Additive Groups of Rings

Research Notes in Mathematics

83

S Feigeistock

Additive groups of rings

Pitman Advanced Publishing Program BOSTON LONDON MELBOURNE

S Feigelstock Bar-II an University

Additive groups of rings

Pitman Advanced Publishing Program BOSTON-LONDON - MELBOURNE

PITMAN BOOKS LIMITED 128 Long Long Acre, Acre, London WC2E 9AN 128 PITMAN PUBLISHING INC 1020 Plain Plain Street, Street, Marshfleld, 1020 Marshfield, Massachusetts Massachusetts 02050 02050

Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto Toronto Feigelstock 1983 © SS Feigelstock 1983

First published published 1983 1983 AMS Subject Subject Classifications: Classifications: (main) (main) 20K99 20K99 (subsidiary) 16, 16, 17 17 Library of of Congress Cataloging in Publication Data Feigelstock, S. Additive groups groups of of rings. rings. (Research notes (Research nQtes in in mathematics; mathematics; 83) 83) Bibliography: p. Includes index. 1. Abelian groups. I. Title. II. Series. Series. 512' .2 82-22412 QA171.F34 1983 512'.2 ISBN 0-273-08591-3

British Library Library Cataloguing Cataloguing in Publication Data Feigelstock, Feigelstock, S. S. Additive groups of of rings.—(Research rings.-(Research notes in mathematics; 83) Group rings 1. Group rings I. Title II. Series 512'.4 QA171 ISBN 0-273-08591-3 ISBN 0-273-08591-3 All rights reserved. No No part partof ofthis this publication publication may may be reproduced, reproduced, stored stored in in aa retrieval retrieval system, system, or or transmitted, transmitted,ininany anyform form or orby by any any means, electronic, mechanical, mechanical, photocopying, photocopying, recording recording and/or otherwise, of the the publishers. publishers. otherwise, without without the theprior priorwritten writtenpermission permission of This book book may may not not be be lent, lent,resold, resold,hired hiredout outororotherwise otherwisedisposed disposed of other than that of by way of trade in in any form of of binding or cover other in which it is published, without the the prior prior consent consent of of the the publishers. publishers.

Reproduced and Reproduced and printed printed by by photolithography photolithography in Great Britain Britain by by Biddies Ltd, Guildford

Preface

R.A. Beaumont Beaumont began investigatingthe the additive additive groups In 1948 1948 R.A. [5][5]began investigating groups of of andand H.S. Zuckerman rings. Shortly thereafter thereafterL.L.Redei, Redei,1.T.Szele Szele[54], [541. H.S. Zuckerman [11] joined in in these these investigations, investigations, totobe be followed followed by by L. L. Fuchs Fuchs [37], [11] joined [371, [38]. Since then progresshas hasbeen beenmade made this branch branchof of abelian group Since then much much progress in inthis group Manyofof the the results appear theory. Many appear in in Chapter Chapter 17 17 of of L.L.Fuchs' Fuchs 1well-known well-known attempt has has been beenmade made here offer an an easily easily accessible book [36]. [36]. An book An attempt here totooffer accessible account accountof of much muchofofthe thework workdone doneononthetheadditive additivegroups groupsofofrings, rings, with with special results not not covered coveredin in [36]. special emphasis emphasis ononresults [36]. This short shortmonograph monograph is far from being beingcomprehensive, comprehensive, and and many many important papers papers have have not been been far from consideredhere hereatatall. all. considered

Although this this work of Although work is is meant meant to to be be self selfcontained, contained.some some knowledge knowledge of abelian group theory is assumed. The prospective reader is especially group theory is assumed. The prospective reader is urged to read 17 of of Fuchs' Chapter 17 Fuchs• book book [36]. urged to read Chapter

gratitude to Strasser for I wish wish to to express express my my gratitude to Professor Professor Elvira Elvira Rapaport Rapaport Strasser introducing to this subject, me to subject, for forher heradvice adviceand andguidance guidance during during the the past past introducing me fifteen monograph. fifteen years, years, and and for forsuggesting suggesting that that I Iwrite writethis this monograph. I also thank thank Professor L. Professor L. Fuchs Fuchs for forhis ~isvaluable valuablecomments comments and and suggestions. suggestions.

Contents

1.

Preliminaries: §1: §2: §3: §4:

2.

Definitions Definitionsand and structure structuretheorems theorems Mult Mult GG Type Type Examples

1: The The ninilstufe 1s tufe of ofaagroup group Nilpotence without conditions, and Nilpotence without boundedness boundedness conditions, and generalized nilpotence generalized nilpoten~e

§2:

25 25 31 31

Other Ring Properties: Other Ring

Semisimple,prime, prime,semiprime, semiprime,simple simple ring,division division ring, ring, §§1: 1: Semisimple, field, field, radical radical ring ring §2: Principal ideal idealand and Noetherian Noetherian rings rings §3: Descending chainconditions conditionsfor for ideals Descending chain §4: Subdirectly Subdirectly irreducible rings rings Local rings §5: Local §6: Rings Ringswith withtrivial trivial left leftannihilator, annihilator,subrings subrings ofofalgebraic algebraic numberfields, fields, and number and semisimple semisimple rings continued continued §]. §7. E-rings E—rings and 1—rings T-rings 5.

10 16

groups of Nilpotent Nilpotentand and Generalized Generalized Nilpotent Rings: Rings: Additive groups §§1:

4.

5 6

Nil1and Ni and Quasi—Nil Quasi -Ni'l Groups: Groups :

§1: Nil groups groups §2: Quasi-nil groups groups 3.

1

4

36 43 50 61 65

68

76

Torsion Free Free Rings: Rings:

§1: §2: §3: §4:

Notation, definitions, definitions, and Notation, and preliminary preliminary results results The Beaumont-Pierce Beaumont—Piercedecomposition decomposition theorem The Torsion Torsion free rings rings with withsemisimple semisimple algebra algebra type type Applications

88 90

98 103

Notation

G[nJ

aa group. group. the the order order of xx e G. G. the torsion part part of G. the torsion G. the p—primary of G, pp a prime. p-primary component component of prime. {nxjx E G}, nn aa fixed fixed integer. {nxlx {x E Gjnx a fixed integer. integer. n a GInx == 0}, n

Ga Ga v(G) N(G) r(G) (S) lS) h(x) hlx) hp{x)

subgroup of of 6, the cx—th a-th Ulm Ulm subgroup G, a an an ordinal. the nilstufe nilstufe of G. the the strong strong nilstufe nilstufe of G. the G. the rank the rank of G. the subgroup of GG generalized by c 6. the subgroup of by S =G. the height height of xx E G. the the p-component h(x). p-component of hlx).

t(x)

the type the type of xx E G. G. for G the the type type of G G G homogeneous. 01 =the = the type type set set of G. {t(x)jx G. {t(x)Ix E G, xx ~ 0} t(x) (t(G)). the p-component of t(x) (tlG)).

G

Gt nG

t('G) t(G) 1(G) T(G) tplx) (tp(G))

[(k11k2,... [(k 1 ,k 2 , ••• ,kn, •.• )J

the the type type containing containing the the height vector vector (k1, (k 1 , k2 , ••• ,kn•···>· Additional notation notationconcerning concerning type which be used used may 2, type whichwill will be maybebefound foundinin[36, [36,vol. vol. 2, pp 108- 111 ] • pp.• 108-111].

Z(n)

cyclic cyclic group group of of order order n.

Z(pCD) Z(p0°)

p—Prüfer p-Prufer group.

End(G) End( G) £nd(G)

the group group of ofendomorphisms endomorphisms of of G. the group group ofofquasi—endomorphisms quasi-endomorphisms of of G. quasi-isomorphism. quasi-equality a a ring.

Q. :!:

R R+

R

group of R. the additive group the

I i~RR

Z

Z

Q

Q

ZP w

lSI SI

"9I isisan an ideal idea 1 in in R" R'.• the ideal ideal in R the generated R generated by S cE RR the ring ring of nxn matrices with components components in R, R, nn aa the matrices with 1nteger. · positive integer. the ring ring of the of integers. integers. the field the field ofofrational rationalnumbers. numbers. the ring of integers.• the ofp—adic p-adi c integers

the first first infinite the infiniteordinal ordinal the cardinality cardinality of aa set s. S. the

Additional symbols whichappear appearininonly onlyone onesection, section, will will be Additional symbols which be defined in the the section section they they appear appear in. defined

11 Preliminaries

§I. §1.

DEFINITIONS DEFINITIONSAND ANDSTRUCTURE STRUCTURETHEOREr'lS THEOREr·1S

in this areabelian, abelian, with with addition the All groups groups considered considered in this monograph monograph are the group operation. group

If GG is is p-divisible, prime,if if G == pG. p-divisible, pp aa prime, pG. If G A group G Definition: A then is said to be divisible. p-divisible p-divisible for forevery everyprime prime p, G said be G It is divisible only if if G It isisreadily readilyseen seen that that GG is divisible if ifand and only G ==nG nG for every every positive integer ri. integer n. prime, ifif group G is p-pure p-pure in G, pp aa prime, A subgroup HH of aa group G Definition: A pkG = pkH for every If HH is p-pure H every positive integer k. If p-pure in GG for positive integer H n pkG = pkH every prime p, then HH is said said to to be be pure pure in in G. pure in in every prime G. Clearly HH is pure if and onlyiiff H n nG if and only nG == nH nH for every every integer integer n. G G G I} of aa group group G is said said totobebeindependent independent A subset {a; li E I) I, kk an arbitrary positive i1' ... ,ik E I, an arbitrary positive integer, integer, and and for .1 n1,. Z, n1a1 = ... implies that 0 = 0. n1, ... ,nk E n1a11 + ... + nkaik == 0 implies n1a11 = .•. = nkaik = =

Definition:

if for if for distinct distinct

By there exists By Zorn's lemma, lemma, there exists aamaximal maximal independent independent set set M in GG containM ing of infinite infiniteand andprime primepower power orders. orders. The The rank G, ing only only elements elements of rank of G, be maximal amongstthe the independent independentsubsets subsets of of GG r(G) == IMI. M0 be maximal amongst r(G) Ml. Let M0 containing only only torsion 1r~0 1 is called called the the torsion torsion containing torsion free free elements. elements. Then 1M01 free free rank rank of G, denoted r0(G). r 0 (G). Let t•lp be be maximal maximal amongst amongst the independent subsets subsets of of G independent containing elements of power of p, containing only elements of order aa power G pp a prime. prime. Then lt~pl is called called the the p-rank p-rank of G, denoted rptG).

r(G), r0(G), r(G), r 0(G),

and

rptG)

are well defined are defined [36, [36,Theorem Theorem 16.3].

G Definition: A A subgroup subgroup BB of of aa group group G is aa p-basic p-basicsubgroup, subgroup, pp aa is prime if: prime if: 1) B is aa direct direct sum sum of infinite infinitecyclic cyclicgroups, groups,and and cyclic cyclicp-groups. p-groups. B 1) is p—pure p-pure in G, and 2) B B 3) G/B is p-divisible. p-divisible. G/B

If If

is called G is aa p-group, p-group,then thena ap—basic p-basic subgroup subgroup of G called aa basic basic . G G subgroup. and all and all

G possesses subgroup [36, Theorem Every group Every group G possesses aa p—basic p-basic subgroup Theorem 32.31, 32.3], p-basic subgroups G subgroups of G are isomorphic [36, [36,Theorem Theorem 35.21. 35.2]. are isomorphic

Ulm subgroup Ga Ga of For every every ordinal a, define the the a-th Ulm Definition: For 1 For every every ordinal a, the a+l G1 = a+lth as follows: G G as n th Ulm G = ii nG. For n<w

Ga+l=(Ga)l. Ga+l = (Ga)l. 8 G GY. n G6 == n

subgroup of G, G, subgroup subgroup subgroup of of G, G,

For 8 aa limit ordinal, the the 6th ath limit ordinal,

Ulm Ulm

y ~ (~ 1 ••••• ~n ... ) many if kn == ~n for all allbut butfinitely finitely manysubscripts subscripts n, and kn = ~n = whenever k = = ~ or The equivalence relation. ~ == ~. is an an equivalence equivalence relation. The n n . class of a height sequence is called a type, will be of a height sequence (k 1 , •.• ,kn•···) called a type, and and will be [(k1,. ..,k,...)]. The type of xx E G, t(x) == [h(x)1. denoted [(kp···•kn•··:>L The type [h(x).J.

if

The type type set set of {t(xfl xx E G, xxI of aa group group G, 1(G) T(G) == {t(x)l 0}. Definition: The $ O}.

If If

= 1, then G G is group with with type IT{G)I is said saidtotobebea homogeneous a homogeneous group type IT(G)I = 1(G). singleton belonging belonging to T(G). t(G) t(G) ==the the singleton

if kn = 00 Definition: A A height heightsequence sequence (k1' ... ,kn, ... ) is idempotent, idempotent, if if , every positive integer integer n. A type it is idempotent idempotent if or ~ for every possesses an an idempotent idempotent height height sequence. sequence. possesses 1

group G G implies that A·group is indecomposable indecomposable ifif G = H HG &KK implies Definition: A H= = 00 or K == 0.

is rigid, rigid, ifif End(G) G Definition: A torsion free free group group G is + a subgroup subgroup of Q .

2

is isomorphic isomorphic to

Definition: Let G and H G Definition: and H be subgroups torsion free free group group be subgroupsofofa adivisible divisible torsion D, 1.1.6, this is not D, in in lieu lieu of of Proposition Proposition 1.1 .6, this not a a very very severe severe restriction on restriction on torsion free free groups groups G, G, H. H. If If there there exists exists aa positive positive integer integer nn such such that G and H nG ~ H, H, and nH ~ G, then G are said to to be be quasi-equal, quasi-equal, G H G ~H. H. are If two groups and A B If two groups A B are isomorphic to quasi-equal quasi-equal groups, groups, G, HH are isomorphic respectively, then A and BB are then A are said to to be be quasi-isomorphic, quasi-isomorphic, A~ B. A

Definition: A G A group group G satisfying, G @K K implies HH = 0 or K = 0, G & H HG 0 is said said totobebestrongly stronglyindecomposable. indecomposable. H of aa group group G is essential essential in Definition: A A subgroup H G for for every everysubgroup subgroup K G, KK ; 0. K (a. 11131 1 1

Hom(GØG, .p E Hom( 11> ~G, G) G)

are

G be any any element element in G of order order to be

aa.. . )) .

1 2J 2

min(j 1 ,j min(j1 ,j2) 2) less G or equal equal . pp Therefore on G less than than or Thereforethe thering ring multiplication multiplication on . are determined defining the the product product of basis with the basis elements elements with the sole sole are determined bybydefining restriction labl lbl) for every every pair pairofofbasis basiselements elements a,b. IabI ~< min(lal, min(IaI, Ibi) In all alIofofthe theabove aboveexamples, examples, the the ring ring multiplications multiplicationswhich whichwere were defined defined are are associative associative (commutative) (commutative) ifif the the product product of of basis basis elements elements is associative associative (commutati ye). (commutative). 1.4.5. 1.4.5.

G aa torsion group. group. G

G , Proposition 1.1.1. It It isisreadily ~ GP, Proposition readilyseen seen that that (4? ppaprime a prime with R+ = GG Hom(G ~ G i.e., every R , Gp) Hom(G~, G)~ n. Hom(Gp Gp, every ring ring R G ) i.e., p p ppaprime a pr1me R with R; = =G Therefore the is aa ring ring direct directsum sum R == f.il RP Gp.. Therefore the p p p a prime p paprime investigation of ofthe thering ringstructures structureswhich whichmay may be be defined defined on on aa torsion torsion group group G, reduces p-primary case, case, pp aa prime. prime. reduces to to the p-primary

G == G

1.4.6.

G G aa p-primary p-primary group, group,

p a prime. prime.

B be aa basic subgroup G be subgroup of G with basis basis {a; I i E I}. B I). Then As in G) ~ = B~, [36, Theorem Theorem 61.1] and and so so Hom(~. G)~ Hom(~. G). As example 1.4.4, G 1.4.4,the thehomomorphisms homomorphisms 11>p from ~ into G are determined determined by by example G with the sole restriction restriction that ,p(a1 A aj) defining ~~>(a; to be be an an element element in G with the sole that for all I. Therefore, Therefore, the the ring l~~> r., j

I

00

G1

G

-+ G1 -+ G -+

G/ G1 .... 0 G/G1 .+ 0

is exact. torsion free group exact. Every Every torsion group is flat, flat,and andsosothe thesequence: sequence: 0.4 G1 ® G2

is exact. exact.

(GIG1) ® G2 -40

This implies the This the exactness exactness of of the thesequence: sequence:

0 -. Hom[(G/G1)

G2,

G] -

G) -. Hom(G1

G2, G)

44.4]. For every [36, Theorem Theorem 44.4]. every 00 ~ xx € G, t(G1 t( G1 ® @ G2) G2 ) i t(x), and and so so Hom(G G2 , G) G) = 0 by by Property Property 1.3.1.4. Hence Hom(G1 1 (X)®G2, T3 Hom[(G/G1) ®.G2, Hom(G2, End(G)). Let T Hom[(G/G G] ~ Hom(~ 2 • G)~ Hom(G maximal be a maximal G) 2 , G] 2 , End{G)). 1 ) @.G 3 be T3 f Ti, element in 1(G), t(x3) = 0 t x3 G, with t(x 3 ) = r3, element T(G), T 3 ~ 'i' iI = 1,2. Let 0 ~ x3 € T3 , and and put x = the the coset coset xx++G1 G1 for every every x € G. Clearly t(x 3) = = t(x 2 ), > t(x3). = t(x t(x3), and t(x 3) ~ t(x 3). If If t(x 3) = 3 ), then

12

13 = = t(x J ~ T t(x 3) == t(x 3) = t(x 2 ) = •2 . By = T'2' By the the maximality maximality of •'2' 2 > t(x2) 3 = t(x3) 2 • •'3 3= 2, = by Property 1.3.1.4) and ) aa contradiction. Hence Hom[ ( G/G (~ G ,G] = 0 by Property 1.3. 1.4) and Hom[(G/G1) = 1 (aG2,G] 2 (A) Hom(G2, End(G)) == 0. A similar so (A) Hom(G 2 , End(G)) argument shows shows that similar argument (B) Hom(G1, End(G)) == 0. (B) Hom(G1 , End(G)) 0. Let xx e: G, and Hom(G, End(G)). There and let let lPpEe: Hom(G, exists a positive such that exists positive integer integer n, and and integers nn1,n2 1 ,n 2 such = n n1x1 + n2x2. Hence by (A) nx flX = x n x . nlP(x) = n lP(x + n lP(x = (A) and and (B). ) ) nq(x) = n1q(x1) + n2(p(x2) 1 1 2 2 1 1 2 2 = 0 by However End(G) is torsion torsion free, free, and and so so lP(x) = = 0, i.e., i.e., Hom(G,End(G)) Hom(G,End(G)) = = 0. G is nil, a contradiction. By 1.2.1, G Byl.2.1, isnil,acontradiction.

Thenext nextresult result will will show The show that that stating statingthe theexistence existenceofofa aunique unique minimal minimal element in in the is superfluous. element the previous previous lemma lemma is superfluous. Lemma 2.1.6: Lemma 2.1.6:

then

1(G) T(G)

Let G G be be aa rank rank two two torsion torsion free free group. group. possesses possesses aa unique unique minimal minimal element. element.

If GG is not If not nil

(G,.) be R2 i 0. Suppose Proof: Let R == (G,·) be aa ring ring satisfying R2 with t(x) i t(y), and x·y x,y e: R R Clearly either t(xy) >> t(x), t(x), 0. Clearly x.y if 0. or t(xy) >> t(y). t(y). Suppose Then Suppose that that t(xy) >> t(x). t(x). Then every every element element in GG is linear combination over the aa linear combination of xx and xy over the rationals rationals and and so so t(x) is the the unique element in GG by Property 1.3.1.2). 1.3.1.2). It It may minimal element by Property may therefore therefore be be unique minimal assumed that (A) R assumed that X·Y == 00 for all x,y e: R with t(x) i t(y). (A) x.y such that Suppose that that IT(G)I There exist exist y,z e: G G Suppose 3. Let xx e: G. There such T(G)I >~ 3. t(x), t(y), are distinct distinct types. t(x), t(y), t(z) t(z) are types. x == ay ay ++ bz, bz, with a,b rational 2 bx.z = = 0~ by numbers. Hence x2 x = ~X·Y + bx-z by (A). Let 00 i x, 0 0 if yy e: R, with t(x) t(x) = t(y). If are independent, = t(y). If x and y are independent, then then every every element element in G over the the rationals, linearcombination combination of x, and y over rationals, and and so so t(x) G is aa linear is the 1(G) by 1.3.1.2). Therefore Thereforeit it the unique unique minimal minimal element element in T(G) by Property Property 1.3.1.2). maybebeassumed assumed thatthere thereexist exist non-zero nx ==my. may that non-zero integers n,m such such that nx my. This yields yields that nmx.y G is torsion This nmx-y == n2x2 n2x2 = 0. Since G torsion free free x.y X·Y == 0, 0, or (B) for all all x,y e: RR with t(x) t(x) == t(y). Clearly (A) x-y ==00 for (A) and and (B) (B) (B) x.y 2 (A) implies implies that a yield that that R a contradiction. Therefore Therefore (A) R2 = 0, and choose 0 x1 2. • IT(G)I ~ 2. Let T(G) = £• T }, and choose 0 i X; e: G, IT(G)I < 1 2 = If t(x) the fact that t(x;) == T;• i = 1,2. Put x == x1 x1 ++ xx2. t(x} = Tl then the that 2 . If = yields that T'22 == t(x2) x2 ==xxx1 yields t(x 2 ) ~> '1 Tl by by Property Property 1.3.1.2). Similarly x2 — x1 t(x) == T2 case possesses aa unique t(x) T 2 implies either case T(G) possesses implies that • 1 ~ T 2 • In either '1 >- '2• minimal minimal element. element. i

An immediate immediateconsequence consequence last two two lemmas lemmas thefollowing: following: An of of thethelast is isthe

13

Theorem2.2.1.7: Theorem 1.7: Let GG be be aa rank rank two two torsion free free group. group. then G G is nil. is ni 1.

If If

IT(G) I > T(G)I >33

rank two two torsion free free groups groups For every every positive positive integer n, there For there exist rank G with T(G) = n, n, [9], and and there there exist exist rank rank two two torsion torsi on free free groups groups with with G infinite type set set [36, [36, vol. vol. 2,2,p.p.112, 112, Ex. Ex. 11]. 11 ). Therefore infinite type Therefore Theorem 2.1.7 is Theorem 2.1.7 not aa statement statement concerning concerning the empty empty set.

Clearly the the classification of to Clearly ofnil nilrank ranktwo twotorsion torsionfree freegroups groups reduces reduces to the case IT(G) I > 22 are are nil.

14

H Lemma2.1.9: 2.1.9: Let GG and H &H. Lemma be torsion free free groups groups with with G &H. be is (associative) nil if if and only if if H is (associative} nil and only is (associative) (associative} nil. H nil.

Then

G G

It may Proposition1.1.6, 1.1.6, that G H Proof: It may be be assumed, assumed, Proposition and H are subgroups subgroups G proof: and that that G H. Suppose of aa group group G, G, and Suppose that not (associative) (associative} nil. G ~H. nil. that HH is not 2 = (H,·J (H,.) be an (associative} (associative) ring with There exists exists aa Let R = be an with R ~ 0. There such that nG ,g22 € GG positive integer integer nn such nG ~ H, nH ~G. For g1 g1 ,g I-I. and nH G. define g1*g2 g1•g 2 = (ng1) (ng 1} •. (ng readilyseen seen that that S S= = (G,*) (G,•} is an an (ng2). It isisreadily 2}. It H (associative) ring. Let h1,h2 (associative) h1 ,h 2 E H such such that h1 -h 2 ~ 0. Then 4 2 nil. (nh = n4h1h2 n h1h2 ~t 0. Hence s ~ 0, and GG is not not (associative) (associative} nil. (nh1)*(nh2) 1 )•(nh 2} = Similarly if if G G nil, then is H. Similarly is not not (associative) (associative) nil, then neither neither is Theorem 2.1.10: Let G be aa rank two torsion torsion free group, Theorem 2.1.10: be rank two group, with IT(Gfl jT(G} I == 3. 3. G G nil iiff and only if if G.; G G2, G. a Then G is not not (associative) (associative} nil and only G1 then either either ~nd(G} 6 is quasi—isomorphic group with IT(G)I > 2, then = Q, or G is quasi-isomorphic group End(G) = Q, If to the the direct directsum sumofoftwo tworank rankone onegroups groups of ofincomparable incomparable types. types. If G End(G} Q, then G is aa rigid rigidgroup, group, and and so so G is rail r.il [36, comment End(G) = [36, comment G = Q, If G6 is quasi-isomorphic preceding Proposition 121.2). If quasi-isomorphic to the the direct directsum sum preceding Proposition 121.2]. groupsofof incomparable incomparable types, then of rank rank one one groups types, then it itisis readily seen, seen, [9, remark preceding Theorem Theorem8.5), 8.5], that IT(G}I remark preceding IT(G)I = 3.

15

§2.

Quasi—nil Quasi-nil Groups: Groups:

Q+ precisely two non-isomorphicring ring structures, structures, the The admits precisely two non-isomorphic the group Q+ admits The group Szele [68, [68, Satz 1] classified and Q. Szele Satz 1] classified the thegroups groups allowing allowing zeroring, and Q. He conjectured precisely two two non-isomorphic non-isomorphic ring every ring structures. structures. He conjectured that every the additive group infinitely many non-nil torsion free free group group G G ~ Q is the additive group of infinitely many non-nil torsion Q This was shownnot nottoto be be true true by non-isomorphic was shown by L. L. Fuchs, Fuchs, non—isomorphicassociative associative rings. rings. This Borho, [12], showed that Szele's Szele's conjecture was not far [38]. Borho, showed that was not far from from being being true, see see Theorem Theorem 2.2.4 2.2.5. true, 2.2.4 and and Corollary Corollary 2.2.5.

Theinvestigation investigation of of nil The nil groups, groups, and and additive groups groups of precisely precisely two two non-isomorphic rings, by Szele, Szele, was was generalized generalized by by Fuchs Fuchs as non—isomorphic rings,initiated initiated by as follows:

if there group G is (associative) (associative) quasi—nil quasi-nil if there are are only A group Definition: A G finitely many non—isomorphic (associative) rings R, with finitely many non-isomorphic (associative) R, R+ == G. G. Fuchs classified the the torsion (associative) Fuchs classified (associative) quasi-nil quasi-nilgroups groups completely, completely, and made madeconsiderable considerableprogress progresstowards towards classifyingthe the torsion torsion free and classifying free and and description mixed (associative) [38]. A complete complete description mixed (associative)quasi-nil quasi-nil groups, groups, [35], [35], [38]. of the the torsion torsionfree free(associative) (associative)quasi—nil quasi-nilgroups groups which which are are not nil nilwas was obtained by Borho, [11]. This obtained by W. W. Borho, This in conjunction conjunction with the the results results ofofFuchs Fuchs yields aa classification classificationofofthe the(associative) (associative)quasi—nil quasi-nilgroups groups which which are are not not nil. Themain mainresults resultsof of this section Borho, [12], [12], [35], The section are are due due to Fuchs, Fuchs, and and Borho, [35], [38]. Thefollowing following technical technical lemma generally useful useful in studying The lemma isis generally studying the the of rings: additive groups groups of Lemma 2.2.1: Lemma 2.2.1:

Let

R be aa ring with R+ ==G. R be G.

Then

G annihilates G11

Gt. Gt.

1 Proof: Let 00 ~ xx € Gt• G', let lxi , and and let lxl = = n. Since yy € nG, Gt, y € G y= = nz, nz, zz € G. Therefore xy xy = x(nz) x(nz) = (nx)z (nx)z = = 0. 0. Similarly yx yx = 0.

Theorem2.2,2: 2.2.2: Theorem

1) 2) 3) 3)

G is quasi—nil. quasi-nil. G G is associative quasi—nil. quasi-nil. G == B ~ 0, B B a G a finite finitegroup, group,

The The following following are are equivalent:

G

Proof: 16

G be aa torsion group. Let G be group.

Clearly

1) 1) • 2):

~ D

divisible torsion aa divisible torsion group. group.

Suppose that G G 3): Supposethat H reduced, 2)•3): isassociativequasi-nil. H is associative quasi-nil. G=H(i)D, G= 2) divisible. Hp is D and D divisible. is aadirect directsummand summand of GG for every every prime prime p, Proposition1.1.1, 1.1.1,i.e., i.e., G Proposition G == Hp (f) KP. For every prime prime pp with Hp fI 0, 0, be an an associative associati ye non—zeroring, let Sp be non-zeroring, with s+p = Theorem 2.1. 1, and Theorem 2.1.1, = H , p be the zeroring on = let TP be the ring direct on KP. The The ring directsum sum Rp = SP@ TP satisfies R; = G, G, R~ I 0, and Rp + Rq Rq for distinct distinctprimes primes p,q. Hence H = 0 for all all but butfinitely finitelymany many primes. It It therefore therefore suffices sufficestotoshow show p . H be aa basic subgroup that Hp is finite finitefor forevery everyprime prime p. Let BBp be subgroup of n.

with basis Every set set of basis {a.li € 1}. = p 1 , i € I. Every U. Put la-1 p 1 1 = associative products with i,j € I,I, products a .. a. la.·a-1 > 1. 1. Choose 0 ~t vv E H. Every element x E G is of the form x =x=b+d+ru+sv+w, b + d + ru + sv + w, bb€B, E B, dED, dE D, Everyelement xEG isoftheform + r1u + w1, = = 1,2 be r, ss E Q, w H. Let i and WE X; = bi + d; + r u + siv + W;, = 1,2 Q, 1 elements of of G elements theabove above form. form. The G written ininthe The products an associative associative ring ring structure Rn on x1 -x 2 = (r1r2 (r1r 2 ++ s 1s 2n)a 1 induce induce an G. R

=

~

and let pwG. Write xx in the the above above canonical canonical form, form, and let xx E pTh. n=l ru ++ sv sv ++ w. w. For integer n, x == b ++ dd ++ ru For every every positive positive integer 2n)a = 2 2 ( p-n x) 2 == p-2n (r2 ÷+ ss27r)a1 (r2 ++ ss2lT)a2n+l. the set set of n)a 2n+l" Let n(p wG) be be the 1 = (r Put

pWG pwG =

PnG, n pnG,

n= 1

{r22 ++ s 2nl obtained integers {r above for each xx E PWG, pwG, and and every p-adic integers obtained above for each positive integer integer n. Then n(pwG) is aa countable countable set setofofendomorphisms endomorphisms of If RRn ~ Rp for two Z(p~) containing the the identity identityautomorphism. automorphism. If two p-adic p-adic then the elements of are the the elements elements of lr(pwG) n(pwG) integers n,p then the elements of ~(pwG) are multiplied by aafixed fixedautomorphism automorphism of Z(p~), i.e., by aa p-adic p-adic unit multiplied by i.e., by However the automorphism must must belong be However the identity identity autornorphism belong to (pwG), and and so a must be chosenfrom froma acountable countableset setofofp-adic p-adic units. units. Therefore chosen the isomorphism isomorphism class Therefore the of RRn is countable. countable. The the p—adic p-adic integers The non-denumerability non-denumerabilityof of the integers yields that there many R there are are infinitely infinitely manynon-isomorphic non-isomorphic associative associative rings rings R with n + r(H) == 1. R+n = G, G, aa contradiction. Hence r(H) 1. .

22

be aa prime Let pp be prime for for which which DDp ~ 0, and and let let 00 ~ dd E Dp • Let D, 0o ~hE d. ++ r.h, b. E B, d1. ED, x1 =b.+ = + d1 / h H. For xx1,x2 1 ,x 2 E G, put x. 1 1 1 1 1 an associative associative ring r; E Q, 1 ,2. The The products xx1x2 r 1r 2d induce induce an Q, i = 1,2. 1x2 = r1r2d structure Rp on on G. G. For primes primes pp ~t q, q, RP Rq. Therefore Dp = 0 for many all but but finitely finitely manyprimes primes p.

+

Conversely, suppose that GG is of Mult GG ""Hom{~. G) Conversely, suppose that of form form 1). t4ult G) X ... Hom{G~. B)(+)Hom{H~. H), since Hom(X,Y) aa torsion torsion group, group, Hom(X,Y) = 0 for X V a If GG is quasi-nil and Y a torsion free free group. group. If quasi-nil then then so so is H. Therefore and only finitely many Hom{H~. H) ""'r~ult HH determines determines only finitely manynon—isomorphic non-isomorphic ring structures on on G. Hom{G~. B) B) ""'Hom(B~B. B) t+) Hom{H~, B) (f) Hom(B~, B) (t) Hom{Hfti, B). The groups B, B·Sl!B, B~ are finite,and andsosothe thefirst first three summandsabove above are are are finite, three surmiands Proposition 1.3.3, is finite, finite. finite. Since r{~H} = 1, Proposition 1.3.3, and and BB is is finite. Hom{H·~. B) is

finite, HH an G == B t:t) H, B For For the the associative case case consider consider G B finite, associative quasi-nil quasi-nil torsion associative torsion free free group. group. Let lSI = m. m. The The ring by products products of elements multiplications multiplications on on G 6 are are determined determined by elements in B, for which thereare areclearly clearly only only finitely finitely many which there many possibilities, possibilities,products productsofofelements elements in B B and products products of of elements elements in H. Since mH mH with elements elements of H, and annihilates B B in every every ring ring on on G, the the products products of elements elements in BB with elements of of H H are determined by the the products elements are determined by products of elements elements of BB with coset coset Theorem 2.2.4.1), representatives of elements of H/mH. Since r(H) < 2, r{H) ~ Theorem 2.2.4.1}, representatives of elements of H/mH many H/mH is finite, finite,and andthere thereare areonly onlyfinitely finitely manyways ways of ofdefining definingproducts products with elements R == (G,·) (6,.) be be an an associative of elements elements of B elements of H. Let R B a1,a22 E H, aa1•a2 b If a1 is ring, and and let al'a B. If 3 ++ b, aa33 E H, bE 1·a 2 == aa3 replaced by element in the the coset coset a1 + mH, mH, i = 1,2, then replaced by an an arbitrary arbitrary element there are are only only finitely above remains the element element ab ab above remains the the same. same. Since Since there finitelymany many Suchcosets, cosets,itit suffices to 6/B is associative such toshow show that G/B associative quasi—nil. quasi-nil. However G/B associative quasi-nil. 6/B ""'HI-I is associative nil. As G is of As in in the G of the the form form 2) 2) with with HH nil. the case case of of form form 1) it suffices Argumentssimilar similar to those 1) it suffices totoinvestigate investigate Hom(~. G). Arguments those used used 1) reduce the investigation investigation to Hom(H~, D) D } reduce the D) ""' (±) Hom(~, D in case case l} p prime P many Since DDp = 0 for all all but butfinitely finitely manyprimes primes p, it suffices totoconsider consider it suffices and so Hom(H~, DP), prime, with with OP ~ 0. Now Now r(HSltt) = 1, and Dr). pp aa fixed prime, Suppose that Suppose that

23

Z(pco). Changing every ~ € Hom(HAH, Dp) is aamap map ~= Changing the sunrand summand .p: ~H i-RH ~ Z(p=). Z(p°°) Z(p=) only only alters alters the the ring ringdetermined determined by by ~ by ring isomorphism. a ring isomorphism. by Therefore it it suffices Z(p°°)) Therefore sufficestotoexamine examine Hom(i-1*l, Hom(H~. Z(p90)). Z(p9j). However Homl~. Z(p~) If u1(x,y) is isomorphic isomorphic to the the additive additivegroup group ofofp-adic p-adicnumbers. numbers. If u1(x,y) and are two ring multiplications up2(x,y) two ring multiplications on on GG with u1 a non-zero non-zero 2(x,y) are p—adicnumber, number,for for all all multiplication, multiplication, then then u2(x,y) == ¶u1(x,y), wu 1(x,y), w aa p-adic xx 1,x2 € G. Multiplication by by pp is is an an automorphism automorphism of Z(p=). G. Z(p©°). Therefore, 2 p—adicunit, unit, define the the multiplications u1(x,y) and pp- Pu 1(x,y), p aa p-adic isomorphic ring structures Every non—zero p—adicnumber number of the isomorphic ring structures on on G. Every non-zero p-adic isisof G. form pkw, k a p-adic unit. Therefore the elements of an integer, w a p-adic Therefore the elements of an k Hom(~. Z(p=)) induce non-isomorphic ring on G, the induce three three non—isomorphic ringmultiplications multiplications on zero multiplication, u1(x,y), and Pu pp1(x,y). zero multiplication, 1(x,y).

and it it suffices If no·t nil, nil,then then H = Q, Q, and sufficestotoexamine examine the the If HH is not multiplications determined ® Hom(Q~. Q). multiplications determined by by Hom(~. D(!) Q) ""Homl~. D) D) D® Q) The second second summand summandisisisomorphic isomorphic to and therefore determines The to Mult Q, Q, and determines two two The same argument employed employed above above to non-isomorphic on G. The same argument non-isomorphicmultiplications multiplications on G. H nil, determines many non—isomorphic show that that Hom(tfSJi, D), H show nil, determines only onlyfinitely finitely many non-isomorphic shows the the same for Hom(~. D). This proves on G, same for proves ring multiplications multiplications on G, shows Theorem2.2. 2.2.7. Theorem 7. I-I

24

33 Additive Additive groups of nilpotent and generalized nilpotent nilpotent rings Thenilstufe nilstufe of The of aa group. group.

§1.

Another generalization groups, also also due due to the following: following: Another generalizationofofnil nil groups, to Szele Szele is is the G ~ 0 be Thegreatest greatest positive positive integer nn such Definition: Let G be a group. group. The Rn ~ 0, that there exists an an associative associative ring R with R+ = G, G, and Rn that there R is called the nilstufe nilstufe of G, denoted v(G). If called the If no no such such positive integer integer nn Thestrong strongnilstufe nilstufe of G, N(G) is defined exists, then then v(G) == ~. The defined as as above,with with the theassociativity associativity of R deleted. above, R

Clearly Clearly the (associative) (associative) nil nilgroups groups are are precisely preciselythe thegroups groups G G N(G) = 1. satisfying (v(G) (v(G) == 1) N(G) The The following following examples examples are aredue duetoto Szele Szele [68, [68, Satz Satz 2]: 2]:

For every every positive positive integer n, Example3.1.1: 3.1.1: For Example group GG satisfying v(G) N(G) = = n. n. v(G) ==N(G) For For every every positive positive integer i, i,

there exists exists a there a torsion free free

let nG. G. be be aa rank rank one one torsion free free group group , n1

[(i,i,...,i,...)].

n+l Put G G== (i;) G.• Clearly Rn+l =0 R for every = 0 forevery ® G.. i=l , + such that R = G. G. For each each 1 < < i < nn choose ei E Gi such ring RR with R+ h(e;) = = (1,i , .•• ,i, ••• ). The The products h(e1) (i,i,...,i,...).

oftype of type

[(i,i, ••• ,i, .•• )].

1

1

=

e.e.

ei ·ejJ = 1

1

! r

e.+. , J

for i+j n for i+j>n

an associative associative ring ring structure < i, i, jj Consequence > t(g·g ) > t(gnJ = t(G ), aa contradiction. t(G1) 1.) == t(h.) 1 n n an ideal in R. Hence RG RG G ,, and similarly G RR c Gn, i.e., i.e., G c G and similarly G G is an ideal R. n nnn n_22n—1 n-l n 2n—1 2n- 1 R Put R== R/Gn. By the induction hypothesis RR 0, or R c G - n 2n 2 . v(G) < n i.e., = 0, 1.e., is nil, Theorem 2.1.2, and so R2 However Gn isnii,Theorem2.1.2,andso R =Gn=O, v(G)~2-l. 1

i

Gn

N(G) as was The The conditions ininTheorem Theorem 3.1.12 3.1.12 do do not notimpose impose aa bound bound on on NtG) was seen 3.1.8. seen in Example Example 3.1.8. nn torsion free rank one one torsion free group, group, with Corollary 3.1.13: 3.1.13: Let G G; , wi th Gi aa rank G == _@ ® G1, 1 =1 are equiva1ent. equivalent. The following following are •.• ,n. The i = = l,i,...,n. n. for all I < i < n. t(Gi) is not not idempotent idempotent for 2) v(G) v(G) < n. ~ n. Proof: 1) by Theorem 3.1.13, and 2): v(G) < 2n-l by Theorem 3.1.13, and so so v(G) v(G) 01 a

an cardinal. an arbitrary arbitrary cardinal.

the latter latter implication 1) ,. 2) and 3) 3) ,. 1), the implication since since Q+ Proof: Clearly 1) 2) and It therefore the of aa field. field. It the additive additive group group of therefore suffices suffices to toshow show that:

is

+

2) .. 3): Let R be for ideals, with be aa ring satisfying satisfying the the DCC DCC for with RR+ ==G. G. For R pfl+lR. Hence Hencethere there exists exists every prime prime p, and and positive integer n, pnR ~ positive integer 2 pn+lR.

· h th such that pos1· t1· ve integer 1nteger at aa positive nn sue

Th ere f ore for f or aa e: G, G, th there pnR == pfl+lR pn+ l R• Therefore ere 51 51

ni-i n n such pna pn(a - pb) 0. Since G exists bb € G such that p G a == ppn+lb, b, or p (a — pb) = 0. is G torsion free, a and so G G torsion a== pb, pb, and p-divisiblefor forevery everyprime prime p. Hence is p-divisible G is divisible, divisible,and and GG ~ Q+, Proposition G Proposition 1.1.3. •

a Observethat that the above theoremadds addstwo twoequivalent equivalentconditions conditions to those Observe above theorem those given in 4.1.3. given in Theorem Theorem 4.1.3.

Corollary non-nil torsion Corollary 4.3.3: Let GG be be aa non-nil torsion free free group. group. The are The following following are equivalent: (associative) ring 1) Every Every (associative) finitely many finitely many ideals. ideals. 2)

R with RiR+ = G, G,

and

R

R2 ~ 00 possesses possesses only only

R2

Every (associative) ring R with R+ = G, Every (associative) R G, and R2 R2 ~t 00 satisfies the the

DCCfor for ideals. ideals. DCC 3)

G ~ Q+ •

the implications 1) 2) and 1) are Proof: Again Again the 1) • 2} and 3) 3) • 1) are obvious. obvious. 2} 2)

.. 3}: 3):

Suppose Supposecondition condition2}2)isissatisfied. satisfied.

By 4.3.2, By Theorem Theorem 4.3.2,

G ~(±) Q+,

a and so G The ring G ~ Q+(t)H. 5i- =H. be the the zeroring with s+ = H. The ring direct Qi-®H. Let SS be 2 ~ 0. If sum sum R R ==Q(t)S satisfies RiR+ ""G, If HH ~ 0, choose G, and R R2 Q®S satisfies O a H. The infinite chain of ideals in R, (a) n... 0 I a € H. The infinite chain ideals (a} ~ (2!a) (2!a} ~ (3!a) (3!a} ~ ••• + is properly properly descending, descending, aa contradiction. Hence H = 0, and G G ~ Q • Q Lemma4.3.4: 4.3.4: Lemma

Let

G be aa torsion group. G be group.

Thefollowing following are The are equivalent:

1) G 1} G is the the additive additivegroup group of ofan an (associative) (associative} ring ringpossessing possessing only only finitely finitelymany many ideals. ideals. G is the (associative) ring ring satisfying the 2) G the additive additive group group of an an (associative} the ascendingchain chaincondition conditionfor for ideals. ascending

n. m nii m

3)

G ==

9 €) i=l j=l a. i=l j=l J

arbitrary cardinal, cardinal, Proof:

.

Z(p~}. 1

p. pi

m,n. aa prime, m,ni 1

a. an positive integers, integers, aj 3

1,... i == l, ...,m; ,m; j == l, ... ,n;.

Clearly 1) 1} • 2}:

{p1}71 for Supposethere thereexist existinfinite1y infinitely many distinct primes Suppose many distinct primes {pi}7=l be an an (associative} (associative) ring with which G ~ 0. Let R be with R+ = G such that RR R pi satisfies chain condition condition for for ideals. Theinfinite infinite chain satisfies the the ascending ascending chain ideals. The chain of 2) • 3}:

52

ideals in ideals

2 2

R, GGp

contradiction. contradict1on. For every every pp

€

kk

G c... c... is is properly ascending, ascending, aa G i=l Pi 1=1 Hence Gp ~t 00 for only only a finite finiteset setofofprimes primes {pi}~=l.

p11

c

0 Gp

1=1 i=l P1i

c ....c c (f)

a... Gp[pk] s ... is {pi}~=l' Gp[p] sc GP[p2 J S··· is an an ascending ascending

chain of ideals each contained contained in Gp. Hence there exists exists aa chain of ideals in R, R, each Hence there positive integer integer n1 ni

n. such that G = such = G [p. 1 ], Pi Pi 1

ni G = () ••• i=l pi i=2 Pi pi i=k m m many Hence Gp 1 00 for only only finitely finitely manyprimes primes {pi }i=l •. Let pp E {P; }i=l, Now and let let DDp be G . and be the maximal maximal divisible subgroup of Gp. Now 0Dp = ® Z(p divisible subgroup ()(a p P p possesses aa properly properly R Suppose an infinite infinite cardinal. cardinal. Then Then R possesses Suppose that that aoc is an co co "" descending chain Z(p=) => ••• , a a..., a descending chain of of ideals @ (D Z(pco) => 0 G Z(p => ••••D=> @ Z(pco) i=k i=l i=2 reduced subgroup subgroup of Gp• contradiction. Gp == Hp (f) Dp , Hp a reduced

00

00

54

)

).

n+l n Proposition Proposition 1.1.10. 1.1.10. For For every integer n, ppnGP ~ ppn+lGP is an an every positive positive integer inclusion of ideals in R. n, such inclusion of ideals R. Hence Hencethere thereexists exists aa positive positive integer = pn+lGP. = pkH®D that pnG = pkG = pkH €) D for every every positive integer Clearly pkG positive integer p p p p

reduced, this implies that Therefore pnHP = pn+lHP. Since Hp is reduced, this implies i.e., for pnH p = 0, i.e., forevery every •l1 - (±) (£) (£) Z(p7), p. pi aa prime, n, ni positive ® Z(p~) ® finite i=l j=l i=l J"=l a3 ex j 1

integers, cxj Proof:

Simi Jartoto the the proof proof of of Corollary Similar Corollary 4.3.6.

Theorem 4.3.10: Theorem 4.3.10: 1)

a finite cardinal, a finite cardinal, i = 1l,...,m, , ••• ,m, j == 1, .•• ,ni.

Let

be a group. G be group. G

Thefollowing following are are equivalent: The

G the additive additivegroup group of ofan an (associative) (associative) ring ringpossessing possessing only only is the G 55

finitely finitelymany many ideals. ideals. m ni m non—negativeintegers. integers, G (f) ® 0 Z(p~). p. pi aa prime, m. ni non-negative G c.!{£)Q+ 0 m, n. ® a a i=l j=l j=lcx. aj l,...,m; cx, a3 arbitrary cardinals, a. aj cardinals. i = = l, .••• m; j == l, •••• ni. 2)

1

1

1

only finitely finitely many Proof: 1) • 2): Let RR be be aa ring possessing possessing only many ideals, ideals. only finitely finitely many with R+ = G. G. Then R = R/Gt is aa ring ring possessing possessing only many ideals, ideals. = R/Gt an arbitrary arbitrary cardinal, with 'It torsion torsion free. Therefore it Co!~ Q+. acx an cardinal. is divisible, I < 0 t a I, xe:R. XE R. Since G/Gt Theorem 4.3.2. Let I
1

J

a, a. aj

finite , ••• ,m; jj = = l,...,n1. 1 , ••• ,n 1 • finite cardinals, cardinals, i == 11,...,m;

An argument argumentalmost almostidentical identical to that An thatused used ininproving provingTheorem Theorem 4.3.1 4.3.1 yields: Theorem4.3.12: 4.3.12: Theorem

be be a group. group. Let G G

The following following are equivalent: The equivalent:

1) G is the the additive additive group group of of an an (associative) (associative) ring ring satisfying satisfyingthe theDCC DCC for G ideals. n ni prime. m, n; 2) GG ... @ Q+ 0 (!) (3 p. a prime, ® a i1 j=l finite a i=l j=l aj

00

non-negative integers. non—negative integers, 56

1

1

1

l,...,m; a. cardinals. ii = = 1, ••• ,m; a, aj arbitrary cardinals,

jj == 1,...,ni. 1 , ••• ,ni . The are easy easy to to prove: prove: Thefollowing following corollaries corollaries are

is the group. G the additive additive group group of of an an 4.3.13: Let GG be be a group. Corollary 4.3.13: G ring with trivial annihilator, satisfying the DCC for ideals (associatjve) ring with trivial annihilator, satisfying the DCC ideals if and only one, and and hence hence both equivalent if and only if if GG satisfies satisfies one, both of of the equivalent ofTheorem Theorem 4.3.10. conditions of Corollary 4.3.14: 4.3.14:

1) 2)

The following following are are equivalent: The

Let GG be be aa non—nil non-nil group. group.

2 1 00 satisfies satisfies the for ideals. Every ring RR with R+ = G, Every G, and R theDCC DCC for ideals. R2 m ni . G ""c±)Q+ ® (±) ® (±) (±) Z(p~) 0 p. aa prime, m, ni m, n. ® Z(p~), 1 1 1 a i=l j=l a. i=l j=l finite 1

1

1

J

non—negative integers, non-negative integers,

a, aj

finite = l,l,...,m; finite cardinals, cardinals, i = ••• ,m; jj = = l,...,n1. l, •.• ,ni.

be an an(associative) (associative)ring ringwith withtrivial trivial annihilator, Lemma Lemma4.3.15: 4.3.15: Let RR be such that R+ = Dm D R is aa ring such D == for all n >m. integer m such <m!a> for ~ m. Clearly <m!a>+ is divisible, and so This obviously implies m!a = 0, or divisible, and so <m!a>+ =D. This obviously implies that m!a D. that KK is aa torsion torsion group. group. Let

K a reduced 3) ~ 1): Let .GG==D(G) cc II and and so so + +.= II+. +.= J[IP(G)]++ and tP(G)+; Therefore J[q>(G)] 'f 0, then q(G) IP(G)+; i.p(G) . Thereforeifif J[p(G)] and so It that if if CG and H H G It isiseasily easily seen seen that are quasi—isomorphic quasi-isomorphic C ~ J[IP(G)]+. onlyiiff H free groups, groups, then then CG is anti—radical anti-radical ifif and and only is. torsion free H the fact that contradicts the that G anti-radical. C is anti—radical. Therefore CG Q. J[tP(GJl+ contradicts c.p(G) is semisimple, implies that R = 0, i.e., semisimple, which which implies J[p(G)] = i.e., q>(G) Hence J[tP(G)l R is semisimple. 75

The above abovetheorem theoremwas was proved different manner in [6, 5.5]. The proved in ina adifferent manner in [6, Theorem Theorem 5.5].

be aa finite Let G be finiterank rankstrongly stronglyindecomposable indecomposable torsion G additive group groupofofa aring ringwith withtrivial trivial left left If G is the the additive group. If free group. G stronglytrivial trivial annihilator then annihilator then G is strongly strongly irreducible, associative associative strongly G field K K left annihilator, annihilator, and End(G) is aa subring subring of ofan an algebraic algebraic number number field [K:Q1 = r(G). satisfying [K:Q] r(G}. Theorem 4.6.19: Theorem 4.6.19:

be aa ring with with ~(R) = 0. By Proof: Let R == (G,~). ~ E Hom(G, End(G)) be + nilpotent ideal ideal of ~(G) Observation the maximal maximal nilpotent G ~~(G)+, p(G) , and the Observation 4.6.2, 4.6.2, G p(G) is nonzero elements elements in ~(G) are zero by by the theBeaumont—Pierce Beaumont-Pierce Theorem. Theorem. The The nonzero for units units in End(G}. End(G),Corollarly Corollarly 4.6.10, which implies that ~(G)+~ p(G) = I+ I .p(G), see every nonzero As in every nonzero ideal II in ~(G), see the proof proof ofofTheorem Theorem 4.6.11. 4.6.11. As the proof p(H) ofTheorem Theorem 4.6.17, ~(H) iss a nonzero nonzero ideal for every every the proof of ideal in ~(G) p(G) p(H) nonzerofully fully invariant H This implies implies that ~(G) ~~(H) invariantsubgroup subgroup H of G. G. This nonzero which in turn yields The same which in yields G and is strongly strongly irreducible. irreducible. The same G ~ H, and so G G argument (1} ~ (3) in inthe theproof proofofofTheorem Theorem argumentused usedtotoprove provethe theimplication implication (1) 4.6.17, shows shows that K == End(G) End(G) is aa field field satisfying satisfying [K:Ql [K:Q] = r(G). End(G)) (G,~), ~ E Hom(G, End( G)) be o. Then Let S S = (G,'4), be aa ring ring satisfying sS22 ~ 0. subring of End(G) and soisis certainly certainly not ~(G) is aa nonzero nonzero subring and so not nilpotent. By Theorem 4.6.11, 9.(S) and so G G is an stronglytrivial trivial By Theorem 4.6.11, ~(S) = 0, and an associative associative strongly left annihilator left annihilatorgroup. group.

Corollary 4.6.20: G be aa finite finite rank Corollary 4.6.20: Let G be rankstrongly stronglyindecomposable indecomposable torsion either G G is free group group with r(G) r(G) = n. n. Then Then either is an an associative associative strongly strongly trivial annihilator trivialleft left annihilatorgroup, group, or or Rn+l = 0 for every ring R with R R+ = G. G. Proof:

§7:

Theorem 4.6.19, and andCorollary Corollary 4.6.12. Theorem 4.6.19,

E—rings and and 1—rings: E-rings T-rings:

In [35, posedthe theproblem problemofofclassifying classifying the the rings In [35, Problem Problem 45], L. L. Fuchs Fuchs posed rings R by R satisfying RR ~ End(R+). Much Muchprogress progress has has been beenmade madeononthis this question by P. Schultz, [59], P. Schultz Schultz [13]. Many P. [59],and and by by R.A. R.A. Bowshell Bowshell and and P. Manyofof their their results are presented here. are presented here.

If If

H R<End(R+), and R+=H(t)O reduced,and R and = H® D with H reduced, and 0D H divisible, then is aa torsion divisible, then either 0D == 0, or 0D ~ Q+ and H torsion group. group.

Lemma4.7.l: Lemma 4.7.1:

76

= a. Then R+ has Let rr0(D) directsummand summand isomorphic has aa direct isomorphic to G) Q+ 0(o) ++ a + • R+ and so R <End(R) and has directsummand summand isomorphic has a direct isomorphic to + End( End ( 1'1'\ \TJ Q ) .,.. n( ®
is finite. is infinite r[Hom(Bk' Bk)] = {k •• rk if rk infiniteand and r~ if rk finite. rk is rk is r[Hom(Bk, jj >> k, rk = 1. Let be a positive integer, Since r[Hom(Bk, r[Hom(Bk' Bk)] > 11 == rp(R+), rp(K)

However

a contradiction. a

Conversely, let let x E R~. H ~ R~. H Conversely, Since hp(x) h(x) == hp(z) == ""• This clearly clearly implies 0, and and so This implies that that y == 0, so xx E H, H,

y E R;,

= ""· hp(y) == hp(x-z) =

i.e., i.e.,

then

Z(pk) (i)H, [36, R+ == Z(pk)

H is p-divisible, H is p-divisible,and and so so

Therefore Then

Lenrna 4.7.2, and so so 4.7.2, and Lemma

""• 0 11

=

+ rp(R ),

a a contradiction.

0 G., G., ® iiEI e:I 1 r(G1) = 1, and Gi where r(Gi) G1 where I. Then is torsion torsion free free for forevery every i e: I. Then I is finite, t(G1) i idempotent for i 'I j, G .... End(G.)+, t{G .) is idempotent finite, t(Gi) ! t(Gj) j, 1 1 1 for all i,j e: I, I, and R R "" (t) End( G1.). ® End(G.). iEI i e:I Lemma4.7 4.7.6: Lemma .6:

ring satisfying Let RR be be aa ring

R with R ... End(R+),

R+ = =

1

1

Proof: Proof:

R+ R+ has directsummand summand isomorphic has a direct isomorphic to

nfl End(G.). Hence 1 i1€! e:I This This is is possible possible only only if if II is finite.

< r(R+) = r( nn End(G 1.)) ~ Now Now = III. III. I El ie:I has aa direct direct summand isomorphic ® End(G.)+(t) Hom(G Hom(G1, G.). R+ has summand isomorphic to to 1., GJ.). 3 1 ie:I i'lj iEI However, r((t) = r(R+) = and = = III. III. Therefore RR"" r(® End(G.J+) ® End(G"!"), 1 1 IEI ie:I ie:I i€I Hom(G., G.) = 00 for for all i,j e: I, I, i 1- j. By Proposition Proposition 1.3.4, 1.3.4, Hom(G., G.)= j. By 1

1

1

1

i,j i,j I, i,je:I,

J

t(G.) all i < t(G1) t(Gi),f.t(Gj) forall i/-j. we j. Since t[End(Gi)+]~t(Gi) $ t(G.) for have that Gi ""End(Gi)+. Since End(G1) have End{Gi) is not not aa zeroring, zeroring, t{Gi) is idempotentfor for all all i e: I, I, Corollary Corollary 2.1.3. For jj e: I, I, let nj be idempotent be the For I, natural projection projection of R+ onto G3. Gj. iI e: I, and aa e: R, the maps maps natural G1 ~ -+ Gj Gi via xx ~ nj(ax) and xx ~ nj(xa) are homomorphisms. homomorphisms. Since = nj{xa) = 0 Hom(G1, for all xx e: Gi, G1, ii 1- j, Hom{G;, Gj) = 00 for i 'I j, j, nj(ax) = for j, G1R c Gi and i.e., RGi: RG1 and so for all all i.e., Gi, and G;R: so G. Gi is an an ideal ideal in RR for Ii e: I. I. Therefore R == (f) is a ring direct sum. G. "" C±> End{ G.) a ring direct sum. ® End(G.) ® G. 1 i€I 1 ie:I i€I ie:I 1

1

be aa torsion group. Thereexists exists aa ring with 4.7.7: be group. There 4.7.7; Let G G G is cyclic. satisfying RR ""End{R+J if and only if G is cyclic.

Theorem Theorem

Proof:

Z(n). Let G == Z{n).

Then

R+ = G

End(G). (Z/nZJ+ = G, G, and Z/nZ — ""End{G).

be torsion group, let R be Conversely, let let G R Conversely, be aa torsion group, and and let be aa ring with with G for which R+ = G satisfying RR ""End(R+). Let SS be ofprimes primes pp for which be the the set of kk

4.7.2, GP 'It 0. 0. By By Lemma Lemma 4.7.2,

R+ = Z{p P). = (±) Z(p

nonzero component nonzero component in is cyclic. cyclic.

for every S every pp e: S, S is aa finite finiteset, set,and and so so R+ R+

G

Theorem4.7 4.7.8: Theorem .8:

RP

p€S pe:S

Since Since the the unity 11 e: RR has

be be the the direct directsum sum of ofrank rank one one groups. groups. There Let G G There exists exists +

+

m m

if and only if if G and only (f)(!) G. End(R )) if a R G == Z(n) Z(n)® ® G1 a ring R with R = G, and RR"" End(R i=1 1 i=l 79

r(G1) r(Gi) = = 1, 1, t(GiJ

with

Let

for i I t(G.) for .$. t(Gj)

j,j,

and

m m satisfying the G the conditions conditions of ofthe thetheorem. theorem. G == Z(n) Z(n) (!) ® ~ Gi satisfying i=I = 1, 1,... 1.4.8, be aa unital unital subring G1, i = be subring of Q with R~ ... Gi, •••,m, ,m, see see 1.4.8, Q m + + R ""End(R ). R (Z/nZ)C!)®{.+) ® R.. R == (Z/nZ) R.• Then R ""G, and R i=I 1

Let

R1 Ri

and put and

R be a ring with ring with

Conversely, let groups, and and

R— "" End( R+).

p p

i E

Gp It 0. 0.

be be a a prime prime for for which which

R+ = G R+ G aa direct directsum sum of of rank rank one one

R+ = CD H, with with = Gt Gt®H,

Now

torsion free one torsion free group group for foreach each Let

t(G.) t(GiJ

l, ••• ,m. i,j == l,...,m. pin; i,j

for all for allprimes primes

= Gi pGi =

Proof:

idempotent,

H == i G), and contradicting Lemma 4.7.2. >1, to 1 , contradicting Lemma 4. 7. 2. Hence

Theorem4.7.9: 4.7.9: Theorem

Let

with ring R R

G be aa group which is is not G be group which not reduced. reduced.

There exists exists aa There

if and only i f

G"" Q+(f) Z(n), G Q4®Z(n),

R+ = G,

and

R R ""End(R+)

nn

positive integer. aa positive integer.

If If

Proof:

+

R ""G,

and

G == Q+(!}z(nJ, + R ""End(R ). R

then

R R == Q(t) Q® (Z/nZ)

is ring satisfying satisfying is aa ring

Conversely, let R be aa ring, with R+ not reduced, satisfying + + + k Z(p R ""End(R ). By 4.7.1 and and 4.7.2, 4.7.2, R ""Q R By Leninas Lemmas 4.7.1 (±) (f) Z(p P), P), — ® pp a prime prime

with

for every 0 0 (R/Rt) —> 0

O —> RR(i Rt 0 -::> > RR ~ R —> R Rt —>

1~1

t~2

U1

----:::> >RR

O

where

11i

Since

Consider the Consider the

is induced induced by by the themaps maps

---i>>0, 0 •

>R/Rt ~i(a

-

b) = ab, ®b)= ab,

iI = 1, 2,3. 2,3.

for which is p-divisible p-divisible for forevery everyprime prime pp for which

Rp ipt 0,

It is readily seen that for each prime p, Rn). Itisreadilyseenthatforeachprime (.j:', {RPIXIRP). p. Rp R p pprime p prime k k and so is aa ring summand Zip Pz for ringdirect direct summand of R, and so Rp ~ Z/p some non-negative non-negative for some integer kp. Hence Rp ® Rp ~ Rp' and and in in fact, fact, every every element element in Rp ~ Rp may written in may bebewritten the form form 1 ~a, the a, with l the tfle unity unity in Rp • Clearly the . the restriction of to is an isornorphism for every prime restriction ~l Rp 5ll Rp an isomorphism every prime p, and U1 so U1 ~l is an an isomorphism. isomorphism. Rt = R®Rt=

1

1

.-.

Again p-divisible for for every ev~ry prime prime pp with Againthe the fact fact that R+/R; R iRt is p—divisible in fact Rp t 0, yields that RR '5!1 {R/Rt) and in fact every every (R/Rt) ~ R/Rt' (RiRt) .x, {R/Rt) (RiRt) ~ (R/Rt) RiRt, and a, element in is of the form with element (R/Rt) •Xl {R/Rt) of the form 1 X1 1 the the unity unity in (RiRt) G) (RiRt) +

+

.

1

86

1

R/Rt. that

This clearly clearly implies This implies that ~p33 is is an an isomorphism, isomorphism, which which now now yields yields or that R ~ 2 is an an isomorphism, isomorphism, or is a T-ring. R a 1-ring.

Question 4.7.30: An An immediate immediateconsequence consequence Theorem 4.7.29 thefact fact that Question of ofTheorem 4.7.29 is isthe if RR is aa 1—ring if T-ring for for which which R+ is not not aa torsion torsion froup, froup, then then 1) r(R+/R;) = 1, 2) t~R+/R~) is idempotent, idempotent, 3) RR; is cyclic cyclic for forevery every prime p, and and 4) 4) for R for which for every every prime prime pp for which RP 1 0, R+/~ is p—divisible. p-divisible. Is every every group 4) the the additive additive group group of aa groupsatisfying satisfying 1) 1) - 4) 1-ring? A positive answer, T-ring? answer, together together with withTheorem Theorem 4.7.25, would would provide aa complete description of complete description ofthe the1—ring T-ring groups. groups.

87

55 Torsion Torsion free rings

Notation, definitions, definitions, and §1. Notation, and preliminary preliminary results: Most the material in in this thischapter chapteris is work Beaumontand and Pierce Pierce Mostof of the thethe work of of Beaumont on torsion torsion free free rings, rings, i.e., i.e., rings [7], on rings with with torsion torsion free free additive additive group, group, (7], R is said it a group property, a ring [8], and and [53]. In general general for n a group property, a R said to to be aa it—ring n-ring if n-group. A A subring SS of RR is said said to to be be aa be if R+ is is aait—group. n-subring if s+ is aan-subgroup it-subgroup of R+. it-subring of R R

if

Very roughly roughly speaking, speaking,the theidea ideaof of this chapter Very chapter isistotoembed embed a torsion torsion * * into the free ring RR into that if if R R* free theQ—algebra Q-algebra RR* = Q® Q® R. R. It It will willbebeshown shown that satisfies certain satisfies certainproperties, properties,then then·considerable considerable information information is is obtained obtained concerning R+. R+.

All rings chapter are ringsininthis this chapter areassumed assumed to be be associative. associative. Let R, T. Then R and SS are be subrings subringsof of aa torsion torsion free free ring 1. R R, SS be such that that S, if ifthere there exists exists aa positive positive integer integer nn such quasi-equal, RR ~ S. torsion free free rings R, SS are quasi-isomorphic, nR C S, and nS nR R. Two Two torsion rings R, quasi-isomorphic, ifthey theyare areisomorphic isomorphictotoquasi—equal quasi-equal rings. R R & S, S. if

=

=

Notation:

d(G) divisible subgroup dlG) = the maximal maximal divisible subgroup of of aa group group G.

Z(R) Z(R) =the = the center center of of aa ring R == {x

€

RI Rl xy == yx yx

for all y for

€ R}.

R* * = Q(x) Thesymbol symbol®® will will often free ring. The R R, R aa torsion torsion free oftenbe be omitted omitted R Q®R, R as aa full full subring in order order to simplify simplify notation, notation, and and in in order order to to view view R as subring of k * * R* • R, will be be written written as as . The elements of R R* R The elements Q,

iI == 1l,...,k. •••• ,k.

divisible if if G G group GG is quotient quotient divisible possesses Definition: A torsion free free group such divisibletorsion torsiongroup. group. a free free subgroup subgroup F F such that G/F is aa divisible

88

Quotient divisible will Quotient divisible willbe beabbreviated, abbreviated, q.d. q.d. * R* Tworings, rings, Definition: R called the the algebraic algebraic type type of R. Two is called of a ring R. R* . * * the same algebratype type1f if R ~ s . A torsion free G admits R,S have have the same algebra free group group G a a multiplication multiplication ofofalgebra algebra type type A, A aa Q—algebra, Q-algebra, ififthere thereexists exists aa ring A ** + + such that R R R R such = G, R ~ A. and = G, R A. Theorem 5.1.1: Theorem 5.1.1: type.

Quasi—isomorphic torsionfree free rings have Quasi-isomorphic torsion have the the same same algebra

ofgenerality generalitywe wemay may consider consider the thecase case R,; S, Proof: Without Without loss of R 5, R,S torsion free i.e.,there there exists exists aa positive positive integer integer nn such torsion free rings, rings, i.e., such that nR c~ S, and nS R. Then R*= R* = QQ@ R R == QQ ~ nR nR ~ nS ~ER. E S* = Q ~ S = R* , Q ~ nS ~ Q ® R = R and so RR* =S = S* • An consequenceofof Theorem Theorem5.5.1.1 is An immediate immediate consequence 1.1 is

Corollary 5.1.2: Let A A be Q-algebra, and and let let G,H be torsion free Corollary be aa Q-algebra, be torsion free groups, G Q.&H. H. Then G admits of algebra algebra type type A if and G admitsaamultiplication multiplication of A if and only if if H only does. H R be torsion free free ring, and P(x1,... Lemma5.5.1.3: Lemma 1.3: Let R be aa torsion and let P P = P(x 1 , ••. ,xn) be a homogeneous polynomial withcoefficients coefficients in such that x1 , ••• ,xn are homogeneous polynomial with in Z, such non—commuting variables. Then RR satisfies the identity PP if non-commuting variables. the polynomial polynomial identity .if R * does. R* and only 1f and only

if

Proof: Suppose that R Suppose that satisfies the the polynomial polynomial identity P. Let R R*. * Since R is aa full there exists exists a positive a 1 , ••• ,an E R. full subring subring of of R* R* there R integer mm such such that rna; E R, i = = l,l,...,n. ••• ,n. Let d == deg deg P. P. Then R* mdP(a , ••• ,a ) = P(ma , ••• ,rna ) = 0. Since R* is torsion free, mdp(al,...,an) torsion free, P(mai,...,man) = 0. 1 1 n * n R The converse converse is is obvious. P(a 1 , ••• ,an) = 00 and R satisfies P. The obvious. G be quasi-isomorphic quasi-isomorphic torsion torsion free Corollary 5.1.4: 5. 1.4: Let G and HH be free groups. groups. Corollary G is a non-zeroring a homogeneous (1) G is the the additive additivegroup groupofof a non-zeroringsatisfying satisfying a homogeneous polynomialidentity identity P == P(xl'··· ,xn) if if and onlyiiff H H is. (2) Every polynomial and only Every ring with G if and with additive additive group group G satisfies PP if and only only if ifevery every ring ring with with H additive group additive group H satisfies P.

Proof:

Corollary 5.1.3. Corollary 5.1.2 5. 1.2and andLemma Lemma 5. 1.3.

Lemma 5.1.5: Lemma

be aa torsion torsion free free ring. Let R be R

R R is T-nilpotent T-nilpotent ififand and only only

89

if

R* * is T-nilpotent. if R T-nilpotent.

that Proof: Suppose Suppose that

R is 1-nilpotent. T-nilpotent.

R

Let

= {ai}i=l

R*. C ~ R* .

There exists exists aa There

such that that miai E R, i = 1,2 positive integer R is integer mi such 1,2, .••. Since R T—nilpotent,there thereexists exists aa positive positive integer T-nilpotent, integer kk such such that (m 1a1)(m2a2 )k ... (mkak) = 0. 0. (mlal)(m2a2)k. ..(mkak)

torsion free torsion free

k nIT

i=l

a. a.1 == 0,

Therefore and

kk

kk

(( iin m1)( mi)( iin a;) = 0. . 1=1 i=l i=1 i=l

Since

**

is

R

R

*

R* R is T-nilpotent. T-nilpotent.

Corollary be quasi-isomorphic quasi—isomorphictorsion torsion free groups. Corollary 5.1.6: 5. 1.6: Let G,H G,H be groups. (1) GG is the T—nilpotent ring ring which is not aa zeroring the additive additive group group of of aaT-nilpotent which is zeroring ififand and only H is. (2) Every ring with with additive additivegroup group GG is T—nilpotent T-nilpotent if and H (2) Every if and H only if every ring with additive group H is T-nilpotent. only if every ring with additive group T—nilpotent. Proof:

Corollary 5.1.2 5.1.2 and and Lema Lemma 5.1.5. 5.1.5.

be aa torsion torsion free free ring, an arbitrary arbitrary ordinal. Lemma ring. a an Lemma5.1.7: 5.1.7: Let RR be R if and only if if R* R is a-nilpotent a-nilpotent if and only R* is.

Then

An easy easy induction induction argument Proof: An argument shows shows that (R*)8 (R*) 6 == (R8)* (R6)* for every every (R*)a = (R0)* ordinal 6. a-nilpotent, then then (R*)a = (Ra)* = 0. The Thereforeifif RR is a-nilpotent, The 8. Therefore converse is is obvious. converse obvious. H be torsion free groups, Corollary 5.1.8: Let GG and H Corollary be quasi—isomorphic quasi-isomorphic torsion a an an ordinal. (1) G G is is the theadditive additivegroup groupofofanana—nhlpotent a-nilpotent ring whicn whicn is not a zeroring if and only if not a zeroring if and only if H is. (2) Every ring with with additive additive group group GG H (2) Every H is a-nilpotent a-nilpotent ififand and only only ififevery everyring ringwith withadditive additivegroup group H is a—ni lpotent. a-nilpotent.

Proof: §2. §2.

Corollary 5.1.2 5.1.7. Corollary 5.1.2and andLemma Lemma 5.1.7.

The Beaumont—Pierce Decomposition Theorem: The Beaumont-Pierce Decomposition Theorem:

The group analogue of ofthe theWedderburn Wedderburn Principal PrincipalTheorem, Theorem, which which The group theoretic theoretic analogue was usedinin Chapter Chapter4,4, section section 5, 5, will was used willbebeproved proved here. here. Since Since the the proof involves several willbe be stated stated before before the the preliminary preliminary results involves severalsteps, steps,itit will leading up uptoto it it are leading are given. given. Theorem5.2. 5.2.1: finite rank Theorem 1: Let RR be be aa finite rank torsion torsion free free ring, R* R* = ~·~N aa R*, * , and the nil nil radical Q—spacedecomposition decomposition of of R*, Q-space R* , with if N the radical of R and S~ aa 90

semisiniple subalgebra of R*. semisimple subalgebra R*. Put S ==RR nti s, N == RR n if. N. Then SS is aa satisfying s* == S, N is the subring the maximal maximal nilpotent subring of R R nilpotent ideal ideal of R, N* and satisfies N* == -if. hasfinite finite index and satisfies N. In addition addition S (+) N has index in R. Lemma5.2.2: 5.2.2: Lemma

Then

~:

R+I (S (t) N) + "" — S~ /S +:

it w be R* onto be the the natural natural projection projection of R* 5R+ -, s+1/S+ 5 via = n (z) ++ s+ for for all zz e: R+.

Proof:

'.P:

I

R). for which Let sS1 there exists y e: R" which x+y e: R}. if for Si there 1 == {x e: S

~

Let

~(z)

S. Define Clearly

~

~

is an an

c ker epimorphism, and and (S ® N)+ ~ ker p. ~. Let zz e: R, R, zz = x+y, x+y, xx e: S, 5, yy e: N. if. Suppose that that zz e: ker p. Suppose ~. Then x e: S, and =zz-- x€R xe:R fln iT= N =N, N, i.e., and so y = r.'\+ r.'\+ + + ++ zz E (S 1+1 N) , and and so so ker .p ~= = (S ':r' N) , or R /(S ++ N) ""S 1/S • Lemma 5.2.3: Lemma 5.2.3:

*

*

* == N. -~. S5 * = -S, 5, and N N —

—

*

It therefore Proof: S is the theunique uniqueminimal minimalQ—algebra Q-algebra containing containing S. It therefore S* R* * 5. Let x suffices to to show show that S is aa full full subring subring of of s. Then xx e: R S x e: S. 5-. such that nx e: R. andso sothere thereexists exists aa positive positive integer nn such and R. Hence The same nx e: R fln S = SS and and so so s+ts+ is aa torsion torsion group. group. The same argument shows shows -N. that NN* = = N. for s such such that that the There is There is aaQ—basis Q-basis {x 1 , ••• ,xm} for the free free F of 5Sgenerated generated by by this basis basis is is aa subring subring of S. F

Lemma 5.2.4: Lemma 5.2.4:

subgroup Proof:

5+

Since s+ is aa full fullsubgroup subgroup of S 5. in S is aa Q-basis Q-basis for ~.

s, maximal independent independent set 5-, a maximal m

a .. kzk with k=l lJ be aa positive integer l, •.•,m. ,m. Let nn be integer such such that ,j,k == 1,... a.1J"k e: Q; Q; i ,j,k {x1 == nz1 for all i,j,k i ,j ,k == 1,... Z nz; , iI == 1,... 1 , •••,m} ,m} naijk e: Z for 1 , •••,m. ,m. Then {X; satisfies satisfies the theconditions conditionsofofthe thelemma. lemma. z1 , ••• ,zm

Observation 5.2.5: Observation 5.2.5:

St/F S~/F

Hence

z.·zJ. = 1

~

= k=1

is aa torsion torsion group. group.

S+/F is aa torsion It therefore torsiongroup. group. It therefore suffices suffices totoshow show Proof: Clearly S+/F 5.2.2 that R+/(S N)+ is aa torsiongroup, group,ororbybyLemma L;mma 5.2.2 that S~/S+ is aa torsion R,, and and so 5, zz e: N. torsion group. group. Let xx e: R. Then xx e: R so x == y+z, y+z, yy e: S, N. such that ny e: S, By 5.2.3 5, and By Lemma Lemma 5.2.3there thereexists existsaapositive positive integer nn such R+/(SmN)+ and nz e: N. Hence nx = ny + nz e: S(+)N, R+/(Sq)NJ+ is a torsion group. nz nx = ny + a torsion group. nz N.

Lemma 5.2.6: Lemma 5.2.6:

For all but many For but finitely finitely manyprimes primes

p,

(S~/F)p, and

(S+/F)p 91

are divisible divisible and are and equal. equal. The proof proof of Lema The Lemma 5.2.6 5.2.6 involves involvesmany many steps, steps, and and will willbebebroken brokendown down notation. The The degree degree of of into aa series series of of claims. claims. First Firstwe we introduce introduce some some notation. nilpotence nilpotence of W willbebedenoted denoted by by t. For pp a prime, prime, kk >>00 an an integer, integer, if will Ik == {x € F s1}. x € S}, SJ, and Jk = = {x € Fl p-k xx € s1.. F II p-k x Claim 5.2.7: 5.2.7: {1) Claim F == 10 I 0 22 IIi1 22 12 I 2 2 ... , (1) F

=Jl =J2 =···•

(2)

F= F = Jo

{3) (3)

IkE !: Jk' 'k

are two two sided sided ideals ideals in F, Ik and Jk are 'k {5) (5) 'k'it Ik. It E ~ IkH ' (4)

{6) (6)

(7)

and Jk.Jt!: E Jk+t' and Jkt ~ I k for for all a 11 k,z k ,t > ~ 0. 0.

Proof:

(1) - (4) (1) (4)

are obviously obviously true. are

Let x1 x1 € Ik, x2 == p1y2 , y1,y2 y1 ,y 2 € 'k' xX22 € I 1 . Then xx11 == pky1 , and x2 x1x2 = k+t Hence x 5, i.e., i.e., x1x2 y1y2 € S, x1x2 € Ik+t . 1x2 = p y1y2 , with y1y2 (6): follows from usedtoto prove prove (5). (5). from the the same same argument argument used {6):

(5):

s. S.

and there there exists zz € N (7): Let xx € Jk. Then xx == pky, y € s 1 , and if such that y-z € R. R. Let x1,... and let let z1, ••• ,zt € N such that p-kxi - zi z1 € R, x1, ••• ,xt € Jk, and

Suppose, inductively, that p-kx. x. 11 12 '1 12 for l~i Then < i11 < i2 0. > 0. = [~ ( I k) ] . k + + + = By By Claim Cla1m 5.2.9, ppkT[p] Tp[p] ""lk+l/[pF nfl lk+l] ""~[Ik+l] = (B)

~(I k)

+ k (pkT1p)Ep] ""Jk+l/[pF n Jk+ll + ""(p T1p) [p] for for all all k > > 0. o. Choosing k ==00 Since and is finite, yields that yields that Tp[p] ""Tlp[p]. Tp[p] E ~ T1P[p], Tlp[p] is finite. T[p] == T1P[p]. For every G, r(Gp) = this implies implies that that TP[p] Hence = r(GP[p]). ~[Jk+ll

r{ TP)

+

= r( T1p) •

=

93

Claim5.2.9 5.2.9 and and{B) (B)yield yield that {pkTP)[p] ~ I~+l/[pF n Ik+l]+ ~ ~(lk+l) Claim 1k÷l' — + + (p2k+l1)[p] 2k+l ) fl for all k > o. 0. IiF n 12k+ 21 ~ {p = ~ 12k+2 I 2k+ 2/£pF TP)[p] for This = ~{I 2 k+ 2 together with the the finiteness finiteness of T [p] yields together with yields that that Tp(p] == {pTP)[p] = This clearly implies implies that Tp is = {p 3Tp)[p] = •.• ... P This

is divisible. is divisible. Hence Tp is aa divisible divisiblesubgroup subgroup of Tlp with implies that Tp = = Tlp' r{Tp) == r{T 1p) K(2.), K(~) such ~ K(~). Tk Ik ++Po~ n Jk. p2.W2n2Jk.

there exists an there an integer integer

it

be aa prime. < ~ it p be prime. Since r{TP) ~< r(T 1p) < suffices to toshow show that d(Tp) == d(T 1P), or since since TP .= Tlp' that By Claim Claim5.2.21 5.2.21 there there exists exists an d(Tlp) ~ d(TP). By an integer K(Q) K(~) such such that

ofLemma Lemma 5.2.18: Proof of

Let

p

97

4 4-.2- + pn R.2 2for kK ~> K(Q), K{R.), nJk n Jk E ~ nJk n Jk E ~ n Ik + p n B. However nn2 Ik 'k ~E Ik, 'k' Claim for all k ~> K(R.). 5.2.19(1), and F. Hence {A) K(i). 5.2.19(1), and n~ ~F. n4Jk ~E Ik (A) n4Jk 'k ++ pR.F for

c oR.{l;) be the canonical By (A) eR.{n 4 J~) ~ eR.: FF ~ F/pR.F be canonical epimorphism. epimorphism. By + + + R. R. + R. + — Jk/lp F n Jk) for all k >> K{R.). Now Now oR.{Jk) == {Jk + p FJ/p F ~ ~ Let

by Claim 5.2.9(2). Similarly, by Claim 5.2.9(2). Similarly,employing employing Claim Claim 5.2.9{1), 5.2.9(1),

{pk-R.Tlp)[pR.]

eR.{I~) ~ {pk-R.Tp)[pR.]. {p,m) 1. (p,m) = 1.

Let nn44 == p3m, pjm, m aa positive integer integer such such that m

= (pk+i_9.T1)[p2._i] Then {pk+j-R.Tlp)[pR.-j] Then n4(pk-R.T 1P)[pR.] =

subgroup subgroup of of (pk-R.Tp)[pR.].

Put

R.Z = = j+l

is isomorphic isomorphic to aa

and let ~ max{K(R.), R.}. and letk >k max{K(.Q),

Then

is isomorphic isomorphic to to aa subgroup subgroup of {pk-R.TP)[pR.], so and so (p ], and k-1 T P == d{Tlp ) , and < r {pk-R. Tp ) • For {pk-1 Tlp ) ~ kk sufficiently sufficiently large large ppk_lT1 rr(pk_lT1p) 1 the claim claim is pk-R. Tp == d{TP ) • Hence d{Tlp ) ~ d ( TP ) and and the is proved. proved. (pk-lTlpJ[p] (p

Finally, Finally,we wecan can prove proveTheorem Theorem 5.2.1.

is of of Theorem Theorem 5.2.1: thatremains remains to to be be shown shown isis that S (+) N is Proof of 5.2.1: All that SC+)N finite Lemma5.2.3. 5.2.3. By By Lemma Lemma 5.2.2 suffices to finite index index in in R, R, Lemma 5.2.2 ititsuffices to show show that S~/S+ is is finite. finite. S~/S+ ~ (S~/F)/(S+/F) = T1/T T1/T eo~ {t). Tlp/Tp' ® p paprirne p a prune Observation 5.2.5 and and Proposition Proposition 1.1.1. Now Now Observation 5.2.5 T1p/Tp is finite finitefor forevery every primes, Lemma 5.2.18, and 00 for all prime p, Lemma Lemma 5.2.18, allbut butfinitely finitely primes, Lemma 5.2.6. 5.2.6. + + S1/S is finite; thus concluding Theorem Hence S~/S+ finite; thus concludingthe theproof proofofof Theorem 5.2.1. .

5.2.1 to groups, Translating Theorem Theorem 5.2.1 groups, we we have: have:

be aa finite finite rank Corollary 5.2.23: G 5.2.23: Let G be rank torsion torsion free free group. group. Then G multiplication ofofsemisimple G c>. H$ K, where HH admits admits aa multiplication semisimple type, type, and and KK is 2 ~ 0, unless K = 0, the of aa nilpotent nilpotent ring N2 the additive additive group group of ring NN satisfying N is G or G is nil. nil. §3.

Torsion free free, rings with Torsion with semisimple semisimple algebra algebra type.

The results of upupto to quasi—isomorphism, The results ofthe theprevious previoussection sectionshow showthat, that, quasi-isomorphism, the classification of of finite finite rank of the the additive additive groups groups of rank torsion torsion free free rings rings the reducesto to two two cases; cases; the the additive groups reduces groups of of rings rings with withsemisimple semisimple algebra algebra type, and the additive additive groups of nilpotent nilpotent rings, and the groups of rings, which which are are not not zerorings. zerorings. type, In this thataafinite finite In this section section aa further further reduction reduction will willbe beobtained obtained by by showing showing that 98

rank free group group admitting admitting aa multiplication multiplicationofofsemisimple semisimple algebra algebra rank torsion torsion free type is quasi-isomorphic quasi-isomorphic to direct sum sum of groups groups admitting type to a direct admitting multiplications multiplications of simple simple algebra algebra type. type. It willfurther furtherbebeshown shown that rank torsion torsion It will that aa finite finite rank free group admitting a free group admitting a multiplication multiplicationofofsimple simplealgebra algebratype typeisisquasi— quasiisomorphictoto aa direct direct sum groupsadmitting admittingmultiplications multiplicationsof of field field type. isomorphic sum ofof groups type. Therefore, the additive additivegroups groups Therefore, upto upto quasi-isomorphism, quasi-isomorphism,the theclassification classification of the of finite type is is settled finiterank ranktorsion torsionfree freerings ringswith withsemisimple semisimple algebra algebra type by the additive additive groups of full by determining determining the groups of fullsubrings subringsofofalgebraic algebraicnumber number fields. fie 1 ds.

be aa finite finite rank Theorem rank torsion torsionfree freering ringwith withsemisimple semisimple Theorem5.3.1: 5.3.1: Let RR be such contains a subring algebra subring sS = s 1 (+) ... .•• fi:)Sm such that S; algebra type. Then RR contains = has simple algebra type, i = = 1, ... ,m, and R+ ;s+ is finite. finite. has simple algebra l,...,m, To prove prove Theorem Theorem5.3.1 5.3.1we wefirst first need: To need: ring with with Let R R be aa torsion torsion free free ring be Q—algebra with unity unity e. Let R; =RnA;. Then Q-algebra with Lemma 5.3.2: Lemma 5.3.2:

and

R+/(Rl

.....

C-:+"J.

.@

f\n) +

R** R

= = A1 ~ 2, and so AA ~ F.

Piecing together the the results results of this Piecing together thissection sectionwe we obtain obtain G be torsion free free group groupofof finite finite rank. G Theorem5.3.11: 5.3.11: Let G Theorem be aa torsion rank. Then G admits multiplication of semisimple typeifif and andonly onlyif: if: semisimple type admits aa multiplication

(1)

G is quotient quotient divisible, divisible, and and

G

k

G C. C±> stronglyindecomposable indecomposable group group which which admits admits aa ® G., with G.1 a strongly i;,;l 1 . such that End(Gi) is. an multiplication of field multiplication field type type End(Gi)' such an algebraic i number field satisfying = n.umber field satisfying [End(Gi): Ql Q] == r(Gi), i = 1, ... ,k. (2)

G

0

Suppose that that G Suppose admits semisimple type. type. Then Then G admits aa multipl1cation multiplication ofofsemisimple G wewemay G is quotient quotient divisible, divisible, Corollary Corollary 5.2.17. 5.2.17. By By Corollary Corollary5.3.3 5.3.3 mayassume assume that GG admits of simple simple algebra algebra type. type. Let RR be be aa ring admits aa multiplication multiplication of

Proof:

102 102

R* a simple Q-algebra. be the the smallest with R+ = GG and R* Z{R*) be simple Q—algebra. Let FF s Z(R*) R* in R. field of be an an F-basis F-basis for R let xxl,...,Xk field of definition definition of of R, and and let R. 1 , ••• ,xk be

Then

kk

_,'Cl) fl (Rn (R n

R R,;=

11=1

F)xi.

*

Put

R. Ri

= =

+

(R (Rnn F)xi (R n F)x., F)xi, and (R

= =

Gi,

and so G. i = l, ..• ,k. Then R. R. ~ FF and admits of type type F, l,...,k. G. admits aamultiplication multiplication of ** 1 1 ** a By Corollary Corollary 5.3.8, R. a. field. By R. = and so F == R. = = 1 = End(G.) 1 and 1 = End(G.) 1 = l,...,k. End(G 1.), i = 1, .•• ,k. The , ••• ,k follows The indecomposability indecomposability of of G., ii == 1l,...,k . * 1 from Corollary Corollary 5.3.10. Now = 1l,...,k. from Now [End(Gi): Q] Q] == dimQRi = r(G1), r(Gi), i = , ••• ,k. 1

1

The converse converse is is obvious. The obvious. 54.

Applications.

Theresults results of of the sectionsofof this this chapter will be The the previous previous sections chapter will be applied applied to to solve problems which arose arose in in Chapter solve problems which Chapter 3, and and to toprove prove Theorem Theorem 4.7.22. k

Proof of of Theorem Theorem 3.2.3: 3.2.3:

be finite rank be aa finite rank torsion torsion free free group, group,

G == @ Gi Let G

i =1 1=1

+

and suppose supposethat that every every ring Ri with R1 R1 Suppose there there and R~ =~ Gi is nilpotent. nilpotent. Suppose exists exists aa ring ring R is not not nilpotent. nilpotent. Then Then by R with R+ = G such that R R Corollary 5.2.23, Corollary 5.2.23, GG &. Hl+lK, HH ~ 0, where HH admits admitsaamultiplication multiplication of semisimple By Theorem 5.3.11(2), & L(T)M, semi simple type. By Theorem 5.3.11 (2), G c.. L® M, with L a strongly L indecomposable group admitting a multiplicationofoffield field type. type. Let indecomposable group admitting a multiplication k. 1 strongly indecomposable, Gi & ~ Gl GiJ"' G.J. strongly indecomposable, i == 1l,...,k; •••• ,k; jj == 1, .•• ,k .. j=l j=l

1

.

1

< k, and By Jonsson's Jonsson's theorem, theorem, [36, Theorem 92.5], Gi j c.. LL for By Theorem 92.5], forsome some 1 ~< i ~ for forsome some 1