The divisor group of a fir

Pub . Ma . UAB Vol . 26 N° 1 Ma a 1982 In oduc ion THE DIVISOR GROUP OF A FIR P .M . COHN Bed o dCollege Regen 's Pa k London, NW1 4NS Rebu el 30 d'oc ub e del 1981 The e ha e been many a emp s o de ine de e minan s on non-commu a i e ings (c .e.g .[13] and . he e e ences quo- ed he e), o which pe haps he mos success ul is he de ini ion o Dieudonné [10], leading o any skew ield K and any n > 1 (excep when n=2 and K=F 2 ) o an isomo phism (1)  GLn(K) ab  - K *ab . Suppose now ha K is ob ained om a ing R by in e ing ce ain ma ices o e R, o ming a se E . The way in which he elemen s o K a e ob ained om R and E was desc ibed in Ch .7 o [3], and we mayask whe he GL n (K) ab can be desc ibed di ec ly in e ms o R and E . Since (1) is an isomo phism o all n, we can limi ou sel es o a sin gle alue o n, o we maysimply ake he limi GL(K)=1im GL n (K) . Ou aim he e is o desc ibe he Whi ehead g oup K 1 (K)~ GL(K) ab in e ms o E ; his can be done unde ai ly gene al condi ions, hough o mo e p ecise esul s we need o ake R o be a i and K i s uni e sal ield o ac ions . In pa i cula , by aking R o be a ee associa i e algeb a we ob- ain an explici exp ession o de e minan e o e a ee ield (Th .5 .2) . To s a e he esul e, le :R -> K be a homomo - phism o any ings and suppose ha e e yelemen o K can be ob ained om he en ies o he o mal in e ses o he ma i ces om a se E , which is mul iplica i e (as de ined below, c . aleo[3], p .249), hen i is no ha d o show ha in- duces an epimo phism o abelian g oups (2)  * : Eab -> K1(K), whe e  E ab  is  he  uni e sal  abelian  g oup  o  E  (Th .2 .2  and Co .) . In gene al he e is no eason o * o be injec i e, bu when K is he uni e sal ield o ac ions o a Syl es e domain R and E he se o al] ull ma ices o e R, hen (2) is an isomo phism . This is p o ed ( o he sligh ly la ge clase o pseudo-Syl es e domains) in Th .3 .1 by cons uc ing an in e se ma)pping o * . Fo a somewha di e en ea men o he  same p obl em see  [121  and al so [61 . Fo Syl es e domains i is di icul o say mo e be- cause li le is known abou ac o iza ion in such ings . Bu when we ha e a i R , o .mo e gene ally a ully a omic semi i (i .e . one in which e e y ull ma ix can be exp essed as a p oduc o a oms) hen a mo e p ecise s a emen is possible . In R de ine apn,íme as a class o s ably associa ed a oms and he d .í .í,&wc g oup D(R) as he ee abelian g oup on al] he p imes, and le U be he uni e sal ieldo ac ions o R , henwe p o e in Th .4 .4 ha K 1 (U)°- U*ab- D(R) x [GL(R)/GL(R)nGL(U)'] . In pa icula , when  R = k < X >  is a ee algeb a, his be- comes U*ab = D (R) x k* (c . Th .5.2 .) . These esul s ha e also been ob ained by G .Ré- ész [12] by a di e en me hod . Ou second main esul is conce ned wi h localiza ion o i s . Le R be a ully a omic semi i and E a mul ipli- ca i e se o ma ices such ha R . is again a semi i , hen R,, is also ully a omic and he di iso g oupo R .. is iso- mo phic o D~(R), he subg oup o D(R) gene a ed by he p imes which su i e in R,, (Th .6 .3) . I am indeb ed o G .M . Be gman o his ex ensi e com- men s on an ea lie e sion, and o G . Ré ész o se e al help- ul ema ks . 1 . No a ion and gene al backg ound Le R be any ing ; we w i e m Rn o he se o al] m x n ma ices o e R and also pu MR o M R 1 and R n o 1 R n . The chanac enía íc o an m x n ma ix is de ined as n - m . I a ma ix is exp essed in block o m (P), we o en w i e his as  P, T ( Q)  o sa e space ;he e T indica es ha he blocks P, Q a e o be w i en as a column, bu a e no hemsel es ansposed . Simila ly o mo e han wo blocks . The d iagonal sum o wo ma ices A, B is de ined as A + B = (A B) . The se o all in e ible n x n ma ices o e R is deno ed by GL n (R) and we embed GL n (R) in GL n+1 (R)  by  he  ule  A  1 - =  A  +  1 .  Fu he  we  pu  GL(R)  _ lim  GL n (R) .  As usual, a ma ix is said o be elemen a y i i di e s om he uni ma ix in a mos oneo -diagonal en- y ; he g oup gene a ed by all elemen a y n x n ma ices is w i en E n (R) and as be o e we pu E(R) = lim`En(R) . Fo any A E m Rn , he leas in ege such ha A = PQ, whe e P E M R , Q E R n is called he inneh hank and -a ma ix is said o be jul'l i i is squa e, say n x n and o inne ank n . In gene al, i A is ull, i need no be he case ha A+ 1 is ull ; i A + I is ull o any uni ma ix I, A is calleds ably bul'l and he se o al] s ably ull n x n ma ices o e R is w i en F n (R) ; we embed F n (R) in F n+1 (R) as o GL(R) and w i e F(R) = lim F n (R) . Some imes we shall need a gene aliza ion This limi always exis sand is an in ege o - -, bu i I n is ull o all n, he inne ank is ac ually non-nega i e (c . [81) . 'We no e ha an n x n ma ix is s ably ull p ecisely when i has s able ank n . Two ma ices A, B a e said o be abeoc ia edi A = PBQ o some in e ible ma ices P,Q . I P, Q can be aken in E n (R), we call A and B E-assoc ia ed . I A + I is (E-) associa ed o B + 1 (whe e he uni ma ices need o he inne ank . The zxabl'e nank o a ma ix A is de ined as lim { k(A + I n ) - n}, whe e k deno es he inne ank . no be o he same size), henwe say ha A and B a e e ably (E-) associa ed . Twos ably associa ed ma ices a e no necessa ily o he same size, bu hey ha e he same cha- ac e is ic, p o ided ha R is a ing wi h in a ian basis numbe (i .e . all in e ible ma ices a e squa e) . By a bíel'd we unde s and a no necessa ily commu a i- e di ision ing ;some imes we use he p e ix 'skew' o em- phasis . I G is any g oup, i s de i ed g oup is deno ed by G'  and  we  w i e  G ab  =  G/G'  o  he  abelianiza ion  o  G . Fo a ield K we w i e K* o he g oup o i s non-ze o elemen s  and  by  abuse  o  no a ion  we  simply w i e  K ab  o K* ab .  I  R'  is  any  ing,  an  R-6 ield  is  a  ield  K  wi h  a homomo phism R  ' K ; i K is gene a ed, as a ield, by he image o R, we speak o an epic R- ield . An m x n  ma ix o e a ield K, o ank , is said o ha e l'eb nullí u m- and nígh nul'l'íxy n- . These nulli ies a e only de ined o e a ield,bu i A is a ma ix o e R and K is any R- ield,we can conside he nu- lli ies o A o e K ; heya e simply he nulli ies o he image o A in K . We shall no epea he de ini ions o i and semi i (c . [31, Ch . 1 o [41, Ch . 4) . We me ely ecall ha e e y semi i R has a uni e sal ieldo ac ions U, ob ained by o mally in e ine all ull ma ices o e R (c . [31, p . 282 .) . Mo e gene ally, i R is any ingandE a se o squa e ma ices o e R, hen a map : R = Sis called a E-ín eh ing ep imonphíbm i is a homomo phismmapping each ma ix o E o an in e ible ma ix o e S and S consis s o he en ies o in e ses o he ma ices  A , AEE . I is easily seen ha his is in ac an epimo phism in he ca ego y o ings . In wha ollows, Ewill gene ally be mul ipl' ica í e,  i .e .  1 E  and  i  A,  B E = - S,  hen  ( A  B)  E E o all C o app op ia e size . Fo any ing R and se E o ull ma ices o e R, he localíza íon Rz : o R by E is de ined as he ing ob ained om R by o mally in e ing all he ma ices in E . Then in any E-in e ing epimo phism : R  ' S, S is clea ly a homomo phic image o RE . Suppo se now ha s is mul iplica i e ; hen each elemen o S may be ob ained as he las componen u n o he solu ion o a ma ix equa ion (1)  A u = 0, A = (Ao,A1, . . .,An) E n R n+1 , whe e (A l .. . .,A n ) EE and u = (1,u1, . . .,un)T .  I  p = u n ,  we say ha (1) is an S-admissible sys em o p and call (Ao,A1, . . .,An-1)  he numenazon,  (A 1  . .,A n )  he denom ina on o p (c . [5], § 4) . I is o en con enien o pu (Al$ . . .,An-1) = A * and A n = A .), hen he nume a o will be (A 0 ,A * ) and he denomina o (A * , A.) . 2 . The calcula ion o K 1 o a localiza ion I is a well known ac ha o any ing R, E(R) = GL(R)'  (c . [1], V .1 .5, p .223), so ha GL(R) ab = GL(R)/E(R) ; by de ini ion his is he Whi ehead g oup K 1 (R) . I we ha e a skew ield K, hen o any non-singula ma ix A o e K he e exis s a E K* such ha A --_ a (mod E (K)), and he e a is de e mined mod K*' . The esidue class o a in K ab is called he I)íeudonné de enmínan o w i en de A . We no e ha by (1), o any skew ield K . Suppose now ha R, S a e any ings, E is a mul i plica i e se o ma ices o e R and :R ' S is E-in e ing epimo phism . Ou objec is o exp ess K 1 (S) in e ms o R and E . In o de o do hiswe need o exp ess ma i- ces o e S as solu ions o sys ems o equa ions o e R (as was done in (1) o § 1 o elem . en s) . P oposi ion 2 .1 Le R, S be any nínga, E a mu.l íplíea í- e ee o6 ma . níeee  o en  R,  andle  :R -- =  S  be a  E-ín en .íng epímonphísm .  Gí en any  P E m S n ,  he e exíb s an ínxegen '> 0  and ma níces (A o ,AA  ) E +m R n+ +m ,  u  =  (u  u*,u,) T E n+ +m S n (3)  A =  , *, ~  o , whene zhe numbena ob eolumne oj Ao,A*,A . and 2í zewíbe he numbenh oj nows oj uo,u*,u,o ane n, ,m, nespee í ely, eueh xha (A *,A  ) CE, a ab A, one non-ze o en y, since he gene al case may be ob ained by adding such ma ices . By ow and column ans o ma ions we can educe e e y hing o he case P = (p o), p E S . Le A u = 0 is a sys em o P, as equi ed . and (5)  u o = I, u . = P . Moneo en, u ís he uníque elemen o5 n+ +m S n sa ís1yíng (4), (5) hon he gí en ma níx A . We shall cal] A an S-admíssíble sys em o P . P oo . The uniqueness o u ollows om (4), (5), since any ma ix in E is in e ible o e S . To p o e he main asse ion we no e ha i i holds o wo ma ices P', P" E m S n , hen i also holds o P = P' + P" . Indeed, i P', P" a e de e mined by sys ems Al u' = 0,  A" u" = 0,  analogous o  A  abo e, hen (as in he case o elemen s, c . [3], p .250), P is gi en by he sys em A¿ A* A,~ 0 0 Í (I,u*,P',u*,P)T =0 . A"  0  -A"  A" A" 1 o  00 * 00 Hence i su ices o p o e he esul o ma ices wi h only 0 be an S-admissible sys em o p, hen A o0 A * A . 0 1 0 ~0 1 n-1 0 0 0 0 1 m-1 u * 0 = p 0 0 0 The equa ion (4) may again be w i en in a o mo C ame 's ule (131, p .251 and [51, § 4) lde shall again cal] (A * , A .) he denomina o o he sys em A,  bu  de ine  he  nume a oA as  (A * ,  -A 0 ) .  This  is  igh associa ed o he nume a o as de ined in [51 (and ecalled in § 1), so he change has no e ec on he conside a ions o [51 excep no a ionally . F om (6) we see ha P is s ably associa ed o e S o i s nume a o , in pa icula i has sa- me cha ac e is ic, and i is in e ible i he nume a o is in e ible o e S, i .e . i he sys em can be chosen so as o ha e a nume a o in E . Mo eo e , when S is a ield, he le and igh nulli ies o P o e S ag ee wi h hose o i s nume a o . Le R, S be any ings, E a mul iplica i e se o ull ma ices o e R and :R  ' S a E-in e ing epimo - phism . Since E is mul iplica i e, i s ma ices a e e en s a bly ull and we may embed E in  F(R) by he ule A F-> A +I ; his allows us o ega d E as a submonoid o F(R), wi h he mul iplica i n AB . Now conside he uni e sal abelian g oup Eab o E ; his is de ined asan abelian g oup ,ab wi h a homomo phism E  ' sab which is uni e sal o all homomo phisms o E  in o abelian g oups . To desc ibe ,ab  explici ly,  le  us  deno e  by  [A1  o  [Al E  when con u- sion is possible, he class o A E E unde s able E-associa- e minan s o ma ices o e pseudo-Syl es e domains : Co olla y . Le R be a pseudo-Sy .C ee e domaín and U í h uní enba .C (íeld oj bnae . íona, hen bo any z ab1y 6u .Cl ma níx A o en R we ha e whene  íz hemap (1) índueed by :R - U andse íh alzen o en U . Fo o e U, A is s ably E-associa ed o a E U, such  ha  a ab  =  se  A .  Hence  [Al  =  aab  =  se  A . Fo Syl es e domains Th .3 .1 has also been ob ained by G .Ré ész [121, by ano he me hod, based on he abo e Co- olla y ( o he case o i s he e is ye ano he p oo in [61 ) . 4 . The di iso g oup o a ully a omic semi i In  o de  o  in es iga e  he  s uc u e  o  U ab  mo e ully we need o assume he exis ence o comple e ac o iza- ions in ou ing R . We ecall ha a squa e ma ix A is calles an a om i i is a non-uni and canno be w i en as'a p oduc o wo (squa e) non-uni ma ices ; i is clea om his ha an a om is necessa ily ull .  A ing is said o be bully a .íomíe i e e y ull ma ix can be w i en as a p oduc o a ini e numbe o a oms, o is a uni . In pa icula , e e y elemen no ze o o a uni hen has a comple e ac o i- za ion in o a oms . Le R be a semi i and U i s uni e sal ield o ac ions . By a Z- al'ue on R we shall unde s and a homomo - phism :GL(U) -> Z such ha (A) > 0 o all A E F(R) . To gi e an example, le us assume ha R is a ully a omic semi i and ecall om (3],p .201 he unique ac o i- za ion p ope y : E e y ull ma i . x o e R is ei he a uni o has a ac o iza ion in o a omswhich is unique up o s able associa ion and he o de o he ac o s . Now le P be an a om and o any A E F(R) de ine (A) = i in any comple e ac o iza ion o A he numbe o a omic ac o s s ably associa ed o P is jus . By unique ac o iza ion his is well-de ined and we ob ain a Z- alue on R by pu ing ( A] .F - [B] F) = (A) - (B) . This is called he 6ímp . e Z- alue associa ed wi h he a om P . P oposi ion 4 .1 .  Le R be a bull'y a omíc semílín and le be any Z- alle on R . Then (i) (P)=0 bo PEGL(R), and (ii) (A)= (A') whene en A, A' a e 6 abl?y aaeoeía ed . P oo .  (i)  Le  P E GL(R),  hen  (P) > 0,  (P -1 ) > 0,  bu (P) + (P -1 ) = (I) = 0, hence (P) = 0 . (ii) Le A, A' be s ably associa ed, say (A + I)U = V(A' + I),  U,V EGL(R) ; since (U) = (V) = 0, we ha e (A) = (A') as claimed . Le us de ine a p íme o R as a classo s ably asso cia ed a omic ma ices . Wi h each p ime p i he e is associa- ed a simple Z- alue i . Mo e gene ally, picic an in ege n i > 0 o each p ime p i , hen w = En i i is a Z- alue, o  i  is  de ined  on  each  ull  ma ix  A :  w(A)  = En i i (A), whe e he sum on he igh is ini e because i (A) = 0 o almos all i . We obse e ha e e y Z- alue a ises in his way ; o i w is a Z- alue on R, le P i be an a om in he class p i and pu n i = w(P i ), hen w and En i i ha- e he same alue on eacha om and hence on al] o F(R)ab, Theo em 4 .2 . be he s ímple Fon, any lam .í .Cy Z- alue, and eon en .bely, e e y Z- alue on R íz o6 h .í~s 6onm . We ema k ha wi h e e y ull ma ix associa ed a Z- alue w A which is simple i is  an  a om,  iz .  wA = En i i ,  whe e  he  i ple Z- alues and n i = i(A) . We can also use Z- alues o semi i s : P oposi ion 4 .3 .  Lex R be a sem .í6 .íA, hen m,íe .í6 and on1y .í6 Zhene .íe a Z- alue w on w(A) = 0 pnee .íeelN ¡ohen A .íó a un .í . P oo . I R is a ully a omic semi i and i ple Z- alues co esponding o he di e en p imes o R, hen w = E i has he desi ed p ope y . Con e sely, when w exis s, ake any ac o iza ion A E F(R) and ac o ize i in o non-uni s in anyway : This p o es Le R be a 6ully a om .íe zem .í6 .ín andle ( i) Z- aluea eon .n .ezpond .íng o he pn .ímeb 01 R . (n i )  o6  non-nega .í e .ín egena,  In i i  íh  a A he e is and only i A a e al] he sim- cha ac e ize ully a omic R .íz 6ully a o R such ha a e he sim- (2)  A = P 1 - .P . Since w(P i ) > 1 by hypo hesis, we ha e w(A) = Ew(Pi) > , and his p o ides a bound on he numbe o ac o s in (2) . By aking a ac o iza ion wi h maximal we ob ain a comple- e ac o iza ion o A . This comple es he p oo . Now ake a ully a omic semi i  R  and le  pi (i E I) be he amily o all p imes . Fo each p i we ha e a homomo phism :  i:  F(R)ab  - =  Z,  and  combining  al]  hese  maps,  we  ha e a homomo phism and hence, by Th .3 .1, F(R)ab ,. ZI . Bu  each  ull  ma ix  maps  o  0  in  almos  al]  ac o s  o  Z I , hence he image lies in he weak di ec powe Z( I ) . Le us w i e D = D(R) o he ee abelian g oup on he p i (w i- en addi i ely) . hen we ha e a homomo phism X : F(R)ab ->D K 1 (U) - D(R) . F om i s cons uc ion he map X is su jec i e, hence so is (3) .  We claim ha i s ke nel is GL(R)/(GL(R) n E(U)) .  Fo any A E GL(R) sa is ies i (A) = 0 o al] i, hence A E ke  X * .  Con e sely,  i  ([ A]  - [ B] ) x* = 0,  hen  ~  = B~ , hence A, B ha e he same a omic ac o s, up o o de and s able associa ion .  Le  A = P 1 - P be a comple e ac o i- za ion and le B be he p oduc (in some o de ) o Q 1 , . . ., Q , whe e Qi i . s s ably associa ed o P i . Replacing A, B by A + I, B + I o sui ably la ge I, we mayassume Q i o be associa ed o P i , say P i = U ¡ Q ¡ V i , whe e U i , V i EGL(R) . Then excep o he o de o he ac o s we can w i e A =  Q 1 . . .Q U 1 . . .U V 1 . . .V = BF, whe e  FEGL(R) .  Hence  A = BF (mod GL(U)')  and so  [Al - [B] _ [ F]EGL(R) .GL(U)' . I ollows ha ke X = GL(R) .GL(U)'/GL(U) - GL(R)/GL(R) n GL(U)' . He e we may eplace GL(U)' by E(U) ; mo eo e , since D is ee abelian,X is spli by D o e i s ke neland we ob- ain Theo em 4 .4 . Ley Rbe a bul .2y a . amíe eemí,Iín w .ízh uní enha2 6íeld ob gnae . íanó U and d .wíeangnaup D(R), hen (4)  K 1 (U)  =  U ab  -  D  xJ [GL(R)/(GL(R)  n  E(U))] . The di iso g oup D inhe i s a pa ial o de ing om R, by w i ing i > 0  whene e  n  is posi i e on  R . Howe e , he o de ing on D is no enough o de ine R wi hin U, as is shown by he ac ha he de e minan o a ma ix o e R is usually a p ope ac ion (¡ .e . has no ep esen a i e in R) . I is also o in e es o compa e Z- alues wi h al a ions (c . [11]) . Clea ly a Z- alue will be a alua ion i and only i Le  A  =  (A 0 ,  A* ,  A .)  be  an  admissible  ma ix  o  p,  hen  an admissible ma ix o p-1 is (Ao+A .,A*,A .), so he condi- ion (5) becomes, a e a sligh ea angemen , ( 6)  (A o + A w ,A * ) > min { (A o ,A * ),  (A,', A * )} . We ecall ha when wo ma ices di e in only one column, say he i s :  A = (A1, . . .,An), B = (B 1 ,A 2 ,- ,A n ),  hen he ma ix ob ained by adding he i s columns and lea ing he o he columns unchanged is called he de enmínan al sum and is w i en A .  B  =  (A 1 +  B 1 ,A 2 - .- A n ) . Wi h .his no a ion we see ha is a alua ion i and only i (7)  (A B) > min { (A), (B)}, whene e he de e minan al sum is de ined (c . [111) . In gene al his  condi ion need no hold,  e .g .  in  k <x,y'>  consi- de he simple Z- alue associa edwi h x . We ha e (xy) = (yx) = 1, bu (xy - yx)-= 0 . Ne e heless he e is a alua ion on he uni e sal ield o ac ions U asso- cia ed wi h x ; o ob ain i we w i e U as a skew unc ion ield K(x ;a), whe e K is he uni e sal ieldo ac ions o  k <y ¡ ¡ i E Z>  and  a  is he shi au omo phism y i I  ' Y¡ +i ( hus  yi  is ealized as  x -i yx i).  0n  K(x ;a) he o de in x is he equi ed alua ion . In e ms o Z- a lues his alua ion is ob ained as he sum o ce ain simple Z- alues, bu his is no a e y e icien way o cons uc ing his alua ion . 5 . The case o ee algeb as To illus a e Th.4.4 we shall conside he case o ee algeb as, whe e i is possible o compu e he second ac- o on he igh o (4) o §4 . We i s p o e a lemma . Lemma  5 . 1 .  Le  k  be a commu a . í e bíeld and  U=k -~ X'~  he uní enaal 6íe .Cd 01 b ae íona  ob  he  linee k-algeb a  k< X > , xhen  E (U) nGL 1 (k)  =  1 . P oo .  Le  A=a+ lEE(U),  whe e  a E - = k ; we ha e o show ha a =1 .  W i e A as a p oduc o elemen a y ma ices o e U and le P be he diagonal sum o al] he denomina o s o he en ies occu ing in hese ma ices . Ou plan will be o ind a k- ield K such ha we can specialize X o alues in K so ha P emains in e ible and A maps o I . Fo each n no di isible by X, he cha ac e is ic o k, we adjoin a p imi i e n h oo o 1, co n say, o k and de ine (1)  K(n) = k(x,y I yx = w n xy) . I is easily seen ha K(n) is hen a skew ield, in ac a di ision algeb a o index n . Le K be an ul ap oduc o he K(n) wi h a non-p incipal ul a il e , anddeno e by x', y', w' he elemen s o K whose componen e a e all x, y, w n  espec i ely,  hen  y'x'  = w'x'y'  and  w' n $ 1  o all n . I ollows ha K is in ini e-dimensional o e i s cen e . We now apply he specializa ion lemma om [41, p .141 . Clea ly he'cen e o K, C say, is in ini e and k <~X :~ is embedded in  KC ,`X'~ so we can specialize  X  o a- lues in K so ha P emains in e ible . I ollows ha o al] bu ini ely many n no di isible by x we ha e a specializa ion om X o K(n) making P in e ible . In each o hese ields K(n) he educed no mmaps each ma ix in E(U) o I, hence a n = 1 o all bu ini ely many n no di isible by X . This s ill lea es in ini ely many alues o n and so is impossible unless a =1 . Fo he ee algeb a  k <X> = R,  e e y in e ible ma ix is a p oduc o elemen a y and diagonal ma ices, i .e . GL(R) = E(R) .k  (by P op .2 .7 .2 o [3],p .95), hence GL(R)nE(U) = E(R) .k'nE(U) = E(R) (k nE(U)) = E(R), by he le  mma . The e o e GL(R)/GL(R)nE(U) = E(R) .k /E(R) - k /k nE(R) k , and so we ind Theo em  5 .2 .  Le .  R = k <X>  be xhe {,Lee  k-alpebna on a se X  and  U  =  k K X ",~  í b  6íe .Cd u Ó  l nac ía nb  and  D (R)  í ó  dí í- ,son gnoup, . . Nen U ab - D(R) xk * This sol es Exe cise 7 .6 .10 o [3] . In many cases i is ue ha  E(U)nGL(R) = E(R),  as in he caseo  k <X> ,  bu by no means always . _Fo a s udy o he gene alcase we e e o Ré ész [121 A he o he ex eme, le K be a skew ield in which e e y non-ze o elemen is a commu a o (c .[6]), le C be i s cen e and conside he ee  K- ing  K C <X>  and i s un¡ e sal ield o ac ions  U = K C -i X :~ .  The ing  R  has a weak algo i hm (c . [3], p .78), hence GL(R) = GL1(R) .E(R), and so GL(R)/GL(R)nE(U) = GL(R) .E(U)/E(U) = GL1(R) .E(U)/E(U) . Now G1 1 (R) =K * nE(U), hence he second ac o on hé igh o (4) in §4 is i ial and so KC <_ X~ ab _ D, whe e D is ee abelian o coun able ank (o o ank IXI i his is la ge ) . 6 . Localiza ion Le R be a semi i and E any se o squa ema i- ces o e R ; i is na u al o .ask unde wha condi ions he lo caliza ion R is again a semi i . This has been answe ed in E 171, whe e i is shown ha R E is a semi i i and only i E  i s  6acxon . comple e,  i . e .  whene e  ABEYE,  hen  he e exis s a ma ix C o e R E such ha (B,C) is in e ible o e R E . We shall show ha 0 when R is ully a omic, hen so is R E and ou aim will be o s udy he ela ion be ween he di iso g oups o R and R E in ha case . An a om A in R and also he associa ed simple Z- alue is called E-ínnele an - i A becomes a uni in R~ . and E-nele an o he wise . Theo em 6 .1 . Le R be a jullu a omíc semí6íx and le . E be a 6acxon comple e he og ma íce~s o en R, hen R E íó aga,Ln a ~ull(! a omíc zemílín, and e eny a om o en R eí hen becomes a uní on nema .Ln~s an a om o en R E . P oo . We begin by p o ing he las pa . Le A be an a om o e  R  and  suppose  ha  o e  R E we  na e  A  =  B 1 B 2 ,  whe e he B i a e non-uni s . Then by C ame 's ule, U i (B i + I)V i = C i (i  =  1,2),  whe e  C i is  a  ma ix  o e  R  and  U i ,  V i E Hence Le be he simple Z- alue de ined by A, ake comple e ac o iza ions o C i , C 2 o e R and le w 1 , w 2 be he Z- alues co esponding o C 1 , C 2 bu coun ingonly E- ele- an a oms . Them by  (1), =w 1 + w 2 .  Bu  w i (C i ) : l and so 2  s  w 1 (C 1 )  +  w 2 (C 2 )  =  (A)  =  1 a con adic ion, an'd his shows ha A is an a om o a uni o e R E . Now le P be any ull ma ix o e R E and w i e (2)  U(P + I)V = A, whe e AEF(R), U, VEGL(R E ) . We can w i e A as a p oduc o a oms say, o e R ; each will be ei he an a om o a uni o e R E , hence P can be w i en as a p oduc o a mos a oms o e R E and his shows R E o be ully a omic . The ac ha R E is ully a omic may also be p o ed as ollows : Deno e by w he sum o all E- ele an simple Z- alues on R, henw is a Z- alue on RE and w(A) = 0 o AEF(R E ) only i A is in e ible (by C ame 's ule), hence he c i e ion o P op .4 .3 is sa is ied . By P op .6 .1 we can de ine he di iso g oups o bo h R and R E ; o desc ibe he mapping be ween hemwe need P oposi ion 6 .2 . Le R be a jul' q a omíe eemíbí and E a jae on comple e he oj ma íceh o en R, ao ha R E agaín jully a omíe . Then (i) any wo a omz o en R ha a eno s ablyaasocía ed o en R a e no e ably azsocía ed 0k)e  R £ ,  unle,sn  bo . h .  beeome un . ó,  (i i )  e e y ma íx  P Theo em  6 .6 .  Lex  R =k< X >  be he linee al'gebna on an in6 i- ní e b e  X,  andle  Xo be a  sub,6 ex o6  X  wí h an ínjíníze complemen .  Deno e b y  E =E (X o ) hehez o 6 all 6ull ma níceb o al?l y copníme o  Xo ,  hen  R E íb a símple pnínc .Lpal ideal domaín . In pa icula , aking . X o o consis o a single elemen , we ob ain a simple PID wi h a single a om, bu no a local ing . Re e ences 1 . H .Bass, Algeb aic K- heo y, Benjamin (New Yo k 1968) 2 . G.M .Be gman and W .Dicks, Uni e sal de i a ions and uni e - sal ing cons uc ions, Paci . J . Ma h .79(1978) 293-337 3 . P.M .Cohn, F ee ings and hei ela ions, .LMS monog aphs No .2 Academic P ess (London, New Yo k 1971) 4 . P.M .Cohn, Skew ield cons uc ions, LMS Lec u eNo es No .27 Camb idge Uni e si y P ess (Camb idge 1977) 5 . P.M . Cohn, The uni e sal ield o ac ions o a semi i I . Nume a o s and denomina o s, P oc . London Ma h . Soc . (3)44(1982) 1-32 . 6 . P.M .Cohn, De e minan s on ee ields, o appea 7 . P .M .Cohn and W .Dicks, Localiza ion in semi i s II . J . Lon- don Ma h . Soc . (2) 13 (1976) 411-418 8 . P .M .Cohn and A .H .Scho ield, On he law o nulli y Ma h . P oc . Camb idge Phil .Soc . 9 . W .Dicks and E .D .Son ag, Syl es e domains, J . Pú e appl . Algeb a 13 (1978) 243-275 10 . J . Dieudonné, les dé e minan s su un co ps non-commu a i Bull . Soc . Ma h . F ance 71 (1943) 27-45 11 . M .Mahda i-Heza ehi, Ma ix alua ions on ings and hei associa ed Skew ields, Resul a e d . Ma h . 12 . G . Ré ész, On he abelianized g oup o uni e sal ields o ac ions, o appea 13 . A .R .Richa dson,Simul aneous linea equa ions o e a di i- sion algeb a, P oc . London Ma h . Soc .(2)18 (1928) 395-420