Journal of Pure dna Applied Algebra 02 )1891( -371 391 9 North-Holland Publishing Company GROTHENDIECK GROUPS OF ABELIAN GROUP RINGS .W.H ,ARTSNEL Jr. Mathemarisch Inslituut. Universiteit van Amsrerdam. The Netherlands Communicated by H. ssaB devieceR 61 July 0891 Let R eb a noetherian ring, dna )R(G the kceidnehtorG group of finitely detareneg modules over .R For a finite abelian group ,n we ebircsed )nR(G sa the direct sum of spuorg .)’R(G hcaE ’R is the form ,,,<IR I/n], erehw n is a positive integer dna Cn a primitive nth root of unity. sA na application, we ebircsed the erutcurts of the kceidnehtorG group of sriap (H. u), erehw His na abelian group dna u is na of H of finite .redro 0. noitcudortnI ehT kceidnehtorG puorg (G )’6‘ fo an naileba yrogetac 6‘ is denifed yb srotareneg and .snoitaler erehT is eno rotareneg ]M[ rof hcae tcejbo M fo .% and eno noitaler ]M[ = [Ml + [Mq rof yreve tcaxe ecneuqes O+M’-+M-+M”+O ni :% teL R eb a tfel nairehteon gnir htiw 1. eW etirw )R(G rof eht kceidnehtorG puorg fo eht yrogetac fo yletinif detareneg tfel .seludom-R roF a puorg n, ew etoned yb nR eht puorg gnir fo n revo .R teL Q eb a etinif cilcyc puorg fo redro n, htiw rotareneg ,r and etoned yb Qn eht htn cimotolcyc -ylop .laimon As ew shall ees ni noitceS 2, eht dedis-owt laedi @R)r(nQ fo QR seod ton dneped no eht eciohc fo 7, and ew tup )e(R = )Q(R = -Xn(/lX[)e(R .)1 esehT era osla tfel nairehteon ;sgnir eht orez gnir is ton .dedulcxe 0.1. .meroehT teL R be a left noetherian ring htiw 1 and 71 a finite abelian group. Then ew have, htiw the above notations =)nR(G @ (R(G n/if)) ’n where ’n ranges over lla subgroups of II for hcihw ’n/n is cyclic. roF a noitpircsed fo eht msihpromosi ew refer ot eht foorp fo eht ,meroeht hcihw is nevig ni noitceS 4. tI is ton ni yna suoivbo yaw decudni yb eht larutan gnir -omoh 371 174 .H .W Lensrm, Jr. msihprom +-RR fin, (R .)’n/n snoitceS 1, 2 and 3 niatnoc emos yrotaraperp .lairetam nI noitceS 5 ew ebircsed eht ruoivaheb fo eht msihpromosi rednu egnahc fo spuorg and egnahc fo rings. 0.2. .meroehT teL 71 be a finite cyclic group of order n. Then where i& denotes a primitive htd root of unity, &[@C 1 /d]) denotes the ideal class group of the Dedekind ring &[a 1 /d], and the direct sum ranges over the divisors d ofn. sihT meroeht is ylriaf etaidemmi morf meroehT 0.1, ees noitceS 7. eroM ,yllareneg rof R a dnikedeD niamod and II etinif naileba ew evah soTR(G @ )))x(WOH( )3.0( X erehw x segnar revo a niatrec tes fo sretcarahc fo 71 and )X(R is a niatrec dnikedeD ;niamod ees 7.4 rof .sliated ehT spuorg )nR(G and ))X(R(G=))X(R(C@E evah larutan gnir serutcurts decudni yb eht rosnet tcudorp revo R and )X(R ,ylevitcepser .fc ,8[ yralloroC 1.11. enO thgim rednow rehtehw )3.0( is a gnir msihpromosi fi noitacilpitlum is denifed esiwtnenopmoc ni eht tcerid sum. gnikcehC eht egami fo eht tinu tnemele [R] fo )nR(G eno sdnif that this is ylno eurt ni eht laivirt esac nehw eht redro fo n is a rewop fo eht citsiretcarahc fo .R roF a noissucsid fo eht noitaler neewteb 0.2 and s’renieR noitpircsed fo )nZ(G rof n cilcyc ,]7[ ew refer ot noitceS 7. snoitceS 8 and 9 era detoved ot eht puorg FSS hcihw was detagitsevni yb Bass ]l[ and nosyarG .]2[ tI is denifed as .swollof teL .Y eb eht yrogetac fo all pairs J-f( ,)u erehw H is a yletinif detareneg naileba puorg and u an msihpromotua fo H rof hcihw u” - Hdi is tnetoplin rof emos evitisop regetni ;n ereh Hdi is eht ytitnedi no H. A msihprom ni /‘. morf ,H( )u ot ,’H( )’u is denifed ot eb a puorg msihpromomoh f : ’H+-H rof hcihw f 0 u = ’u 0 f. ehT riap ,H( )u is dellac a permutafion module fi H admits a Z-basis detumrep yb u. teL P eb eht puorgbus fo eht kceidnehtorG puorg )Y.(G detareneg yb eht sessalc fo all noitatumrep .seludom nehT =FSS .P/)?.(G 0.4. .meroehT eW have sFSS Ona ,n[[Z(CI 1 /n]). sihT meroeht is devorp ni noitceS 8. nI noitceS 9 ew niatbo an tsomla etelpmoc noitpircsed fo FSS as an naileba ,puorg using sdohtem morf ciarbegla rebmun .yroeht Rings ni this repap era syawla desoppus ot evah a tinu ,tnemele and seludom era tfel .seludom yB Z and Q ew etoned eht gnir fo sregetni and eht dleif fo lanoitar ,srebmun .ylevitcepser citeroeht-teS ecnereffid is detoned yb ,- and ytilanidrac yb #. Grofhendieck groups of obelian group sgnir 175 1. ehT kceidnehtorG puorg of ]n/l[R nI this noitces n setoned a evitisop .regetni yB [Z ]n/l ew etoned eht subring fo Q detareneg yb 1 ,ni and fi M is an naileba puorg ew tup M[ 1 ]nI = [Zz@M .]n/l fI R is a ,gnir neht ]n/l[R is a gnir cihpromosi ot -Xn(/]X[R .]X[R)I and eht tnemele 1 )n/l(@ fo ]n/l[R is ylpmis detoned yb .n/l roF yna eludom-R M ereht is a larutan eludom-]n/l[R erutcurts no ,]n/l[M and eht rotcnuf morf eht yrogetac fo -R seludom ot eht yrogetac fo seludom-]n/l[R mapping M ot ]n/l[M is .tcaxe 1.1. .noitisoporP teL R be a left noetherian ring. Then R[l/n] is a left noetherian ring, and )]n/l[R(G is isomorphic ot ,H/)R(G where H is the subgroup of )R(G generated by alf symbols [Ml, htiw M ranging over the finitely generated seludom-R for hcih.w .O=M*n Proof. ehT gnir R[ l/n] is tfel nairehteon esuaceb yreve tfel laedi fo R[ l/n] is fo eht mrof ,]n/l[a erehw a is a tfel laedi fo .R ecniS eht rotcnuf ]n/l[M-M morf eht yrogetac fo seludom-R ot eht yrogetac fo R[ 1 seludom-]n/ is ,tcaxe and ecnis M[ 1 /n] = 0 fi n M= 0, ereht is a puorg -omoh l msihprom 1 : )]n/l[R(G-H/)R(G mapping eht tesoc fo ]M[ dom H ot [M[ .]]n/l teL ylesrevnoc N eb a yletinif detareneg ,eludom-]n/l[R and tel M eb a yletinif detareneg eludombus-R fo N hcihw setareneg N as an .eludom-]n/l[R yB a -thgiarts drawrof tnemugra eno shows that lM[( dom )H E H/)R(G sdneped ylno no N, and that ereht is a puorg msihpromomoh ,U : H/)R(G+)]n/l[R(G rof hcihw )]N[(p = lM[( dom )H ni eht noitautis just .debircsed oT evorp 1.1 ti won seciffus ot kcehc that .i and p era esrevni ot hcae .rehto fI N,M era as ni eht noitinifed fo ,u, neht eno ylisae sevorp that ,]n/l[M:zN os Ap is eht ytitnedi no .)]n/l[R(G teL won M eb a yletinif detareneg ,eludom-R and tel MO eb eht egami fo M rednu eht larutan map M*M[l/n]. nehT ]M[(.Eup dom )H 104A[(= dom ,)H so ot evorp that .?u, is eht ytitnedi no H/)R(G ti seciffus ot wohs that ]M[ = ]OM[ dom H. ehT lenrek L fo eht larutan noitcejrus M-MO is nevig yb =L MER{ : ,ZEi3 iz0 : n’*x=O}. ecniS R is ,nairehteon L is yletinif ,detareneg os L*& =0 rof emos k E ,E kz0. eroferehT ew evah ]M( - ]OM[ = [L] E H, as .deriuqer sihT sevorp .1.1 1.2. .yralloroC teL R be a left noetherian ring and g a finite cyclic group of order n. Then ew have, htiw the notations of the introduction ))e<R(G 2 ,HV)o,(R(G 176 H. W. Lenslra. Jr. where H is the subgroup of ))o,(R(G generated by lla symbols [Ml, htiw M ranging over the finitely generated seludom-)q(R for hcihw =M*n 0. Proof. etaidemmI morf 1.1. 2. snoitartliF of seludom-xR teL g eb a etinif cilcyc puorg fo redro n, htiw rotareneg ,T and etoned yb @,, eht htn cimotolcyc .laimonylop fI f is yna gnir msihpromomoh morf pZ ot C hcihw is evitcejni nehw detcirtser ot Q, neht eht lenrek ffo is detareneg yb .)T(& ecneH eht laedi &i)r(,,@ fo QE seod ton dneped no eht eciohc fo .7 teL R eb a .gnir tI swollof that eht dedis-owt laedi is tnednepedni fo eht &(r)R@ eciohc fo .r eW enifed eht gnir )Q(R yb )e(R = .eR)s(n@/eR ehT gnir )Q(Z is a niamod cihpromosi ot ,]&(Z erehw ,& sa etoned evitimirp htn toor fo .ytinu Its dleif fo snoitcarf yam eb deifitnedi htiw .)Q(Q ehT puorg fo units fo )Q(H sniatnoc e ni a larutan .yaw roF yrartibra ,R ew evah )Q(R I R @z .)Q(’Z As an ,eludom-R )Q(R is eerf no )n(p ,srotareneg erehw pc is eht noitcnuf . relufEo ,ecneH fi R is tfel nairehteon neht os is .)Q(R fI ’Q is a puorgbus fo g, neht ereht is a larutan noisulcni )&(R C .)Q(R 2.1. .ammeL teL Q be a finite cyclic group of order n, and suppose taht pk divides n, where p is prime and k E k L 1. Then ni )Q(Z ew have (,,n 1 - )a =p, where o ranges over the elements of Q of order pk. Proof. ehT rebmun fo hcus CJ #slauqe ,‘-it- and yeht era sorez fo -’PX 1 tub ton fo ’-kPX - 1. ecniS )Q(Z is a niamod this seilpmi that n(x-@=(XPk_ ~)/(xP’-‘_ I)= ‘.&iv-I d i=O ni ,]X[)g(H and eht derised tluser swollof fi ew etutitsbus 1 rof X. sihT sevorp 2.1. 2.2. .ammeL teL Q be a finite cyclic group of order n. Denote, for every prime p gnidivid n, by ep the p-primary subgroup of Q. teL further R be a ring and M an eludom-)e(R for hcihw .O=M*n Then there is a finite chain of seludombus-)g(R hcusMfoO=~M1.--31M3oM=M thatforeveryic{l,2,...,t} thereisaprimep gnidivid n for hcihw gp acts trivially on -iM iM/i and -;M(.p )lM/i = 0. Proof. fI n ,~p---2p1p= htiw ;p ,emirp neht ni eht niahc fo seludombus-)e(R M>prM>p~pzM~... M~p..-2p1p3 = Mn = 0 kceidnehforG groups of naileba group sgnir 771 yreve tneitouq M;p...zp~p/M~_;p...2pIp is detalihinna yb emos emirp gnidivid n. ecneH ti seciffus ot evorp eht ammel rednu eht dedda noitpmussa that p*M=O, erehw p is a emirp gnidivid n. teL pk = # ,pe and tel ul,a2, . ,.. ta eb eht stnemele fo Q fo redro ;# OS t =pk -#- .‘ .tliIOrof,M*);a-1(...)2a-l()~o-l(=iMtuP.p@forotarenegasi;ahcaenehT esehT era seludombus-)e(R fo M, and M=(l 1()1a- 1$~.)20- ,O=M*p=M.)rrc- yb 2.1, Mo>Ml .Ml>***> hcaE eludom Mi_ t/M; is detalihinna yb 1 - a;, and ecnis ;o setareneg pe this seilpmi that pe stca yllaivirt no M;-t/M, rof yreve ni ,2.1( . . . . .)t oslA O=)M/I-M(*p ecnis =M-p 0. sihT sevorp 2.2. woN tel n eb a etinif naileba .puorg factor group fo n is a puorg fo eht mrof A ,’n/n erehw n’C IC is a .puorgbus eW stress that owt rotcaf spuorg R/R’ and Z/R” fo R era ylno ot eb deredisnoc lauqe fi II’= R” as spuorgbus fo n. ehT tes fo cilcyc rotcaf spuorg fo is detoned yb R X(R). teL and tel R eb a .gnir nehT ereht era larutan evitcejrus gnir -omoh @eX(n), smsihprom RR*RQ+R(Q), and this selbane us ot yfitnedi eht seludom-)@?I htiw eht seludom-n?f detalihinna yb rek as ew lliw od ni eht .leuqes (Rn-+R(e)), 2.3. teL be a finite abelian group, R a ring, and ,’Q E‘& ,)R(X .”Q#’Q Lemma. R Suppose taht M is an eludom-xR hcihw is htob an eludom-)&(R and an -)”Q(R module. Then =M*p 0 for some prime number p gnidivid ’Q# or .”Q# teL Q’= and Q”= gnignahcretnI ’g and Q”, fi ,yrassecen ew yam Proof. R/R’ R/R”. assume that esoohC EQ gnicalpeR o yb a elbatius rewop ew nac R”a 71’. R”- R’. eveihca that EH rof emos emirp rebmun p. ehT egami d fo rc ni ’Q neht has redro n’ p, so p sedivid .’o,# ecniS M is a eludom(Ii and rc E rek ,)”Q-X( eht noitca fo rc no M is .laivirt R“ = ecneH M is, as a ,eludom-)e(Z detalihinna yb 1 - b. gniylppA 2.1 ot pk )I= ew dnif that osla p setalihinna .M sihT sevorp 2.3. roF ,)R(XEQ etoned yb ,m eht lenrek fo eht gnir msihpromomoh QR-+Q(Q). ecniS )g(Q is a ,dleif ,QI is a lamixam laedi fo and morf Qn, -rc 1 l gm @ Erc rek (R+Q) rof( rc E that pm # ’gm rof @ # .’o, ecneH eht esenihC redniamer meroeht n) we see seilpmi that eht denibmoc map 178 H. W..L ensrra.J r. is a evitcejrus gnir .msihpromomoh tI is osla ,raenil-Q and ot evorp that ti is an -osi msihprom ti seciffus ot wohs that eht snoisnemid-Q # 71 and )n(~EVC )Q#(v era eht .emas teL ii eb eht tes fo puorg smsihpromomoh morf IC ot eht tinu .elcric nehT rek/n )ro )X(XE rof all x E .ii ,ylesrevnoC rof hcae EQ )n(X ereht era ylesicerp (V # )Q tcnitsid x E 5 rof hcihw Q = rek/n .)x( ecneH C meg (pc # )Q = # ii = # n, and )4.2( is an .msihpromosi ecniS nZ is deniatnoc ni Qn, ti swollof that eht map ZR+~~~~,Y(~) )Q(Z is ,evitcejni os fi ew tup ep = rek ))Q(Z-~&( = ,cm 17 XZ neht ew evah ngs~(n) pp = 0. 2.5. .ammeL Let R be a ring, II a finite abelian group, and M an Rn-module. Then there is a finite chain of Rn-submodules M = MO> MI 3 . -. > M, = 0 of M such that rof each iE{1,2,..., t } there exists Q E X( 71)f or which M; - I /Mi is an R(,o)-module. Proof. etirW X(R)= (QI,.o~, . . ..Q(} and esoohc M;=p,,--.p,,M rof O~ist. ereH eht ep era as ,evoba and is deredisnoc as a eludom-r& aiv eht suoivbo map M ER+RR. ecniS eht snoitca fo and no ,etummoc eht era seludombus-nR fo R Zn M M; M. rehtruF MO= M, and M, = 0 ecnis .--,Qp c,gp )nc~egn p9 =O. ,yllaniF hcae Mi- I/M; is detalihinna yb pQi and is erofereht a eludom revo rof 1 I sihT R@Z(& = R(@i), is t. sevorp 2.5. 3. noitatoN for eht proof of meroehT 0.1 nI this noitces ew establish eht noitaton desu ni noitceS 4. yB ew etoned a etinif R naileba puorg and yb a tfel nairehteon .gnir daetsnI fo yletinif“ detareneg R ”eludom ew ylpmis etirw .”eludom“ ehT class fo an eludom-nR ni is M G(RR) detoned yb [M, n]. As ni eht suoiverp ,noitces ew etoned yb eht tes fo cilcyc rotcaf spuorg fo X(R) R, and rof E@ ew yfitnedi eht seludom-)e(R htiw eht seludom-nR detalihinna yb X(R) eht lenrek fo eht larutan evitcejrus gnir msihpromomoh Using 1.2, ew RR+R(Q). lliw weiv eht puorg as gnieb denifed yb srotareneg and ;snoitaler eno G(R<e>) rotareneg rof hcae eludom-)e(R eno noitaler [M, (e)] M, [M,(g)] = [M’, (Q)] rof hcae tcaxe ecneuqes fo ,seludom-)e(R and eno + [M”, (Q)] O-M’*M*M”-0 noitaler rof hcae eludom-)Q(R htiw roF ew [M, (Q)] = 0 M (#e)*M= 0. ,o’~X(n), redisnoc as gnieb deddebme ni ni eht suoivbo .yaw sihT G(R<@>) @esxcn,G(R(~)) swolla us ot add slobmys htiw tcnitsid .S’Q [M, (Q)] yB )R(P ew etoned eht tes fo emirp srebmun gnidivid # R. fI is a emirp ,rebmun p neht R, is eht yramirp-p trap fo R. erehT is a lacinonaC msihpromosi R s eP primellp, and is laivirt-non fi and ylno fi teL S eb a tes fo emirp ,srebmun and tel nP p E P(R). etoned eht puorgbus fo detareneg yb ,c7 rof nehT and RS R PE s. ~5 ns@ RP(~)-s, eht etisopmoc fo eht lacinonac maps secudni a gnir msihpromomoh R~RS -+ R RR-RR. ehT rotcnuf morf eht yrogetac fo seludom-RR ot flesti decudni yb this gnir kceidnehtorG spuorg of naileba group sgnir 971 msihpromomoh is detoned yb Ns. ,suhT fi M is an ,eludom-rrR neht NsMis lauqe ot M as an ,eludom-R and eht snoitca fo ns no NsM and M ,edicnioc tub s-)~t~rr stca yllaivirt no NM. ecneH NsMa M over nR fi and ylno fi S_)~(PZ stca yllaivirt no M. rehtruF ew evah NsNrMz nsN TM rof yna owt sets fo emirp srebmun S and T and yna eludom-nR M. teL Q E ,)n(X and tel S again eb a tes fo emirp .srebmun ecniS ereht era lacinonac snoitcejrus SQ*Q+CT ew yam redisnoc so, as an tnemele fo .)n(X ,oslA SQ is, as a puorgbus fo lauqe& ot eht egami fo ns rednu eht lacinonac map X-+Q;SO se E .)sn(X ecniS eht margaid fo larutan maps and snoisulcni ~RCSRR+-RR iA 1 Rbs) c R(e) is evitatummoc ew ees that rof yreve eludom-)e(R M eht eludom-nR NsM is yllautca an .eludom-)s@(R sihT kramer lliw yalp an laitnesse elor ni noitceS 4. 4. foorP fo meroehT 0.1 nI this noitces ew establish eht msihpromosi .)rrR(Gz))e(R(G_@ teL .)n(X~o, eW mialc that ereht is a puorg msihpromomoh 0_oc : W(e))-G(Rn) rof hcihw Mdv =)l>e< -‘,,,c,, I)tc(p@-s)=[NsM, II]. oT evorp this, ew evah ot wohs that this assignment stcepser eht snoitaler gninifed G(R(e>), .fc noitceS 3. sihT is ylniatrec eurt rof eht snoitaler arising morf trohs tcaxe secneuqes fo ,seludom-)e(R ecnis Ns is ylsuoivbo an tcaxe .rotcnuf oS ti seciffus ot kcehc that fi (#,o)*M= 0, neht C -( ,MsN[.)s-)@p(#)l ]CI =O. SC&J) yB ammeL 2.2 ew yam assume that pe stca yllaivirt no M and that p*M = 0, rof emos p E P(Q). nehT -),eN ~fV+u M, so rof yreve SCP(e) we evah M~~(-sN~M~~~-)~(PNsN~MsN revo RR and [NsM, n] = [Ns- (p}M, .ln 180 H. U'. .ammeL Jr. suhT ew dnif x -( ,MsN[.)s-’Qif(#)l ]n )&CS = 1 -( .9-)Q(P(t#)I ,f&N[( ln - -SNI M}p{ 4) = 0, SC4PLPES as .deriuqer sihT sevorp that o,pc is llew .denifed gninibmoC eht maps ppc ew niatbo a puorg msihpromomoh v .)nR(G‘))e(R(C,~@GO: erofeB gninifed a map ni eht rehto noitcerid ew evorp a .ammel 4.1. Lemma. Let Q‘, Q”E X(x), andsuppose that M is an Rx-module which is both an R(&)-module and an R(@)-module. Then we have in eht group .))Q(R(G,~(x& teL S eb yna subset fo .)n(P eW evorp that S sdleiy eht emas noitubirtnoc no Proof. htob sides. sihT is ylniatrec eurt ,fi no hcae ,edis S sevig eht orez noitubirtnoc ro on noitubirtnoc at all. oS esoppus ti sevig a orez-non noitubirtnoc no eht tfel hand ,edis :.e.i ,??&CS [NsM, I>&< .ot* fI won ~5 =&, neht &(P=S =lQ'JcP(g"), so S sevig no htob sides eht -irtnoc noitub [NsM, <Q$>], as .deriuqer fI ;o_t#j@ neht gniylppa ammeL 2.3 ot NsM, ,je Q: ew dnif that rof emos emirp gnidivid ie# ro .&# nehT p*NsM=O p p E S seilpmi that #( )$g =f&N* 0, gnitcidartnoc ruo noitpmussa = I’(@$), so p*NsM = 0 that sihT sevorp 4.1. [NsM, <@i>] +O. fI is an eludom-nR hcihw rof emos )n(X~’g is an eludom-)@(R ew tup M =)M(y C [NSM, I)&< E .NeWIn@~P )+&CS yB ammeL 4.1 this ylno sdneped no ton no eht eciohc fo .’8 fI M is an M, R(Q’)- eludom and M'cM is an ,eludombus-nR neht M’ and M/M’ era ,seludom-)&(R and )M(Y = ?M(w + .?M/M&t )2.4( woN tel eb yna .eludom-nR yB ammeL 2.5 ereht stsixe a etinif niahc fo sub- M seludom M=Mo>MI>--.>M,=O (4.3) Grolhendieck groups of abelian group rings 181 hcus that hcae &A/t-M is an eludom-),o,(R rof emos ,)n(Xe;e .r....,2,l=i eW redisnoc eht noisserpxe ,i, -iM(Y .)iM/I )4.4( yB .)2.4( this noisserpxe seod ton egnahc fi eht niahc )3.4( is decalper yb a -enifer .tnem tI is osla raelc that eht noisserpxe seod ton egnahc fi )3.4( is decalper yb an tnelaviuqe ,niahc .e.i a niahc M=Me>M; >--a >M:=O rof hcihw A4i_ t/M _,i&MG )i&M/t rof emos noitatumrep o fo { 1,2, . . . . t ] and all ei { 1,2, . . . . I }. ecniS yb s’reierhcS meroeht yna owt sniahc evah tnelaviuqe ,stnemenifer ew edulcnoc that eht noisserpxe )4.4( ylno sdneped no M, tel us etoned ti yb .)M(/y fI MC’M is a eludombus neht gninibmoc a niahc rof M’ and a niahc rof M/M otni eno rof M ew ees that )M(y = )’M(y + .)’M/M(y ecneH yr is evitidda rof trohs tcaxe secneuqes and erofereht secudni a puorg msihpromomoh /v : WW-+ Qe@n,GWW) detoned yb eht emas ,rettel rof hcihw fi Mi, ;e era as .evoba oT edulcnoc eht foorp fo meroehT 0.1 ti won seciffus ot kcehc that pc and U,I era esrevni ot hcae .rehto tsriF ew redisnoc I+u~. teL Q E ,)n(X and tel M eb an .eludom-)e(R nehT M(Dcw <@>I) = ()OIU! - MiNI*)s-‘““(*)l )lX = ()p;cs - )1 ,MsN[(W.)s-)Qif(# .)]?I hcaE is an ,eludom-&(R os NsM Y(VSM )ln = C ,MSNTN[ ((~s)r)l TcP@) .l)m(WnW = C TCS ecneH ew dnif = [Nmfl, 1>,em( = W, ,I>@ H. W. 182 Lenstra, Jr. erehw ew evah desu eht laivirt ytitnedi c =)S-WP(W)l_( l 99&=*fi s. WPCSCT I 0 fi .)e(P#T sihT sevorp that V/CD is eht ytitnedi no .))g(R(Gjn(~cp@ yllaniF ew kcehc that ppw is eht ytitnedi no .)nR(G tI seciffus ot evorp that ,M[(yp( )]n = [M, R] rof yreve eludom-nR M hcihw is an eludom&(R rof emos ,)II(XEQ ecnis yb ammeL 2.5 eht puorg G(Rn) is detareneg yb eht sessalc fo eseht .seludom roF hcus M and Q ew ,evah using eht emas ytitnedi as :erofeb W(M )4 = &II4C [NM, (es>l) = c c t-11 st* - 7-) [NrNsM, ]n l IJ&CS FcS = -((jQ&F I)#(flQ)-T) c (- lrJ4wz[*)’S-’Q~c“~* S. FcScfW 4 = ,fP ln = WmM , as .deriuqer sihT setelpmoc eht foorp fo meroehT 0.1. .5 egnahC fo spuorg dna egnahc fo sgnir nI this noitces ew etagitsevni eht ruoivaheb fo eht msihpromosi fo meroehT 0.1 rednu egnahc fo spuorg and egnahc fo rings. ehT noitaton decudortni ni noitceS 3 sniamer ni .ecrof ,tsriF tel R eb a tfel nairehteon ,gnir rr and ’tr etinif naileba ,spuorg and ‘rr+rr a puorg .msihpromomoh sihT msihpromomoh secudni a ,rotcnuf hcihw ew etoned yb F, morf eht yrogetac fo seludom-’rrR ot eht yrogetac fo .seludom-nR ecniS F is ,tcaxe ti sevig esir ot a puorg msihpromomoh G(RS)-G(Rn), M+l-[FM .ln teL ,)’n(X~’e and tel Q eb eht egami fo eht desopmoc map .’e+’n-n nehT Q yam eb deredisnoc as an tnemele fo .)n(X ecniS R@)cR(e'), ereht is a larutan luftegrof ,rotcnuf hcihw again yam eb detoned yb F, morf eht yrogetac fo seludom-)’o,(R ot eht yrogetac fo .seludom-)e(R fI ni noitidda )Q(P = ,)Q(P neht a map [M -l>’e( MF[ I>@( G(R<e’>)-G@(e)), is .decudni If ?e& * ,9Q tel G(RW))-+G(R(e)) eb eht orez map.