig dxeagk lr iytgd lecen R-d `xwpy,lecen-r mb la`,beg `ed R m beg `ed R m`.7 ilwiv lecen `ed M -y xn`p Rm = M -y jk m M miiw m`.mixvei m ly xyi mekqe miytg dltkn ly mibyend,xzei illk ote`a.(m i"r xvepd).mibega miliawnd miybenl dneca mixcben milecen-r,md (ziteqpi` dnver zeidl mb leki m) mixvei m lr miiytegd milecen R-d aigxp hrn df `yep gztl ick.mixehwe miagxnl xzeia minecd miniieqn mipaena ly {x i } i I M dveaw ozpda :milecenl zix`pil dxabl`n zeqiqa zexcbd dnk i I r ix i dxevdn xai` df {x i } i I ixa` ly ix`pil sexv M milecen R-a mixai` -pd e`) xvepd lecen-zzd.mincwn ly iteq xtqnl hxt,i I lkl r i = 0 xy`k,edfy xexae) dveawd ixa` ly mix`pild mitexivd lk sqe` `ed {x i } i I i"r (yxt wxe m` i I r ix i = 0 m` zixi`pil dielz izla z`xwp dveawd.(lecen-zz,mpn` dielz izla `id m` M lecend ly qiqa `id {x i } i I -y xn`p.i lkl r i = 0 m`.envr M `ed dici lr yxtpd lecen-zzde zix`pil dgkedl zpzip uipiihy ly dtlgdd znl,lecen-d `ed M -e welig beg `ed D m` -y jk lecen-zz N M m` :mixehwe miagxnl zgken `idy ote`d eze`a weica lr xzi.dnverd dze` M -l miqiqa ipy lkl okl.b N, a f` a N, b a / N,sqepa.dcy lrn ixehwe agxnk bdpzn welig beg lrn lecen oaen lka hrnk,ok lkl if` R begd ly dnverd on dlecb M lecend ly dnverd m`y `ceel dyw `l `ed R begde) ziteq xvep M -y dxwna la`.dnverd dze` yi M -l miqiqa ipy mr) R mibeg miniiw ik ze`xdl ic,z`f ze`xdl ick.oekp gxkda `l df (iteqpi` xac.(libxz) R R-l (begk gxkda `l la`) lecen-r-k itxenefi` R-y jk (dcigi,edf.milecen-r ly mfitxenend ly byend zxcbdl irah ote`a epze` `ian df,zexg` milina,lecen-r ly dpand mr slgznd,zexeag ly mfitxenened heyt.mfitxenefi` `ed jitd mfitxenened,libxk.zix`pil-r dwzrd idef :ze`nbec ly mfitxenened zihnehe` `ed (zewilg) zeitelig zexeag oia mitxenened lk.1.(milecen-q) milecen-z ly -F lk mdly mfitxenened,oaenk,`id mixehwe miagxn oia zix`pil dwzrd.2.milecen,sqepa.mfitxenened `ed ι : N M oekiyd f` lecen-zz N M m`.3 r(a + N) = i"r lecen ly dpan,irah ote`a,zyxei M/N zixeaigd dxeagd.milecen ly mfitxenened `id mb a a/n zirahd dlhdd.ra + N M lr f` mibeg ly mfitxenened f : R S-e lecen-s `ed M m`y xak epi`x mb caer xacd eay dxwna dzr lthp.lecen-r ly dpan mb,zihnehe`,xcben 4
oneqne) M ly qt`nd `xwp ρ ly oirxbd,m lecen-r ly dbvd ρ m`.jetdd oeeika ann(m)-y oeeik.il`iaixh oirxb yi dbvdl m` on`p `xwp M lecend.(ann(m) `ed I ann(m) m`.iccv-ec l`ici` edf mibeg ly mfitxenenend ly oirxb `ed `ceel lw) lecen-r/i mb M -y lawp ρ I (r/i) = ρ(r) xicbp edylk iccv-ec l`ici`.(ahid xcben dfy ze`xabl` 1.3 byen edf.beg lrn dxabl` ly byend `ed igkepd qxewa miifkxnd mibyend cg` lrn A dxabl` iaihi`ehpi` ote`a.cg` dpana bege lecen ly mibyend z` alynd mr ayiizn begd ly dpandy ote`a beg ly dpan,sqepa,el yiy lecen-r `id R beg qxewae,miaihhenew mibeg lrn k"ca,xcben dxabl` ly byend.lecen- R-k dpand f :-e,iaihhenew beg `ed R m`,zilnxet.dcy `ed R-y dxwna cwnzp igkepd S-y xn`p.lecen R ly dpan lawn S-y xak epi`x,mibeg ly mfitxenened R S `ed Z(S) := {s S : u S, su = us} xy`k,im(r) Z(S) m` dxabl`-r `id r(ab) = (ra)b = a(rb)-y jkl dlewy l"pd dxcbddy ze`xl dyw `l.s ly fkxnd lrn ixehwe agxn `id S,dcy `ed R eay aeygd dxwna.r R lkle a, b S lkl ixehwe agxnk dim R (S) < m` R lrn iteq cninn `id dxabl`dy xn`pe,r.(jtidl `l j`,ziteq zxvep dxabl`dy dcaerd z` xxeb iteq cniny,al eniy),(mipzyn seqpi`a ile`) minepiletd beg `id R beg lrn dxabl`l zipepwd dnbecd `l `"f - ziaihhenew `id mipeyd mipzynd ly dltkndy migipn gxkda `l xy`k ly dpn `id R lrn dxabl` lk,xac ly ezin`l.i, j lkl X i X j = X j X i -y migipn bega hiap.x = {x a : a A} xicbp R lrn A dxabl` ozpda :R lrn minepilet beg,if`.x a a i"r rawpd f : R[X] A (mibeg ly) mfitxenenedae R[X] minepiletd,(mibegk) R[X]/ ker(f) = A (lirl ephhivy) mihxcphqd mfitxenenedd ihtyn itl.ze`xabl`-r ly mfitxenefi` edfy `vei zexcbdd on ziciine :ze`xabl`l ze`nbec `vei R lkl 1 Z(R)-y oeeikne lecen-z `ed (dcigi mr) beg lky oeeik.1 1 Z 1 R i"r zrawpd (mibeg ly) mfitxenenedd zgz R-a Z ly dpenzdy.dxabl`-z xak mvra `ed R-y `vei Z(R)-a dlek zlken oekiyd i"r,aey) dxabl`-r `ed R[X] minepiletd beg,epxaqd xaky itk.2 dnbec milawn epgp` ziteq dveaw `id X-y dxwna.(1 R 1 R[X] i"r xvepy.r lrn iteq cninn dppi` j`,(r lrn) ziteq zxvepy dxabl`l ly mfitxenened yi,iccv-ec l`ici` `ed I A-e dxabl`-r `id A m`.3 ozep df mfitxenenedy `vei Z(A)/I Z(A/I)-y oeeik.a/i-l R-n mibeg dxabl` `ed A ly dpn beg lky,`id dpwqnd.dxabl`-r ly dpan A/I-l `l) minepilet ibeg ly zepnd weica od R lrn ze`xabl`d,xnelk.r lrn.(mipzyn seqpi`a ile`) R lrn (miaihhenew `l ipzni` minepilet beg ly dxexa `l dpnk dxabl`-r ly xe`zdy,oaenk.4 daxd xe`z,miax mixwna,odl yiy zeiarh ze`xabl` oiadl zxfer gxkda zevixhn ibeg ly ef `id epzpigan xzeia daeygd dnbecd ile`.heyt xzei oeeik.f lrn beg `ed M n (F ) f` dcy `ed F m`y xak epi`x.(dcy lrn) -F idefy `vei M n (F ) ly fkxna F ly oekiy `id f fid n dwzrddy.n 2 cninn dxabl` 5
.D = F f`,f lrn zicnin seq welig zxabl` D-e zixabl` xebq dcy F m`.5 beg D ik) D ly zitelig dxabl`-zz idef,f (a)-a opeazp a D m`?recn jk n dfi` yiy `vei F lrn iteq cnin D-y oeeikn.dcy edf okle,(welig jk F lrn f,wixt i` mepilet yiy `"f.zix`pil mielz {1, a, a 2,..., a n }-y izexixy did a-y oeeik.f (a) = F zixabl` xebq F -y oeeikne,f(a) = 0-y.D = F edf :xexa ote`a dxabl`-k `ed K f`,zecy zagxd `id k K-e dcy k m`.6.k-a ltkd mr ayiizn K-a ltkde,k lrn ixehwe agxn -eewd zxabl` `id R lrn dxabl` ly zenqxetnd ze`nbecd zg`,ziq`lw.7 {1, i, j, k} i"r lecenk zxvepd dxabl`-r-d idef.oehlind ly H mipeipxh i 2 = j 2 = k 2 = 1.ij = ji = k :miniiwnd :zeiyeniy ze`eeyn cer milawn ep` o`kn jk = j 2 i = i = kj.ik = i 2 j = j = ki dfa la` - C lrn zicnin-ec dxabl` mb idef) R lrn 4 cninn dxabl` idef xicbp,ziy`x.(iaihhenew `l) welig beg `id H-y d`xp dzr..(jynda oecp a + ib + jc + kd a ib jc kd -e -e dwzrd q q = qq = f` q = a + ib + jc + kd m`e.q q dpnqpe,dnvr lr H-n r = q q 2 onqp m`y `vei. q 2 epnqpe,iynn xtqn `ed okle,a 2 + b 2 + c 2 + d 2 df j`, r = q 1 e`) r = q 1 xnelk,rq = ±1 okle,rq R la`. rq = 1 f`,kc = ijc = jic = kc,okl,qi = iq f` q Z(H) m`y al miyp.(avnd `l okl.b = 0 mby lawp qj = jq-a hiap m`e,d = 0,dnec ote`a.c = 0 okle.z(h) = R.miiynnd lrn 3 cninn dxabl`,dglvd `ll,oehlind ytig zeax mipy jyna beqn dxabl` ly cnindy `ed igkepd qxewa gikepy miaeygd mihtynd cg` mby) qeipaext ly mqxetn htyn,ok lr xzie,(!) mly reaix zeidl aiig df lrn zeicnin-seq welig ze`xabl` oi` miynnd lrny d`xn (gikep eze`.mipeipxheewle akexnd dcyl hxt miiynnd onqp :`ad ote`a lal xyt` mipeipxheewd ly zyxetn dip.8.r = {( a b b ā ) } : a, b C 6
( r 0.r 0 r ) oekiyd i"r M 2 (C) ly beg-zz `ed R-y d`xn dxiyi dwica witqi okl.dxabl`-r `id R-y `vei M 2 (C) ly fkxna lken df oekiyy oeeik -n`a `id,z`f ze`xl zg` jxc.mipeipxheewl itxenefi`,begk,r-y ze`xdl df la`,(welig zxabl` idefy xexa ik) lirl epxkfdy qeipaext htyn zerv r 2 = 1 d`eeyna hiap.zexiyi z`f ze`xl xyt`e,lit mr aeaf cinydl enk :d`ad ze`eeynd zkxrn z` lawp.r R xear a 2 b 2 = 1 ab + āb = 0 ā 2 b 2 = 1 e` (dpey`xd d`eeyndn) a = ±i-e b = 0-y milawn epgp` dipyd d`eeyndn xy`k, b 2 = 1 r 2 `veie (ziyilyl ddf dpey`xd d`eeynd f`e) a ir-y.b = 1-e,a = 0,lynl,xgap.a = ir i = ( i 0 0 i ) ( 0 1, j = 1 0 ) xicbp k = ij = ( 0 i i 0 ) -e -ynd mb jke,zniiwzn mipeipxheewd zxcbda dpey`xd d`eeyndy xexa f`.dipyd d`ee -xnd lrn ixehwe agxn dppi` zncewd dnbeca epipay R dxabl`dy al miyp.9 lynl,(miakexnd lrn ixehwe axnk M 2 (C) ly agxn-zz dppi` `"f) miake ( ) ( ) i 0 1 0 lrk H lr aeygl lkep,z`f lka,cvik. / R, la` R, 0 i 0 1,q = a + bi + cj + dk dxevdn `ed H-a illk xai`?c lrn ixehwe agxn -xtqnd mr c + di z`e a + bi z` ddfp.(a + bi + (c + di)j)-k z`f aezkp lecend C lrn) ixehwe axgnky lawpe,ote`d eze`a mibveind miakexnd mi (0, i),(i, 0) zebefd mr mipeipxheewd ly mixveid z` ddfp.c 2 itxenefi` H -xnd ly irahd oekiyd ik zmiakexnd lrn dxabl` `l - oaenk - idef,(0, 1)-e mr slgzn epi` i-y meyn) fkxna lken epi` jk mibvend mipeipxheewa miake (k mre j zeixefph zeltkn 1.4 -ly,zixefphd dltknd `ed ze`xabl` ly xywda ea weqrpy xzeia aeygd byend lr dltkn lrk dilr aeygl didi oekp,xee`xa zxeaga oecp xy`k - qxewd seq z`xw ynnl ic.(xee`xa zxeag -dxeag dxicbn `id mieqn xywdae) ze`xabl`- R-d sqe` milecen-r ly zixefph dltkn idn xicbdl - oey`x xac - epilr,eply dxhnd z` mb md zixefph epltky milecend m`y,d`xp ipyd alya.(iaihhenew beg R xy`k).dxabl`-r ly dpan xicbdl xyt` zixefphd dltknd lr f` ze`xabl`-r 7
-enew mibeg lrn) milecen ly mixywda dlecb zeaiyg mb yi zeixefph zeltknl `l okle,ziaihhenew dxabl` icenila xzei dlecb ezeaiygy wxt edf j`,(miaihh zewzrd ly xwga `ed zeixefph zeltkn ly xewnd.df qxewa ewnerl ea weqrp irah ote`a zevv el`.(zeix`pil-ihlen zewzrd,zeillk xzia,e`) zeix`pil-ia milkd cg`e,(zexg` zeil`ivpxtic zeipaze zehppinxhc,lynl) dfilp`ae dxabl`a.(lecb xzei agxn lr) zeix`pil zewzrdl dirad mevnv `ed oda letihl miaeygd on mikaeqn xzei zvw miagxn lr mb) zeix`pil zewzrdy `ed ef dyib ly oexzid ic wqr od zeix`pil-ia zewzrdy cera,ahid mipian epgp` (miixewnd miagxnd ly dxyi dltkn lr zix`pil-ia dwzrd ozpda `id,ok m`,eply dxhnd.jaeqn xzei agxnn zix`pil dwzrda efd dwzrdd z` ipepw ote`a,silgdl,milecen bef -xeneneda cieviy,milecend bef ly zixefphd dltknd didi xzei lecbd agxnd.lecb eplgzd dpnn zix`pil-iad dwzrddy jke,ekezl dxyid dltknd ly (ipepw) mfit ote`a l"pd oeica eiykr lthp.(!hehxy jixv o`k) zixefphd dltknd jxc zwxtzn.ilnxet xzei `id m` zix`pil-ia `id f : M N P dwzrd M, N, P milecen-r ozpda f x0 (y) :=-e f y0 (x) := f(x, y o )-y `"f) cxtpa mipzynd on cg` lka zix`pil-r M N epnqpy,lecen-r miytgn epgp`.y 0 lkle x 0 lkl zeix`pil-r od f(x 0, y) -ia dwzrd lkle,p lecen-r lkly jk,lr i : M N M N zix`pil-ia dwzrde f : M N P (dcigi gxkda) zix`pil dwzrd zniiw f : M N P zix`pil miniiwnd i dwzrde M N lecen-r `vnp m`y al miyp,ziy`x. f i = f -y jk mfitxenefi` ick cr cigi M N gxkda if`,zyxcpd zilqxaipe`d dpekzd z` df bef xear zilqxaipe`d dpekzd n f`,l"pk bef cer P, j m` ixdy.milecen-r ly ĩ : P M N zix`pil-r dwzrd yiy lawp (i zix`pil-iad dwzrdl qgia) eze`ae,ĩ j i = ĩ j = i dzr. j : M N P dwzrd mb yi dixhniqne xzep,dzr.milecen-r ly minfitxenefi` md j-e ĩ-y `vei okl, j ĩ j = j ote`d :ixnbl zihxcphq `id dipad.zeyxcpd zepekzd lra M N lecen ly meiw `ceel z` B onqp.{(m, n) M N} mixveid lr iytegd lecend-r z` MN-a onqp (m + m, n) (m, n) (m, n), m M, n N (m, n + n ) (m, n) (m, n ), m M, n N (rm, n) r(m, n), r R, m M, n N (m, rn) r(m, n), r R, m M, n N :i"r xvepd MN ly lecen-zzd xicbpe M N -a (m, n) ly dpenzd z` m n-a onqp.m N = MN/B xicbpe lynl.zix`pil-ia wzrd idefy dxcbdd on zexiyi xexae,i(m, n) = m n i((m + m, n)) = (m + m ) n = m n + m n = i(m, n) + i(m, n) (m+m, n) (m, n) (m, n) = miiwzn M N -ay jkn raep irvn`d oeieeyd xy`k z` miiwn (M N, i) befdy `ceel xzep.(m + m, n) (m, n) (m, n) B ik,0 -y oeeik,zix`pil-ia dwzrd f : M N P ozpda.epyxcy zilqxaipe`d dpekzd zix`pil-ia f -y oeeik.mn-l ix`pil ote`a f z` aigxdl xyt`,iyteg lecen MN.yxcpk,i(M N) lr f mr zcklzny f : MN/B P dwzrd yi okle,f(b) = 0,m M xear) (m, n) mixai`d i"r lecen-r-k xvep M N -y oeeiky al miyp illk xai`.(n N,m M,aey) (m n) mixa`id i"r xvep M N -y xexa (n N 8
mixwna la`,m n dxevdn miiehia ly iteq ix`pil-r sexv,oaenk,`ed M N-a,lynl.mixveid xear ze` wecal ic zixefphd dltknd ly zepekz gikedl ick,miax dze` xcibdl witqn P edylk lecen-r-l M N -n zix`pil-r divwpet xicbdl ick dveawd-y gipdl daiq lk oi`y oeeik :xdfdl yi,z`f mr.m n mixveid lr cg` dxwn.daeh `id dxcbddy `ceel yi l"za `id {m n : m M, n N} m`,df dxwna.welig-beg `ed R eay dxwnd `ed ef diral lw oexzt yi eay {v i u j : i I, j J} f` R lrn N -l qiqa {u j } j J -e R lrn M -l qiqa {v i } i I.l"za `idy wecal yi j`,zyxet dveaw idefy xexa.r lrn M N -l qiqa heyt xzei.icnl rbiin wqr edf j`,miicia dwicad z` zeyrl,oaenk,xyt` -y jk f : M N R zix`pil-ia dwzrd yi s : I J R lkly al miyl f : M N F zix`pil divwpet yi f`y oeeikn witqi df.f(v i,, u j ) = s(i, j) -i`l lewy l"pk s divwpet lkl oekp dfy dcaerde, f(u i, v j ) = s(i, j) zniiwnd.zix`pil zelz wecal k"ca xyt` zixefphd dltknd ly zeiqiqad zepekzd z`,illk ote`a ynzydl sicre,cgeina minirp mpi` miax mixwna miaeyigd j` - xiyi aeyig zxfra,lynl.'eke m n mixveid zveaw lr epl yiy divnxetpi`a,zilqxaipe`d dpekza mfitxeefi`d z` zepale dipal zexiyi zybl xyt` M N = N M -y gikedl ick xn`p) milecen-r-k N M -l itxenefi` M N -y al miyl xzei lw la`.miicia dpekzd z` yi M N -l okle (zix`pil-ia mb - oaenk - `idy,e(m, n) = (n, m) i"r dpekzd itl,zix`pil-ia f : N M P ozpda :(N M -l qgia) zilqxaipe`d -iad divwpetd f e : M N P zix`pil divwpet yi M N ly zilqxaipe`d oeeikne,f = f e i e 1 `vei mfitxenefi` e-y oeeikne,f e = f e i zix`pil.zraep dprhd,zix`pil-ia i e 1 -y milqxaipe` milecen zepal zxyt`n zixefphd dltknd ly dipal ddf dipa zewzrdl `l`,zeix`pil-ia zewzrdl wx `l (zeni`zn zeix`pil-ihlen zewzrde),`l`.m N P mipnqn zeix`pil-3 zewzrd ly dxwna,lynl.zeix`pil-ihlen witqi :M N P hwiae`d z` ycgn zepal jxev lk oi`,xac ly ezin`ly,ok m`,idz.zeix`pil-3 zewzrdl qgia ilqxaipe` `ed (M N) P -y ze`xdl zix`pil-ia dwzrd xicbp p P lkl.zix`pil-3 dwzrd f : M N P Q ghaenk, f p : M N Q zix`pil dwzrd zniiw if`.f p (m, n) = f(m, n, p) zxcbend F : (M N) P Q dwzrddy xexa.m N ly zilqxaipe`d dpekzdn, F : (M N) P Q yi okle,zix`pil-ia `id F (m n, p) = f p (m n) i"r xexa ilqxaipe`d hwiae`d zecigin.zyweand divpwetd weica `idy,zix`pil.(m N) P = M N P -y mr ztlgzn `idy dcaerd `id zixefphd dltknd ly ztqep ziyeniy dpekz gikep mipeniqd zegep myl.( i I M i ) N = i I (M i N) `"f,mixyi minekq mixwna enk.oihelgl ddf illkd dxwnd j`,mixaegn ipy ly xyi mekq xear z`f d`xp,lynl.xzei zelw mikxc yi j`,xiyi ote`a oeaygd z` zeyrl ozip,mincewd (M 1 M 2 ) -l qgia) zilqxaipe`d dpekzd z` yi (M 1 N) (M 2 N)-l ik f(m 1, m 2, n) = f` zix`pil-ia dwzrd f : (M 1 M 2 ) N m`y al miyp.(n f 2 : M 2 N -e f 1 : M 1 N zeix`pil-ia zewzrd yi okl.f(m 1, 0, n) + f(0, m 2, n) jk f i : M i N yi M i N ly zilqxaipe`d dpekzpn,dzr.f = f 1 + f 2 -y jk elek oeicd jldna.yxcpk, f 1 + f 2 jxc zwxtzn f -y lawpe, f i jxc zwxtzn f i -y.ze`xabl` ly zixefph dltknl,epghady itk,dptp dzr lrk A, B lr aeygp m`.f reaw dcy lrn 'eke A, B,ze`xabl` epl zepezepy gipp 9
.A B zixefphd dltknd z` xicbdl lkep,f lrn (milecen) mixehwe miagxn zayiizny ltk zlert xicbdl epilr dxabl`-f -l zixefphd dltknd z` jetdl ick -ihlen `id (a, b, a b ) (aa bb ) dwzrddy al miyp.ix`pil-f -d dpand mr zixefphd dltknd ly dpekzdn,okl.(a B-l A B A B-n dwzrdk) zix`pil A B A B = la`.dl dni`znd A B-l A B A B zix`pil dwzrd yi dxewn (A B-l xn`p) (A B) (A B)-n zix`pil dwzrd lke (A B) (A B) mfitxenenedd mr dakxd i"r heyt) (A B) (A B) lr zix`pil-ia dwzrda mr zayizny (A B) (A B) A B zix`pil-ia dwzrd yiy `"f.(ipepwd jxc zwxtzn `ide,eplgzd dpnn (a, b, a b ) (aa bb ) dwzrdd.(a, b, a, b ) (a b, a b ) (aa bb ) `ed ltkd (A B) (A B) ly mixveid lr,zexg` milina (a b, a b ) (aa bb ) raep zexcbdd on zexiyi.iaiheaixhqic `ed ltkd,zix`pil `id dwzrddy oeeikne l`nyne oinin ltkl qgia ilxhiip `ed ik) dfd ltkl qgia ilxhiip xai` `ed 1 1-y mixveid lr mb iaih`iveq` ltky oeeikne,(iaiheaixhqic ltkdy ep`xde,mixveid lr A-a ltkd zlerty oeeik.lirl dxcbedy ltkd zlertl qgia beg `ed A B-y eplaiw epxcbdy ltkd mixveid lry xexa,milecen-f -k odly dpand mr zeayiizn B-ae A B-y `vei.elek begl oekp df okle,lecen-f -k ely dpand mr ayiizn A B lr.dxabl`-f ok` `id f` (welig beg lrn e`) dcy lrn mixehwe miagxn md B-e A m`y xak epi`x,{a i } i I xy`k {a i b j } i I,j J `ed dcyd eze` lrn ixehwe agxnk A B-l qiqa od a a 1-e b 1 b zewzrdd okl.dn`zda B-le A-l miqiqa {b j } j J cigi ote`a aezkl ozip A B-a xai` lk,dzr.a B-a A lye B ly mipekiy f i,j (a i b j ) = i,j j a j b j = j (a j 1)(1 b j ) dxeva `l) A lrn lecenk A B-l qiqa `ed {1 b j } j J -y `"f.a j = i (f i,ja i ) b j xy`k,c(a b) = (c 1)(a b) i"r A lrn lecen `ed A B-y xexa la`,yxetna z`f epxn`,mekiql.a lrn iteq cninn A B f` F lrn iteq cninn B m`,hxta.(c A lkl dltknd `"f) AB = BA f` A B-a odly mipekiyd mr B z`e A z` ddfp m` :mixveil qgia oekp df ik,(ziaihhenew `id ze`xabl`d ly a b = (a 1)(1 b) = (1 b)(a 1).a b = b a-y oaenk,dprh oi` la` agxnk wx `l,dxabl`k) zilqxaipe` dpekz yi A B dxabl`l ik oiivl aeyg :yxetna dze` aezkp,jynda ef dprhl wwcfpy oeeik.(ixehwe j : B A B-e i : A A B eidi.f dcy lrn ze`xabl` A, B dpiidz 1.1 htyn md g : B T,f : A T -e dxabl`-f `id T -y gipp.miirahd mipekiyd :y jk ze`xabl`-f ly minfitxenened 10
.f F = g F.1.f(A)g(B) = g(b)f(a).2.hj = g-e hi = f -y jk h : A B T,cigi mfitxenend miiw if` 11