On extensions of pseudo-integers

Publicacions Ma emá iques, Vol 37 (1993), 387-401 . ON EXTENSIONS OF PSEUDO-INTEGERS A bs ac HEATHER RIES An abelian g oup A is pseudo ee o ank k i he p-localiza ion o A is isomo phic o he p-localiza ion o Zk o all p imes p, Le . A p - Z k o all p imes p . I k = 1, we call A a g oup o pseudo-in ege s . We may assume, in his case, ha Z C_ AC and ha o any p ime pi he e is a maximal exponen i so ha p ? i E A . The g oup A is hen gene a ed by { P 2 .  pi is a p ime} and we w i e A= ( __ 1` ) . We say a pseudo ee g oup o ank k is pi comple ely decomposable i i can be w i en as he di ec sum o k g oups o pseudo-in ege s . I A is pseudo ee o ank k, B is pseudo ee o ank l, and B >--> E -» A is an ex ension in Ex (A, B) hen E mus be pseudo ee o ank k -+ l .  In his pape , we conside Ex (P, P) o P = (-~-) and P= (- 1 ~) g oups o pseudo-in ege s . We de- 17 Pia e mine when i is possible, in e ms o he de ining exponen s o P and P, o Ex (P, P) o con ain ce ain ex ensions (which we'll call non i ial) whe e E is comple ely decomposable as a pseudo ee g oup o ank 2 . We ind ha Ex (P, P) con ains such ex ensions i and only i i < Ti almos e e ywhe e and i < i o an in ini e numbe o p imes . 0 . In oduc ion . In [1] Casacube a and Hil on in oduce he concep o he ex ended genus o a nilpo en g oup N, deno ed EG(N) . I is de ined o be he se o isomo phism classes o nilpo en g oups M so ha he p-localiza ions o M and N a e isomo phic o all p imes p, Le . M p - N p o all p imes p . I A is a ini ely-gene a ed abelian g oup, hey show ha o s udy EG(A) i is only necessa y o examine EG(7G k ) whe e k is he o sion ee ank o A . The ex ended genus o 7G is comple ely desc ibed in [2] . The e i is indica ed ha i A E EG(7G) hen A is simila in many ways e 7G and hence A is called a g oup o pseudo-in ege s . We will adop his e minology and, acco dingly, i A E EG(7Z k ) we will say ha A is pseudo ee o ank k . As in [1], we de ine a pseudo ee g oup 388 H . RIES A o ank k o be comple ely decomposable i i is he di ec sum o k k g oups o pseudo-in ege s, Le . A = ®A ¡ whe e each A i is a g oup o -i pseudo-in ege s . In his pape , we in es iga e Ex (P, P) o P and !' g oups o pseudo- in ege s . In [5], he s uc u e o Ilom(P, P) and Ex (1', P) as abelian g oups is iden i ied, while in [4] i is shown ha , o ce ain P and P, Ex (P, P) con ains ex ensions P }--> E -> P whe e E is no comple ely decomposable as a pseudo ee g oup o ank 2 . I e e we should no e ha o any ex ension P >--> E ~> P o P by P, E mus be pseudo ee o ank 2 Since p-localiza ion is exac and Z p is a p .i .d . In his pape , we de e mine when i is possible o Ex (P, P) o con ain ex ensions P » E -> P whe e E is a ce ain ype o comple ely decomposable pseudo ee g oup o ank 2, Le . E= Bl ® B2 o pa icula g oups o pseudo-in ege s B l and B2 . In Sec ion 1, we de ine he ype o comple ely decomposable ex ension (which we will call non i ial) ha we hope o ind by making clea ou es ic ions on Bl and B2 . We desc ibe, in Sec ion 2, when P embeds in o Bl ® B2 o a bi a y g oups o pseudo-in ege s P, B l , and B 2 . Since we wan he quo ien esul ing om such an embedding o be isomo phic o P (and hence o sion ee), in Sec ion 3 we cha ac e ize he o sion subg oup o he quo ien esul ing om a gi en embedding o P in o Bl ® B2 . Finally, in Sec ion 4, we ha e a heo em which allows us o de e mine when such a quo ien is in ac isomo phic o P . We hen ind exac ly when Ex (P, P) con ains non i ial comple ely decomposable ex ensions in e ms o de ining cha ac e is ics o P and P . These esul s we e comple ed as pa o my disse a ion a SUNY- Bingham on . I would like o exp ess my deepes app ecia ion o my ad iso , P o esso Pe e Hil on, o his in aluable guidance and cons an encou agemen . 1 . Pseudo-In ege s and Ex ensions . Fo A, B abelian g oups, we s a e an in e p e a ion o Ex (A, B) which may be ound in [3] . The ex ensions B >--> El -» A and B >-~ E2 -> A a e said o be equi alen i he e exis s 0 : El --> E2 so ha he ollowing diag am commu es : B >-4 E l -» A II 1 , II B >--> E 2-j> A Since 0 is necessa ily an isomo phism, he abo e ela ion is an equi a- lence ela ion and Ex (A, B) is ega ded as he se o equi alen e classes o ex ensions . I may be shown o ha e an abelian g oup s uc u e wi h ON EXTENSIONS OF PSEUDO-INTEGERS  38 9 ze o elemen he equi alen e class o he ex ension B >--> B®A -» A whe e B embeds na u ally in o B®A and B®A p ojec s na u ally on o A . In his pape we shall be conce ned wi h ce ain ex ensions o he o m P - E -» P whe e P and P a e g oups o pseudo-in ege s and E is necessa ily pseudo ee o ank 2 . Howe e , be o e we conside pa icula ex ensions in Ex (P, P) we will no e some esul s p e iously ob ained on he algeb aic s uc u e o Hom(P, P) and Ex (P, P) as abelian g oups . We i s in oduce no a ion o be used h oughou he pape . In [2] i is demons a ed ha i P is a g oup o pseudo-in ege s hen we may assume 7L C_ P C Q and o any p ime pi he e is a maximal exponen i > 0 so ha 1 E P . Mo eo e , as may be easily shown, P is hen P ;' gene a ed by he se {~4 i 1 pi is a p ime} and he elemen s o P a e P ;~ ep esen ed by educed ac ions 6whe e b has p ime powe ac o s pi i , wi h li <_ i . We will hence o h deno e P by (1~) and assume all P ¡ ac ions men ioned o be educed . We will o en employ he ollowing esul (also om [2]) conce ning he isomo phism p oblem o g oups o pseudo-in ege s . Theo em . Assume P = ( 1 ) and H = (k . ) a e g oups o pseudo- P¡ ~  p ;2 in ege s . Then P - H i and only i i = ká almos e e ywhe e . Now Suppose P= ( i ) and P = ( i ) a e g oups o pseudo-in ege s . P¡  P¡ The abo e heo em implies ha o de e mine Hom(P, P) one need only conside he case whe e i > i o in ini ely many i and he case whe e i <_ i e e ywhe e . The ollowing esul s a e hen ob ained in [5] which show ha Hom(P, P) is ei he i ial o ano he g oup o pseudo- in ege s . Theo em . I i > i o in ini ely many i hen Hom(P, P) = 0 . Theo em . I i <_ i o all i hen Hom(P, P) =(~ .) whe e li = i - i . A special case o a heo em p o ed o pseudo ee g oups o a bi a y ank hen desc ibes Ex (P, P) . Theo em . Suppose P and P a e g oups o pseudo-in ege s .  Then Ex (P, P) =V® (Q/7G)" whe e V is a Q- ec o space o ank c and l is he ank o Hom(P, P) . 39 0  H . RIES In his pape , we shall conside ex ensions P >-> E -» P in Ex (P, P) whe e E is comple ely decomposable as a pseudo ee g oup o ank 2, Le . E= B l ® B2 o Bl and B 2 g oups o pseudo-in ege s . We will some imes e e o E i sel as he ex ension . Now i is clea ha one such ex ension is always p esen , Le . he ze o ex ension P >-~ P®P -» P in Ex (P, P) . So we wish o de e mine when Ex (P, P) con ains o he , less ob ious comple ely decomposable ex ensions . We he e o e de ine a comple ely decomposable ex ension P >-~ Bl ® B2 -» P o be i ial i P = BI, P = B2 o P= B2, P = Bl and non i ial o he wise . As we see om he ollowing heo em, ou de ini ion gua an ees ha a non i ial ex ension will no belong o he ze o elemen o Ex (P, P) . Hence ou goal will be o desc ibe when Ex (P, P) con ains non i ial comple ely decomposable ex ensions . Theo em 1 .1 . I P >--> Bl ® B2 --» P is non i ial in Ex (P, P) hen he ex ension is also nonze o in Ex (P, P) . P oo . Suppose P >--> Bl © B2 --» P does belong o he ze o elemen in Ex (P, P) . This implies ha P >--> Bl ® B2 --» P is equi alen o he na u al ex ension P >--4 P ® P -» P . Hence B l ® B2 is necessa ily isomo phic o P ® P . Now in [4] i is shown ha he decomposi ion o a pseudo ee g oup o ank 2 mus be unique, Le . i P ®P -Bl ® B2 hen P= Bl, P = B2 o P= B2, P - B l . Thus we ha e a con adic ion since we assumed ou ex ension o be non i ial . 2 . Embeddings . Since we wish o ind ex ensions o he o m P >--> Bl ® B 2 ---» P, we mus i s de e mine when i is possible o embed P in o Bl ® B2 . We no e ha since P, B l , B2 a e subg oups o Q, o desc ibe an embedding o P in o Bl ® B2 we need only indica e he o de ed pai o which 1 E P is sen . Fo , i 1 >-4 ( i , 2 ) in B l ® B2 hen b ~-, ( lb , 26) o any bE P . We shall conside only embeddings o he o m 1 ~--> (~  -) whe e i > 2 ul =?~ 0 and u2 =~ 0 as o he wise, ei he he quo ien has o sion (and hence could no be isomo phic o P) o we a e led o a i ial ex ension . Fo suppose ha P >--> Bl ® B2 --» P is an ex ension in Ex (P, P) so ha P embeds in o Bl ® B2 wi h he embedding 1 H ( i , 0) o some ul =,A 0 . Now Bi©Ba = Bl/P ® B2 and B1 p B2 = P . Bu B l /P is a o sion g oup and hence mus be i ial which indica es ha B l - P . Now B2 mus hen be isomo phic o P and we see ha he ex ension is i ial . O cou se an embedding o he o m 1 ~--> (0, 2) o some u2 ~ 0 yields a simila esul . ON EXTENSIONS OF PSEUDO-INTEGERS  391 The nex wo heo ems add ess he p oblem o embedding P in o Bl B2 . The i s desc ibes when a speci ic homomo phism om P o Q ® Q is an embedding and he second shows when i is possible, in e ms o he exponen s o P, Bl, and B 2 , o embed P in o BI ® B2 . Fo n E 7G and pi a p ime, le p , (n) be he usual pi- alua ion o n, Le . he highes powe o p i ha di ides n . Then o in Q, we de ine pi ( ) = pi (u) - p i ( ) . Finally, le P = (  ), Bl = (-~), and B2 = (- 4 i ) ep esen he g oups pe y  p ;  p : o pseudo-in ege s . Theo em 2 .1 . The embedding o P in o Q® Q de ined by 1 ~-- (~, 2 ) is an embedding o P in o Bl®B2 i and only i i- p J i) _< mi and i - p i ( 2) < ni o all i . P oo . Now 1 H (ul u2 ) es ic s o an embedding o P in o B l ®B2 i l ' 2 and only i ~ H ( u i , ~7) E B l ®B 2 o all i . Bu ~ E Bl /  pi ip i 2p~  ipi p i 1 p ) > - mi ~ i p i ( ) - i > - mi ~_? i - Pi(1) <- mi . - Simila ly,  E B2 ~ i i - p i ( 2 ) < ni . No e ha , in pa icula , 1 ~--> (1,1) de ines an embedding o P in o B l ©B 2 i and only i i < mi and i < ni o all i . Theo em 2 .2 . The g oup o pseudo-in ege s P embeds in o Bl ® B2 i and only i i < mi and i < ni almos e e ywhe e . P oo . Assume P embeds in o Bl ® B 2 and le 1 ~--> (VI , 12 ) be an i 112 embedding .  Since p, ( ) = 0 and p , (  ) = 0 o all bu a ini e numbe o i, i is clea om Theo em 2 .1 ha i < m i , i < n i almos e e ywhe e . I we suppose ha i < mi and i <n i almos e e ywhe e, we can easily cons uc an embedding . Le S i = {j 1 j > m j } and S2 = {k 1 k > nk} and conside he embedding o B in o Q ® U de ined by 1 H (  p '-m', 11 pk k-nk ) . I can be eadily e i ied ha i - jESl kES2 j -m j k - mk p ~ ( j p j  ) < mi and i - p , (~ pk  ) _< ni o all i and hence, jES1  kES2 by Theo em 2 .1, we ha e an embedding o P in o BI ® B2 . 3 . To sion Subg oups o Quo ien s . Recall ha ou goal is o de e mine when i is possible o ind non- i ial comple ely decomposable ex ensions in Ex (P, P) Le . ex ensions o he o m P >-4 BI ®B 2 --» P whe e ce ain condi ions a e placed on B l , B2 . Since, o such an ex ension o exis , he quo ien B1PB2 mus 39 2  H . RIES be isomo phic e P, we mus de e mine when i is possible e embed P in o Bl ® B2 so ha he esul ing quo ien is o sion ee . Hence in his sec ion we cha ac e ize he o sion subg oup o B l PB2 wi h espec e a gi en embedding . This cha ac e iza ion, gi en in he nex wo heo ems, is essen ial e ou de e mina ion o he exis en e o comple ely decom- posable ex ensions in Ex (P, P) . Gi en an embedding o P in o Bl®B2, we will use Q o ep esen he esul ing quo ien o Bl ® B2 by P, Le . Q = B1 P B2 . We will ep esen he o sion subg oup o his quo ien by TQ . Also, we will assume P= (pii ), B l = (pmi ), and B2 = (p ) . Theo em 3 .1 . Suppose P embeds in o B l ® B2 wi h he embedding de ined by 1 i-~ (1, 1) . Then TQ- ®7L/pi' - "Z whe e li = min(mi, ni) . pi P oo . We i s show ha TQ = {( b ° b )1P B1nB2} . Suppose [( b , b )] is a cose in TQ .  The e mus exis an in ege k so ha k(bi , 62) E im P= {(x, x) 1 x E P} (since 1 H (1, 1) gi es he embedding) . Hence {(b , b)IbE B 1 nB2 } ( b, , 62) _ (k, k) o some xE P and [( bi , b2 )] E  P  . Now EB1nB2 asume [( b , b )] 1S a cose in {(b ")Ib P  } .  Slnce b(ab, ab ) _ (a, CL) E im P in Bl ® B2, [(b , ' b)] E TQ . So TQ = {(b , b )12PP B1nB2} -B 1 nnB 2 .  Now Bl n B2 = (  ) whe e pi li = min(mi, ni) . Hence B1 P nB2 =  p (--L . )IP -- e7G/pli - i i  7G since a o sion i  pi abelian g oup is he di ec sum o i s pi- o sion subg oups . No e ha li - i = min(mi, ni) - i >_ 0 o all i, since i <_ mi and i <_ ni o all i by Theo em 2 .1 . Now we use Theo em 3 .1 o de e mine he o sion subg oup esul ing om an a bi a y embedding . Theo em 3 .2 . Suppose P embeds in o B l ® B2 wi h he embedding h- de ined by 1 ~--> ( i , 2 ) . Then TQ = ®7L/p i  i Z whe e lz = min(mi + Pi pil l ),ni+ pi( ))  2 P oo . Conside he ollowing commu a i e diag am 1 H u1 u2 VI ' 2 P ,--, Bi ® B2 11 ® ú2 P ,- Bi ® Bá 1 1-4 (1,1) whe e B' = ( m z ) wi h m' = mi + p i ( i ), B2 = (i,) wi h n2 - Pi  p ? i ni + p i (2 ), and  i ®  2  : B l ®B 2 - Bi ® B2 is he isomo phism de ined as mul iplica ion by úi in he i s coo dina e and by 22 in he second . No e ha 1 H (1, 1) does de ine an embedding o P in o B' ® B2 . Fo , since 1 - ( i , 2 ) is an embedding o P in o Bl ® B2, i < mi + p , ( V, ) = m i ' and i <_ ni + p i ( 2) = n' o all i by Theo em 2 .1 . Thus 1 ~--~ (1, 1) is an embedding o P in o Bi ®BZ also by Theo em 2 .1 . Le Q'  Bl P ®B2 b e he quo ien esul ing om his embeddin g . Now Q = Q' since l ® 2 - induces an isomo phism on he quo ien s . ul u2 Hence TQ = TQ' -- ®7G/pi i - 'Z whe e l2 = min(mi,ni) by Theo em Pi 3 .1, and he heo em is p o dd . The heo ems in he emainde o his sec ion a e no necessa y o p o e ou e en ual esul on he exis en e o comple ely decomposable ex ensions in Ex (P, P) . Howe e , gi en ha P, Bl, and B2 a e g oups o pseudo-in ege s so ha i is possible o embed P in o B l ® B2, he heo ems p o ide insigh in o he na u e o he o sion subg oups o Q = B, @B2 which occu wi h espec o di e en embeddings . Fo , gi en one such o sion subg oup, we a e able o de e mine all o he o sion subg oups which i is possible o ob ain . We also ind, in e ms o he exponen s o P, B l , and B2, when i is possible o embed P wi h a o sion ee quo ien . Mo eo e , we desc ibe unde wha condi ions a gi en embedding will ha e a quo ien ha is o sion ee . Again, we le P= ( p i i ), B 1 = (p- i ), and B 2 = ( p l i) ep esen he g oups o pseudo- in ege s . Theo em 3 .3 . Suppose P embeds in o Bl ® B2 wi h TQ - ® 7G/pk i, Pi ki >_ 0 . Le ®7L/pk i wi h ki >_ 0 be a o sion g oup . The e exis s an pi embedding o P in o B1®B2 so ha TQ- ® 7G/pk i i and only i ká = kz Pi almos e e ywhe e . ON EXTENSIONSOF PSEUDO-INTEGERS  39 3 P oo .- Le 1 ~--> (ul u 2 ) be an embedding o P in o B 1 ® B2 so ha V1 i2 TQ - ®7G/pk i7G . We know ha i - p i ( l) G_ m i and i - p i ( 2) _< ni p= o all i (Theo em 2 .1) . Also, by Theo em 3 .2, ks = li - i whe e li = min(m i + p i (~ ), ni + p i Now assume 1 H (b , b) is an embedding o P in o B I ® B2 so ha TQ- ® 7 G/pi 7L . As abo e, k' = l2- i whe e l2 = min(mi+ p i (bi ), ni + Pi 394  H . RiEs p, (l» .  Since p, (b) = pi (6) = pi ( ) = pi (Yz) = 0 o almos 2  ,  2  1  2 all i, l i = lz = min(mi, ni) almos e e ywhe e .  Hence k i = k? almos e e ywhe e . Now suppose ki = ki almos e e ywhe e . Le Sl = {i  ki > ki}, S2 = {i  k i > ki}, u =  p~~ - ~~, and =  pi  i . No e ha i iES1  iES2 i E SI U S2 hen p , ( ) = kz - ki and o he wise pi (z) = 0 . We can show ha 1 H ( uul  uu2) is an embedding o Pin o Bl ®B 2 . Fo i i ¢ Sl US2, i 2 i - p, ( ) - i - pi ( i) < mi and i - p, ( 2) - i - p, (  ) < ni . I i E SIUS2, i- p i ~ - i - pi( ) - (ki -k i) - i - p,( ) - ki+ min(mi+ pi( ) , n i + pi 2)) - i < - p i( ) - ki+mi+ pi( )  mi- VI <_ mi as kz >_ 0 . An analogous a gumen shows ha i - p i ( 2) <_ ni . Hence, we ha e an embedding by Theo em 2 .1 . We see ha he o sion subg oup o Q wi h his embedding is ®7G/pk'Z . F om Theo em 3 .2, TQ= ®71/p?i - i whe e l2 = min(mi + Pi  pi p, ( i) ni + p i (l )) = l i + pi ( ) "  I i 151 U 52, hen li = li and l? - i = li - i = ki = k2 . I i E S l US 2 , li = l i + pi ( ) = l ¡ +(k2 -k ki + i and hence lZ - i = k' . Co olla y 3 .4 . Le P, Bl, and B2 be g oups o pseudo- in ege s so ha P embeds in o Bl ® B2 .  Then TQ is ini e o all embeddings o P in o Bl ® B2 o TQ is in ini e o all embeddings o P in o Bl B2 . Mo eo e , i TQ is always ini e hen P embeds in o Bl ® B2 wi h o sion ee quo ien . Fo he nex heo ems we es ablish he ollowing no a ion . Le Q+ (P) = {i 1 i > 0} and B, P = (pes) whe e si = max(mi - i, 0) . Theo em 3 .5 . Suppose P embeds in o B l ® B2 .  The e exis s an embedding o P in o B l ® B2 so ha Q is o sion ee i and only i a+(Bi P) n u+ (B2 P) is ini e . P oo . : Suppose o,+(Bi P) n Q+(B2 P) is in ini e . Le 1 H ( 1 ~l, u 2 ) 1 2 be an a bi a y embedding o P in o B1 ® B2 . Selec pi so ha i E o - +(Bl P) n Q+(B2 P) and pi( i) = pi( 2) = 0 .  Then no e ha x= (  ui lp, i+1 ,  2P i 1 ) E B l ® B2 . Ce ainly x  im P CB l ® B2 bu pix is he image o - i ~ i - E P in Bl ® B2 . Hence Q has o sion . p ; Now assume ha Q+(Bl P) n Q+(B2 P) is ini e .  Le Si = {i i > mi o i > ni} and no e ha , by Theo em 2 .2, SI is ini e . Le S2 = {i 1 i < mi and i < ni} and also no e ha S2 is ini e since ON EXTENSIONS OF PSEUDO-INTEGERS  395 Q+(B 1 P) nQ + (B2 P) is ini e . Now i i 11 S1 U S2 hen i < mi, ni and i = mi o i = ni . Le 1 H--> (!'-, l~ 2 ) be an embedding o P in o B I ® B2 . VI 2 By Theo em 3 .2, TQ - ®7Gpi i -T iwhe e li = min(m i + pj ( i ), ni + Pi VPi ( 2 )) .  Le S3 = {i  p j ( i) =,A 0 o p, ( 2)  0} so ha i i ¢ S3 hen li = min(mi, ni) . Now i i 1 Si U S2 U S3 hen li = min(mi, n i ) = i and hence TQ has only a ini e numbe o non i ial p-componen e . Thus he e exis s an embedding o P in o B l ® B2 so ha he o sion subg oup o he quo ien is i ial (Theo em 3 .3) . So gi en ha P embeds in o B l ®B 2 wi h o sion ee quo ien , we know ha i < mi, ni o almos all i and Q+(Bl P)no,+(B2 P) is ini e . The ollowing heo em enables us o decide when a gi en embedding will ha e a o sion ee quo ien in he special case whe e i _< mi, ni o all i and o,+(B I P) n Q+ (B2 P) is emp y . Al hough we will no do so explici ly he e, we could use a commu a i e diag am as in he p oo o Theo em 3 .2 o ex end he esul o he mo e gene al case . Theo em 3 .6 . Suppose P embeds in o B l ® B2 wi h o sion ee quo- ien . Assume also ha i < m i , n i o all i and Q+(Bl P) n + (B2 P) is emp y . Le 1 i--> (u' , u 2 ) de e mine an embedding o P in o B l ® B2- V1 2 Then Q has o sion i and only i he e exis s pi so ha pi 1 ul o yI holds and p i 1 u2 o y2 holds whe e yl is he condi ion ha i E Q+(BI P) and pi( l) < mi - i , y 2 is he condi ion ha i EQ+ (B2 P) and VPi( 2) < ni - i . Now suppose Q has o sion and le [(b , b )] be a non i ial cose o ini e o de . Hence he e exis s k E 7G (k > 1), - E P so ha k(' 221) d  bl b2 ( cul  cu2) EBl ®B2 . SO á l = u- and ' 22 =  12 , whe e we may asume d i d 2  bi kd l  b2 kd 2 kd o be educed (i i we e no , we would eplace kd wi h i s educed o m) . Since ou chosen cose was non i ial, k  P and hence he e eX1S S pi so ha Pi(kd) > i . NOW (b1 , 2)_ (d l  kd 2) E Bl ® B2 and Q+(B l P)n Q+ (B2 P) is emp y so p i 1 ul o Pi 1 u2 . We assume Pi 1 u l and no e ha i pi 1 u2 also, he e is no hing mo e o p o e . O he wise, since pi(kd) > i and kd E B 2 , we Na e i E u' (B2 P) and pi ( 2) < ni - i, Le ., we ha e condi ion y2 . Simila ly, i we had assumed pi 1 u2 hen we could conclude ha pi 1 ul o y l is ue . Le pi be a p ime so ha p i 1 ul o yI holds and p i 1 u2 o y2 holds .  I p i 1 u l and pi 1 u2 hen u l = piul and u2 = piu2 .  Now x = ( ~Í , - -) E B l ® B2 since we assume l~l, u2 a e educed and P ; 2 i P ; 2  i 2 i < m i , ni o all i . No e ha x 1 im P in B I ® B2 bu pix is he image o  in B l ® B 2 . Thus [x] is a non i ial o sion elemen in Q . P :