Additive Groups of Rings (Chapman & Hall CRC Research Notes in Mathematics Series)

S Feigelstock Bar-II an University

Additive groups of rings

Pitman Advanced Publishing Program BOSTON-LONDON - MELBOURNE

PITMAN BOOKS LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto

© S Feigelstock 1983 First published 1983 AMS Subject Classifications: (main) 20K99 (subsidiary) 16, 17 Library of Congress Cataloging in Publication Data Feigelstock, S. Additive groups of rings. (Research nQtes in mathematics; 83) Bibliography: p. Includes index. 1. Abelian groups. I. Title. II. Series. 512' .2 82-22412 QA171.F34 1983 ISBN 0-273-08591-3 British Library Cataloguing in Publication Data Feigelstock, S. Additive groups of rings.-(Research notes in mathematics; 83) 1. Group rings I. Title II. Series 512'.4 QA171 ISBN 0-273-08591-3 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers.

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

Preface

In 1948 R.A. Beaumont [5] began investigating the additive groups of rings. Shortly thereafter L. Redei, T. Szele [541. and H.S. Zuckerman [11] joined in these investigations, to be followed by L. Fuchs [37], [38]. Since then much progress has been made in this branch of abelian group theory. Many of the results appear in Chapter 17 of L. Fuchs 1 well-known book [36]. An attempt has been made here to offer an easily accessible account of much of the work done on the additive groups of rings, with special emphasis on results not covered in [36]. This short monograph is far from being comprehensive, and many important papers have not been considered here at all. Although this work is meant to be self contained. some knowledge of abelian group theory is assumed. The prospective reader is especially urged to read Chapter 17 of Fuchs• book [36]. I wish to express my gratitude to Professor Elvira Rapaport Strasser for introducing me to this subject, for her advice and guidance during the past fifteen years, and for suggesting that I write this monograph. I also thank Professor L. Fuchs for ~is valuable comments and suggestions.

Contents

1.

Preliminaries: §1: §2: §3: §4:

2.

Definitions and structure theorems Mult G Type Examples

1: The ni 1s tufe of a group §2: Nilpotence without boundedness conditions, and generalized nilpoten~e

25 31

Other Ring Properties: §1: Semisimple, prime, semiprime, simple ring, division ring, field, radical ring §2: Principal ideal and Noetherian rings §3: Descending chain conditions for ideals §4: Subdirectly irreducible rings §5: Local rings §6: Rings with trivial left annihilator, subrings of algebraic number fields, and semisimple rings continued §]. E-rings and T-rings

5.

10 16

Additive groups of Nilpotent and Generalized Nilpotent Rings: §

4.

5 6

Ni 1 and Quasi -Ni'l Groups : §1: Nil groups §2: Quasi-nil groups

3.

4

Torsion §1: §2: §3: §4:

Free Rings: Notation, definitions, and preliminary results The Beaumont-Pierce decomposition theorem Torsion free rings with semisimple algebra type Applications

36

43 50 61 65 68

76

88 90

98 103

Notation

a group. the order of x e G. the torsion part of G. the p-primary component of G, p a prime. {nxlx E G}, n a fixed integer. {x E Gjnx = 0}, n a fixed integer. Ga v(G) N(G) r(G) lS) hlx) hp{x)

the the the the the the the

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

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

[(k 1 ,k 2 , ••• ,kn, •.• )J

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

Z(n)

cyclic group of order n.

Z(pCD) End( G) £nd(G)

p-Prufer group. the group of endomorphisms of G. the group of quasi-endomorphisms of G. quasi-isomorphism. quasi-equality a ring.

Q. :!:

R R+

a-th Ulm subgroup of G, a an ordinal. nilstufe of G. strong nilstufe of G. rank of G. subgroup of G generalized by S =G. height of x E G. p-component of hlx).

the additive group of R.

I

~

Z Q

ZP w

lSI

R

" I is an idea 1 in R" • the ideal in R generated by S c R the ring of nxn matrices with components in R, n a positive 1nteger. · the ring of integers. the field of rational numbers. the ring of p-adi c integers • the first infinite ordinal the cardinality of a set s.

Additional symbols which appear in only one section, will be defined in the section they appear in.

1 Preliminaries

§I.

DEFINITIONS AND STRUCTURE THEOREr·1S

All groups considered in this monograph are abelian, with addition the group operation. Definition: A group G is p-divisible, p a prime, if G = pG. If G is p-divisible for every prime p, then G is said to be divisible. It is readily seen that G is divisible if and only if G = nG for every positive integer n. Definition:

A subgroup H of a group G is p-pure in

G,

p a prime, if

H n pkG = pkH for every positive integer k. If H is p-pure in G for every prime p, then H is said to be pure in G. Clearly H is pure in G if and only i f H n nG = nH for every integer n. Definition: A subset {a; li E I} of a group G is said to be independent if for distinct i1' ... ,ik E I, k an arbitrary positive integer, and for n1, ... ,nk E Z, n1a11 + ... + nkaik = 0 implies that n1a11 = .•. = nkaik = 0. By Zorn's lemma, there exists a maximal independent set M in G containing only elements of infinite and prime power orders. The rank of G, r(G) = IMI. Let M 0 be maximal amongst the independent subsets of G containing only torsion free elements. Then 1r~0 1 is called the torsion free rank of G, denoted r 0 (G). Let t•lp be maximal amongst the independent subsets of G containing only elements of order a power of p, p a prime. Then lt~pl is called the p-rank of G, denoted rptG). r(G), r 0(G),

and

rptG)

are well defined [36, Theorem 16.3].

Definition: A subgroup B of a group G is a p-basic subgroup, p a prime if: 1) B is a direct sum of infinite cyclic groups, and cyclic p-groups. 2) B is p-pure in G, and 3) G/B is p-divisible.

If G is a p-group, then a p-basic subgroup of G is called a basic . subgroup. Every group G possesses a p-basic subgroup [36, Theorem 32.3], p-basic subgroups of G are isomorphic [36, Theorem 35.2].

and all

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

subgroup of G, subgroup of G,

Ga+l = (Ga)l. G8 = n GY.

For 8 a limit ordinal, the

ath

Ulm

y ~ (~ 1 ••••• ~n ... ) if kn = ~n for all but finitely many subscripts n, and kn = ~n whenever kn = ~ or ~n = ~. is an equivalence relation. The equivalence . class of a height sequence (k 1 , •.• ,kn•···) is called a type, and will be denoted [(kp···•kn•··:>L The type of x E G, t(x) = [h(x).J. Definition: The type set of a group G, T(G) = {t(x)l x E G, xI 0}. IT{G)I = 1, then G is said to be a homogeneous group with type t(G) =the singleton belonging to T(G).

If

Definition: A height sequence (k1' ... ,kn, ... ) is idempotent, if kn or ~ for every positive integer n. A type t is idempotent if , possesses an idempotent height sequence.

0

Definition: A·group G is indecomposable if G = H & K implies that H = 0 or K = 0. Definition: A torsion free group G is rigid, if End(G) a subgroup of Q+ .

2

is isomorphic to

Definition: Let G and H be subgroups of a divisible torsion free group D, in lieu of Proposition 1.1 .6, this is not a very severe restriction on torsion free groups G, H. If there exists a positive integer n such that nG ~ H, and nH ~ G, then G and H are said to be quasi-equal, G ~H. If two groups A and B are isomorphic to quasi-equal groups, G, H respectively, then A and B are said to be quasi-isomorphic, A~ B. Definition: A group G satisfying, G & H @ K implies is said to be strongly indecomposable.

H = 0 or K = 0,

Definition: A subgroup H of a group G is essential in for every subgroup K < G, K ; 0.

G if H n K ~ 0

Definition: Let G be a torsion free group, and let D be a divisible torsion free group such that G is an essential subgroup of D; such a D exists, Proposition 1.1.6. An endomorphism ~ E End(D} is a quasi-endomorphism of G, if there exists a positive integer n such that n~ e: End(G}. Definition: A group G is bounded if there exists a positive integer n such that nx = 0 for all x e: G. Definition: A subgroup H of a group G is fully invariant in ~(H) ~ H for every endomorphism ~ of G.

G if

Several structure theorems on abelian groups play a vital role in studying the additive groups of rings. These results will be stated here without proof. However, the reader will be referred to the proofs in Fuchs' book [36). Proposition 1.1.1: .Let Proof:

G be a torsion group.

Then

G=

@

p a prime

Gp.

[36, Theorem 8.4].

Proposition 1.1.2: Let T be a torsion group. T is a direct summand of every group G with Gt = T if and only if T is the direct sum of a divisible group, and a bounded group. Proof:

[36, Theorem 100.1]

Proposition 1. 1.3:

D~

mQ+ m a

Proof:

m

p a prime

A group (I)

D is divisible if and only if a, ap' arbitrary cardinal numbers.

Z( p"") ,

ap

[36, Theorem 23.1]. 3

Proposition 1. 1.4. G = Z(pk) $ H, Proof:

Let

G be a non-torsion free group.

for some prime

p,


t(x), or t(xy) > t(y). Suppose that t(xy) > t(x). Then every element in G is a linear combination of x and xy over the rationals and so t(x) is the unique minimal element in G by Property 1.3.1.2). It may therefore be assumed that (A) X·Y = 0 for all x,y e: R with t(x) i t(y). Suppose that IT(G)I ~ 3. Let x e: G. There exist y,z e: G such that t(x), t(y), t(z) are distinct types. x = ay + bz, with a,b rational numbers. Hence x2 = ~X·Y + bx-z = ~ by (A). Let 0 i x, 0 i y e: R, with t(x) = t(y). If x and y are independent, then every element in G is a linear combination of x, and y over the rationals, and so t(x) is the unique minimal element in T(G) by Property 1.3.1.2). Therefore it may be assumed that there exist non-zero integers n,m such that nx = my. This yields that nmx-y = n2x2 = 0. Since G is torsion free X·Y = 0, or (B) x-y = 0 for all x,y e: R with t(x) = t(y). Clearly (A) and (B) yield that R2 = 0, a contradiction. Therefore (A) implies that IT(G)I ~ 2. Let T(G) = £• 1• T2}, and choose 0 i X; e: G, t(x;) = T;• i = 1,2. Put x = x1 + x2 . If t(x} = Tl then the fact that x2 = x- x1 yields that T 2 = t(x 2 ) ~ Tl by Property 1.3.1.2). Similarly t(x) = T 2 implies that • 1 ~ T 2 • In either case T(G) possesses a unique minimal element. An immediate consequence of the last two lemmas is the following: 13

Theorem 2. 1.7: Let G be a rank two torsion free group. then G is ni 1. For every positive integer G with T(G) = n, [9], and infinite type set [36, vol. 2, not a statement concerning the

If

IT(G) I > 3

n, there exist rank two torsion free groups there exist rank two torsi on free groups with p. 112, Ex. 11 ). Therefore Theorem 2.1.7 is empty set.

Clearly the classification of nil rank two torsion free groups reduces to the case IT(G) I < 3. It is easy to show that the following are necessary conditions for a rank two torsion free group to be a non-nil group [60, p. 204 ]: 1) 2)

3)

IT( G) I IT(G) I IT(G)I

1, 2, 3,

i.e., G homogeneous, t(G) must be idempotent. T(G) must consist of.one minimal type and one maximal type. T(G) must consist of one minimal type, and two maximal types.

One of the maximal types must be idempotent. A classification of the nil rank two torsion free groups with will be given. First some preliminary results are required.

Definition: of G is n(G)

[10], and [60]. {a E

Let G be a torsion free group.

IT(G)j

3

The nucleus

Ql ax E G for all x E G}.

Lemma 2.1.8: [60, Lemma 2.2]. Let G be a torsion free group with r(G) > 1. If End(G) = n(G) then G is ni 1. Proof: Suppose that End(G) = n(G), and let R be a ring with R+ =G. Suppose there exist x,y E R with X·Y ~ 0. Left multiplication by x is an endomorphism of G, hence there exists 0 ~ a E n(G) such that X·Z = az for all z E G, and so x·z ~ 0 for all 0 ~ z E G. Choose z E G such that x and z are independent. Right multiplication by z is also an endomorphism of G, and so X·Z = bx, 0 ~bE n(G). Hence bx = az contradicting the fact that x and z are independent. The above lemma is actually a consequence of [36, Proposition 121.2] and is in fact contained in the statement preceding [36, Proposition 121 .2], namely that rigid groups of rank> 2 are nil.

14

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

Then

G

Proof: It may be assumed, Proposition 1.1.6, that G and H are subgroups of a group G, and that G ~H. Suppose that H is not (associative} nil. Let R = (H,·J be an (associative} ring with R2 ~ 0. There exists a positive integer n such that nG ~ H, and nH ~G. For g1 ,g 2 € G define g1•g 2 = (ng 1} • (ng 2}. It is readily seen that S = (G,•} is an (associative) ring. Let h1 ,h 2 E H such that h1 -h 2 ~ 0. Then (nh 1 )•(nh 2} = n4h1h2 ~ 0. Hence s 2 ~ 0, and G is not (associative} nil. Similarly if G is not (associative) nil, then neither is H. Theorem 2.1.10: Let G be a rank two torsion free group, with jT(G} I = 3. Then G is not (associative} nil i f and only if G.; G1 2, then either ~nd(G} = Q, or G is quasi-isomorphic to the direct sum of two rank one groups of incomparable types. If End(G} = Q, then G is a rigid group, and so G is r.il [36, comment preceding Proposition 121.2). If G is quasi-isomorphic to the direct sum of rank one groups of incomparable types, then it is readily seen, [9, remark preceding Theorem 8.5), that IT(G}I = 3.

15

§2.

Quasi-nil Groups:

The group Q+ admits precisely two non-isomorphic ring structures, the zeroring, and Q. Szele [68, Satz 1] classified the groups allowing precisely two non-isomorphic ring structures. He conjectured that every non-nil torsion free group G ~ Q is the additive group of infinitely many non-isomorphic associative rings. This was shown not to be true by L. Fuchs, [38]. Borho, [12], showed that Szele's conjecture was not far from being true, see Theorem 2.2.4 and Corollary 2.2.5. The investigation of nil groups, and additive groups of precisely two non-isomorphic rings, initiated by Szele, was generalized by Fuchs as follows: Definition: A group G is (associative) quasi-nil if there are only finitely many non-isomorphic (associative) rings R, with R+ = G. Fuchs classified the torsion (associative) quasi-nil groups completely, and made considerable progress towards classifying the torsion free and mixed (associative) quasi-nil groups, [35], [38]. A complete description of the torsion free (associative) quasi-nil groups which are not nil was obtained by W. Borho, [11]. This in conjunction with the results of Fuchs yields a classification of the (associative) quasi-nil groups which are not nil. The main results of this section are due to Fuchs, and Borho, [12], [35], [38]. The following technical lemma is generally useful in studying the additive groups of rings: Lemma 2.2.1:

Let R be a ring with

R+ =G.

Then G1 annihilates

Gt.

Proof: Let 0 ~ x € Gt• y € G1 , and let lxl = n. Since y € nG, y = nz, z € G. Therefore xy = x(nz) = (nx)z = 0. Similarly yx = 0. Theorem 2.2,2: 1) 2) 3)

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

Proof: 16

Let G be a torsion group.

Clearly

1) • 2):

~

The following are equivalent:

a divisible torsion group.

2)•3): Supposethat G isassociativequasi-nil. G=H(i)D, H reduced, and D divisible. Hp is a direct summand of G for every prime p, Proposition 1.1.1, i.e., G = Hp (f) KP. For every prime p with Hp I 0, let Sp be an associative non-zeroring, with s+p = Hp , Theorem 2.1. 1, and let TP be the zeroring on KP. The ring direct sum Rp = SP@ TP satisfies R; = G, R~ I 0, and Rp + Rq for distinct primes p,q. Hence H = 0 for all but finitely many primes. It therefore suffices to show p . that Hp is finite for every prime p. Let Bp be a basic subgroup of n.

with basis {a.li € 1}. Put la-1 = p 1 , i € I. Every set of p 1 1 associative products a .. a. with la.·a-1 < min(la-1, la-1), i,j € I, 1J 1J1 J induces an associative ring structure S on Hp' [36, Theorem 120.1] and remarks following the theorem. Suppose that I is an infinite set. For every positive integer n, choose distinct elements ik € I, k = l, ... ,n.

H ,

..

For 1,J € I,

.

def1ne

ai·aj

ro

=·.. a;

i f i I j or i = j = ; k, k = 1 , ... , n i f i = j, and ; ¢ {il' i 2 , ... ,in}

These products induce an associative ring structure Un on Hp with annihilator of rank precisely n. Hence Un +Urn for distinct positive integers n,m. Again the ring direct sums Un@ Tp' TP the zeroring on KP, yield infinitely many non-isomorphic, associative ring structures on G. Therefore BP is finite, which implies that Hp = BP (f.) Ep, with EP a divisible group. Since HP is reduced, HP = BP, and so HP is finite. 1): Let G = B(+l D, B a finite group, D a divisible torsion group. By Lemma 2.2.1 every ring R with R+ = G satisfies RD =DR= 0, and for every prime p, B2p -c Bp {+) Dp . It may therefore be assumed that G is a p-group. Now t·1ul t G .,. Hom(~. G) ... Hom(B'!IlB, G), [36, Theorem 61.1]. Let r(B) = m. Then r(B~) = m2 • Let IBI = pn. Every ~ € Hom(B~. G) is a homomorphic map ~= 85 1. Choose three independent elements bi E G, i = 0,1 ,2, with o0 E Q, b1 ,b 2 E H. Put G = G3Q. Now G may be viewed as a vector space over Q, with a natural embedding of G into G such that b0 , b1 , b2 are contained in a subspace G1 = b0Q 4) b1Ql+l b2Q of G. Let A = (aij) E M2{Q), i ,j = 1,2 be an arbitrary 2x2 matrix over Q. The products for

i

for

i = 0 or j = 0

,j

E {1 ,2 }

induce an associative Q-algebra structure (Gl)A on G1 . A Q-algebra structure AA on G may be obtained by defining all products of an element in G with. an element in a complement of G1 to be zero. The fact that b0Q ~ G yields that the above products induce an associative ring structure RA with R: = G. Hence every matrix A E 14 2(Q) determines an associative ring RA' with RA+ = G. Let 0 1 fl. = {a .. ), 0 1 B = (B .. ) E M2(Q} with lJ lJ RA.,. R8 , and let ~= RA ~ R8 be a ring isomorphism. ~ is also an automorphism of G. Since H is reduced, ~(b 0 ) E Q, i.e., ~(b 0 ) = r 0b0 , 18

of

r 0 E Q. Now ~ extends naturally to an algebra isomorphism ~: AA ~ A8 • Since ~ restricted to (G1 )A is a regular linear transformation 2

q): (Gl)A ~ (~ 1 J 8 , q}(bi) = .E riJ.bJ., riJ. E Q, J=O

i = 1 ,2;

j

= 1 ,2.

Put P = (r-J.) E M2(Q), i ,j = 1 ,2. The last identity can be written as 1 r 0A = PBP t . Therefore the determinant 1r0A1 = r 02 1AI = IPI 2 IBI, and so IAI = s 2 1BI, sEQ. It suffices to show that the equivalence relation in Q, x ~ y if and only if x = s 2y, s E Q, determines infinitely many equivalence classes. Suppose that the above relation partitions Q into finitely many equivalence classes. For x E Q let x be the equivalence class of x. There exist finitely many rational numbers r 1 , ••• ,rk such that for every rational number r, there exists 1 ~ i ~ k such that r = ri. For every integer n, the set of square roots of elements in n is precisely the field Q(lin). Therefore there are at most k quadratic number fields, a contradiction. Therefore r(HJ ~ 1. If H is not nil let S be the zeroring on Q+, and let T be a non-zeroring on H. Put R = S (-f) T. Then (R2 )+ = (T 2) c H is not divisible, contradicting Lemma 2.2.3. Conversely, if G is nil or if G ""Q + , then G is clearly quasi-nil. Suppose that G = Q+ (f) H. H a nil rank one torsion free group. Let R = (G,•) be a ring, and let nH be the natural projection of G onto H. Define axb = nH(a·b) for all a,b E H. This composition defines a ring structureon H. Since H isnil, nH(a·b)=O forall a,bEH, i.e., H2 ~ Q+. This coupled with the fact that, Q+, the maximal divisible subgroup of G, is an ideal in R, yields that R2 ~ Q+. Choose 0 f b1 E Q, 0 f b2 E H. The multiplication in R is determined completely by the products b .• b. = cx .. b1 , ex •• E Q, i,j = 1,2. For every matrix 1 J lJ lJ A = (cxij) E f,12 (Q) the above products induce a ring structure RA on G. As above, if two such rings RA and R8 are isomorphic, then IAI = s 2 1BI, s E Q, and so there are infinitely many isomorphic classes of ring structures on G. However if the ring RA is required to be associative, then the equalities (bibj)b 1 = bi(bjb 1 ), and (b 1bi)bj = b1 (bibj) yields that 19

(A)a;jall = ailajl and (B) alialj = a;jall for i ,j = 1,2. Consider two cases a) a11 = 0, b) a11 1 0. a) Suppose a11 = 0. Putting i = j in equalities (A), (B) yields that

a~ 1 = a~; = 0, and so ail = a1; = 0 for i = 1 ,2. Therefore A = (~ :) , a € Q. Conversely every such matrix determines an associative ring RA. Again map G into a vector space t' = b1Qffi b2Q. The products in RA induce a Q-algebra structure GA on G, with RA a subring of GA. The mops

b1

~ .- 1b1 ,

~ b2

b2

induce an algebra isomorphism

~·

GA ~ G(~

~)

Since G~ ~ RA' the restriction of ~ to RA is an isomorphism

~: RA ~ R(~ ~)

Therefore every associative ring

isomorphic to l~ ~)

R with

R+ = G is

or R(~ ~) •

b) Suppose that a11 ~ 0. Without loss of generality it may be assumed -1 b • Choosing i = 1, that a 11 = 1, because b1 may be replaced by a 11 1 j = 2 in (A) yields that a12 = a21 = a, and so choosing i = j = 2 in

:~

i.

(A) yields that a22 = Hence if RA is associative, then A=(: The maps b1 ~ b1 , b2 ~ b2 - ab 1 induce a group automorphism of G. Now b~ = b1, (b 2 - ab 1) 2 = 0, and b1(b 2 - ab 1) = (b 2 - ab 1)b 1 = 0. Hence ~= RA ~ R(~ ~) is a ring isomorphism. Corollary 2.2.5:

Let G be a torsion free group.

1) G is the additive group of precisely two non-isomorphic (associative) rings if and only if G ~ Q+. 2) G is· the additive group of precisely three non-isomorphic associative rings if and only if G ~ Q+~ H, H a nil rank one torsion free group. In this case every associative ring R with R+ = G is isomorphic to one of the rings

\g g) ,

Theorem 2.2.4, and G 20

R(~ i~

g) , R{g

~)

constructed in the proof of

the additive group of infinitely many,

•

non-isomorphic, non-associative rings

R.

3) Otherwise G is either nil, or the additive group of infinitely many non-isomorphic associative rings.

The proof of Theorem 2.2.4 remains valid considering only commutative rings R with R+ = G by choosing symmetric matrices A in the construction of the rings RA. Therefore: Corollary 2.2.6: A non-nil torsion free group G is the additive group of only finitely many non-isomorphic commutative, associative rings if and only if G ~o+, or G ~Q+(!)H, H a nil rank one torsion free group. Theorem 2.2.7: Let G be a mixed group. and only if either: G = B(t)H, B a finite group, group. 1)

G is (associative) quasi-nil if

H a torsion free (associative) quasi-nil

2) G = B(f) D@ H, B a finite group, D a divisible torsion group with Dp = 0 for all but finitely many primes p, H = Q+, or H a nil rank one torsion free group, satisfying pH = H for every prime p with Dp I 0. Proof: Suppose that G is a mixed (associative) quasi-nil group. Decompose Gt into B@ D, B reduced, and D divisible. G has a direct summand Z(pk), 0 < k < =, for every prime p for which BPI 0, Proposition 1. 1.1, Proposition 1.1.4, and Proposition 1 .1.3. The direct sum of the ring of integers modulo pk on Z(pk) and the zeroring on a complement of Z(pk) yields an associative ring Rp with R; = G. For primes pI q, Rp Rq. Hence Bp = 0 for all but finitely many primes p. The same argument used in proving the implication 2) • 3) in Theorem 2.2.2 shows that B is finite. Therefore G = B~D@H, H a torsion free group, Proposition 1.1.2. Clearly H must be (associative) quasi-nil.

+

Suppose that D I 0. Let p be a prime for which Dp I 0. If pH I H then for every positive integer n, pnH is a proper subgroup of H. Therefore there exists an epimorphism ~= H ~ Z(pn) (the composition of the canonical homomorphism H ~ H/pnH and a projection of H/pnH onto Z(pn)). Let dE D with Jdl = pn, and view ~ as an epimorphism of H onto the subgroup of D generated by d. Let X; E G, X; = b; + d; + hi, b; E B, 21

di E 0, hi E H, i = 1 ,2. Define x1 -x 2 = k1k2d, With kid = ll>(h;), i = 1 ,2. These products induce an associative ring structure Rn on G, in fact Rn3 = 0. Now (R2n)+ ~ Z(pn), and so Rn Rm for positive integers n ~ m. Hence there are infinitely many non-isomorphic associative rings with additive group G, a contradiction. Therefore pH = H for all primes p for which Dp ~ 0.

+

To show that required. Let

r(H) =

some facts about the endomorphisms of

Z(p~)

Z(p~)

positive integer n) p-adic integer.

For every positive integer n,

define

n(an)

This action of n on a set of generators of Z(p~) extends naturally to an endomorphism n E End(Z(pj). In fact all the endomorphi sms of Z(pj are of this type, [36, vol. 1, p. 181, Example 3]. The automorphisms of Z(p~) are clearly p-adic units, [36, vol. 2, p. 250, Example 3]. Suppose that r(H) > 1. Choose independent elements 0 ~ u, 0 ~ v E H. Every element x E G is of the form x = b + d + ru + sv + w, b E B, dE D, r, s E Q, and WE H. Let X; = bi + d; + r 1 u + siv + W;, i = 1,2 be elements of G written in the above form. The products x1 -x 2 = (r1r 2 + s 1s 2n)a 1 induce an associative ring structure Rn on G. Put pwG

=

~

n pnG, n=l x = b + d + ru + sv ( p-n x) 2 = p-2n (r2 +

and let x E pwG.

Write

x in the above canonical form,

+ w. For every positive integer n, s 2n)a 1 = (r 2 + s 2n)a 2n+l" Let n(p wG)

be the set of

p-adic integers {r2 + s 2nl obtained above for each x E pwG, and every positive integer n. Then n(pwG) is a countable set of endomorphisms of Z(p~) containing the identity automorphism. If R ~ R for two p-adic n p integers n,p then the elements of ~(pwG) are the elements of n(pwG) multiplied by a fixed automorphism of Z(p~), i.e., by a p-adic unit However the identity automorphism must belong to (pwG), and so a must be chosen from a countable set of p-adic units. Therefore the isomorphism class of Rn is countable. The non-denumerability of the p-adic integers yields that there are infinitely many non-isomorphic associative rings Rn with R+n G, a contradiction. Hence r(H) = 1. 22

Let p be a prime for which Dp ~ 0, and let 0 ~ d E Dp • Let 0 ~hE H. For x1 ,x 2 E G, put x. =b.+ d. + r.h, b. E B, d1. ED, 1 1 1 1 1 r; E Q, i = 1 ,2. The products x1x2 = r 1r 2d induce an associative ring structure Rp on G. For primes p ~ q, RP Rq. Therefore Dp = 0 for all but finitely many primes p.

+

Conversely, suppose that G is of form 1). t4ult G ""Hom{~. G) ... Hom{G~. B)(+)Hom{H~. H), since Hom(X,Y) = 0 for X a torsion group, and Y a torsion free group. If G is quasi-nil then so is H. Therefore Hom{H~. H) ""'r~ult H determines only finitely many non-isomorphic ring structures on G. Hom{G~. 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 so the first three summands above are finite. Since r{~H} = 1, Proposition 1.3.3, and B is finite, Hom{H·~. B) is finite. For the associative case consider G = B t:t) H, B finite, H an associative quasi-nil torsion free group. Let lSI = m. The ring multiplications on G are determined by products of elements in B, for which there are clearly only finitely many possibilities, products of elements in B with elements of H, and products of elements in H. Since mH annihilates B in every ring on G, the products of elements in B with elements of H are determined by the products of elements of B with coset representatives of elements of H/mH. Since r{H) ~ 2, Theorem 2.2.4.1}, H/mH is finite, and there are only finitely many ways of defining products of elements of B with elements of H. Let R = (G,·) be an associative ring, and let al'a 2 E H, a 1·a 2 = a3 + b, a3 E H, bE B. If a1 is replaced by an arbitrary element in the coset a1 + mH, i = 1,2, then the element ab above remains the same. Since there are only finitely many such cosets, it suffices to show that G/B is associative quasi-nil. However G/B ""'H is associative quasi-nil. Suppose that G is of the form 2) with H nil. As in the case of form 1) it suffices to investigate Hom(~. G). Arguments similar to those used (±) Hom(~, D } in case l} reduce the investigation to Hom(H~, D) ""' p prime P Since Dp = 0 for all but finitely many primes p, it suffices to consider Hom(H~, DP), p a fixed prime, with OP ~ 0. Now r(HSltt) = 1, and so 23

every ~ € Hom(HAH, Dp) is a map ~= ~H ~ Z(p=). Changing the summand Z(p=) only alters the ring determined by ~ by a ring isomorphism. Therefore it suffices to examine Hom(H~. Z(p9j). However Homl~. Z(p~) is isomorphic to the additive group of p-adic numbers. If u1(x,y) and u2(x,y) are two ring multiplications on G with u1 a non-zero multiplication, then u2(x,y) = wu 1(x,y), w a p-adic number, for all x1,x2 € G. Multiplication by p is an automorphism of Z(p=). Therefore, the multiplications u1(x,y) and p- 2Pu 1(x,y), p a p-adic unit, define isomorphic ring structures on G. Every non-zero p-adic number is of the form pkw, k an integer, w a p-adic unit. Therefore the elements of Hom(~. Z(p=)) induce three non-isomorphic ring multiplications on G, the zero multiplication, u1(x,y), and Pu 1(x,y). If H is no·t nil, then H = Q, and it suffices to examine the multiplications determined by Hom(~. D(!) Q) ""Homl~. D) Hom(Q~. Q). The second summand is isomorphic to Mult Q, and therefore determines two non-isomorphic multiplications on G. The same argument employed above to show that Hom(tfSJi, D), H nil, determines only finitely many non-isomorphic ring multiplications on G, shows the same for Hom(~. D). This proves Theorem 2.2. 7.

24

3 Additive groups of nilpotent and generalized nilpotent rings §1.

The nilstufe of a group. Another generalization of nil groups, also due to Szele is the following:

Definition: Let G ~ 0 be a group. The greatest positive integer n such that there exists an associative ring R with R+ = G, and Rn ~ 0, is called the nilstufe of G, denoted v(G). If no such positive integer n exists, then v(G) = ~. The strong nilstufe of G, N(G) is defined as above, with the associativity of R deleted. Clearly the (associative) nil groups are precisely the groups satisfying (v(G) = 1) N(G) = 1.

G

The following examples are due to Szele [68, Satz 2]: Example 3.1.1: For every positive integer n, group G satisfying v(G) = N(G) = n.

there exists a torsion free

For every positive integer i,

let G. be a rank one torsion free group n , oftype [(i,i, ••• ,i, .•• )]. Put G= (i;) G.• Clearly Rn+l =0 forevery i=l , ring R with R+ = G. For each 1 < i < n choose ei E Gi such that h(e;) = (1,i , .•• ,i, ••• ). The products

ei ·ej =

e.+. , J

for

i+j -< n

0

for i+j > n

n induce an associative ring structure R on G. Now e~ =en~ 0, and so Rn ~ 0, i.e., v(G) = n. Clearly v(G) ~ N(G) < n+l, and. so N(G) = n. ~

i, j

! r

~

Example 3.1.2: For every positive integer n, G satisfying v(G) = N(G) = n+l. For every positive integer i,

let Gi

there exists a mixed group

be a rank one torsion free group

25

of type

[(co,i,i, •.• ,i, ••• )].

n Put G = Z(2co) (i) G;·

Clearly Rn+l ~ Z(2co)

i =1

for every ring R with R+ = G. However Z(2co) annihilates R, and so Rn+ 2 = 0, i.e., N(G) ~ n+l. Choose 0 ~a € Z(2co), and ei € Gi with h(ei) = (co,i ,i, ••• ,i, ••• ). The products

r·j

ei·ej =

for i+j

~

n

a

for i+j = n

0

for i+j

>

n

for ~ i, j < n induce an associative ring structure R on G. Now n+l e 1 = a ~ 0. Hence n+l = v(G) ~ N(G) ~ n+l, and so v(G) = N(G) = n+l. Conjecture 2.1.4 can be restated as follows: For every group G, v(G) = 1 i f and only i f N(G) = 1. It is not true in general, that v(G) = N(G) for every group G. Examples will be given following Corollary 3.1.6. Theorem 3.1.3:

Let G be a torsion free group with R+ = G.

be a nil associative ring with

r(G) = n, and let R

Then Rn+l = 0.

Proof: Let 0 ~ x € R, and let m be the smallest positive integer such that xm = 0. Suppose that m > n+l. Then there exist integers m-1 · al' ••• ,am-l' not all zero, such that I: a.x 1 = 0, i.e., i =1

am-lx

m-1

m-2 - -

a xm-1 = m-1

i aix .

I:

i =1

If a1

= 0,

1

= 1 , ••• ,m-2,

then am- 1

~

0,

and

o. Since G is torsi on free xm-1 = 0, a contradiction.

Therefore a;

~

0 for some

. 1

i.e., a xmm-2

=

~

i ~ m-2.

m-2 i+l Now 0 = a xm = - I: a.x m-1 i=l 1

m-3 . 1 I: a.x 1 + • Repeating the above procedure m-3 more time . 1 1= 1

yields that a1xm-l = 0, with a 1 ~a. Hence xm-l = 0, a contradiction. Therefore xn+l = 0 for all x € R. By the Nagata-Higman Theorem 26

[48, p. 274], R is nilpotent. Let d be the smallest positive integer satisfying Rd = 0. For every positive integer k, let (O:Rk) = {x E Gl xy = 0 for all y E Rk}. Clearly (O:Rk) is a pure subgroup of G for every positive integer k, and the chain 0 < (O:R1) < (O:R 2 ) < .•. < (O:Rd-l) = G is properly ascending.

d

~

n+l,

Hence

[36, vol. 1, p. 120, Ex. 8 (b)]. r(G) = n
t(g·g ) > t(gnJ = t(G ), a contradiction. 1 n n Hence RG nc G , and similarly G R c Gn, i.e., G is an ideal in R. n n_2n-l n 2n- 1 Put R= R/Gn. By the induction hypothesis R 0, or R c G - n 2n 2 . n However Gn isnii,Theorem2.1.2,andso R =Gn=O, 1.e., v(G)~2-l. The conditions in Theorem 3.1.12 do not impose a bound on NtG) as was seen in Example 3.1.8. n Corollary 3.1.13: Let G = _@ G; , wi th Gi a rank one torsion free group, 1 =1 i = l, •.• ,n. The following are equivalent. t(Gi) is not idempotent for all I < i < n. 2) v(G) ~ n. Proof: 1) • 2): v(G) < 2n-l by Theorem 3.1.13, and so v(G) < n by Corollary 3.1.4.

1)

30

If t(Gi) is idempotent for some 1 ~ i ~ n, then additive group of a subring of Q, 1.48, and so v(G) = m.

2) •1):

§2:

Gi

is the

Nilpotence w1thout boundedness conditions, and generalized nilpotence.

In the previous section, groups G were considered for which there exists a positive integer n such that Rn = 0 for every (associative) ring R with R+ = G. In this section the bound on the degree of nilpotence will be deleted. Afterwards, we will go a step further by considering the generalized nilpotent rings which were introduced by Levitzki [50]. Theorem 3.2.1: lW.J. Wickless) [74J: Let G be a group. Every (associative) ring R with R+ = G is nilpotent if and only i f G = D~ H with D a divisible torsion group, and H a reduced torsion free group such that every (associative) ring S with s+ = H is nilpotent. Proof: Suppose that every (associative) ring R with R+ = G is nilpotent. If Gt is not divisible, then G has a non-trivial cyclic direct summand, Proposition 1.1.4 and Proposition 1.1.3, and so a non-nilpotent associative ring structure may be defined on G. Therefore G = D~ H, D divisible, and H torsion free, Proposition 1.1.2. Clearly every (associative) ring S with s+ = H must be nilpotent. If H is not reduced, then Q+ is a direct summand of G, Proposition 1.1 .3 and [36, Theorem 21.2], and so a non-nilpotent (associative) ring structure can be defined on G, a contradiction. Conversely, suppose that G = D~ H, D a divisible torsion group, H torsion free, and such that every (associative) ring S with s+ = H is nilpotent. Let R be an (associative) ring with R+ =G. Clearly D is an ideal in R, and R = R/D is nilpotent. Therefore there exists a positive integer n such that Rn = 0, or Rn c D. Hence R2n c o2 = 0, Theorem 2.1.1. Theorem 3.2.1 reduces the study of groups allowing only nilpotent (associative) ring structures to the torsion free case. An immediate consequence of Corollaries 3.1.4 and 3.1.6 is the following: Observation 3.2.2: (associative) ring

Let G be a torsion free group with r(G) < R with R+ = G is nilpotent if and only if

m.

Every

31

(v(G)

< ~)

N(G)

< ~.

The proof of the following theorem of W.J. Wickless [74, Corollary to Theorem 2.1] depends on results .of R.A. Beaumont and R.S. Pierce [7], (8], and will be given in Chapter 5 section 4. Theorem 3.2.3: Let Gi be a finite rank torsion free group such that every associative ring Ri with R:1 = G.1 is nil potent, = l ••.. ,k. Then every k associative ring R with R+ = (f) G.1 is nilpotent. i=l Corollary 3.2.4:

Let Gi

be a finite rank torsion free group with k

i = l , ••• , k • Then v((i;) G.) T is an associative ring satisfying R+ = G, and R2 '! 0. Since T is an ideal in R, T =<X>. Clearly K = T+ = (x). Therefore K is not associative nil. The above argument, interchanging the roles of H and K, yields that H is cyclic. Corollary4.2.2: Let G=H(+IK, HIO, K-!0 beanassociativestrongly principal ideal ring group. Then H and K are cyclic. Proof: It suffices to negate that H and K are both associative nil. R = (G,·) be a ring satisfying R2 '! 0. 1) R= and and

=

Suppose that R2 K. There exist h0 E H, k0 E K, such that • Let h E H. Since h € R, there exists an integer n, x E R2 such that h = n(h 0 + k0 ) + x. However x E K. Hence h = nh 0 , H is cyclic, contradicting the fact that H is associative nil.

2) Suppose that R2 ! K. For all g1 ,g 2 E G, define g1 * nH the natural projection of G onto H. Then S = (G,*) satisfying s 2 =H. The argument employed in 1) yields that contradicting the fact that K is associative nil. Theorem 4.2.3: 1) 2) 3)

Let

Let G '! 0 be a torsion group.

g2 = nH(g 1·g 2 J, is a ring K is cyclic,

The following are equivalent:

Either G is cyclic, or G""' Z(p)@ Z(p), with p a prime. G is a strongly principal ideal ring group. G is an associative strongly principal ideal ring group.

1) • 2): Non-trivial cyclic groups are clearly strongly principal ideal ring groups. Suppose that G = (x 1 ) (f) (x 2 ) with lx;l = p a prime, i = 1,2. Let R be a ring with R+ = G, and R2 '! 0, and let I be a proper ideal in R. Then III = 0, or p, and so I is generated by a single element. We may assume that R 'I <X;>, i = 1,2. Hence <X;>+= (xi) for i = 1 ,2. This implies that

~:

43

X· X.

J

l

k.x., 0 -< k.l < p if i = j, i l l {. = 0 if ; ~ j, ; ,j = 1,2.

= 1,2, either k1

~

or k2

0

~

0.

Put I = <x 1 + x2>. Suppose that k1 ~ 0. Let r,s be integers such that rk 1 + sp = 1. Then rx 1(x 1 + x2) = rk 1x1 = (l-sp)x 1 = x1 • Hence x1 € I, and so (x 1 + x2 ) - x1 = x2 E I. Therefore I = R. If k2 ~ 0, then the above argument, reversing the roles of the indices 1 ,2 again yields I = R. 2) • 3): It suffices to show that G is not associative nil. case by Theorem 2.1.1.

This is the

3) • 1): Suppose that G is an associative strongly principal ideal ring group. . If G is indecomposable, then G = Z(pn), p a prime, 1 -< n -< ~. Corollary 1.1.5. If n = ~. then G is divisible, Proposition 1.1.3, and so G is nil, Theorem 2.1.1, a contradiction. Hence G is cyclic. Suppose that G = H(t) K, H ~ 0, K ~ 0. By Lemma 4.2.1, either H and K are both cyclic, or both associative nil. If H and K are both associative nil, then they are both divisible, and so G is nil, Theorem 2. 1.1, a contradiction. Therefore G = (x 1) (f) (x 2 J, with lx; I = n1 , i = 1,2. If (n 1 , n2) = 1, then G is cyclic. Otherwise let p be a prime divisor of

m.

(n 1 ,n 2). Then G = (y 1) (t) (y 2 )@ H, with IY; 1 = p 1 , i = 1,2, and 1 ~ m1 ~ m2 • Since lY 1) (t) (y 2 J is neither cyclic nor associative nil, H = 0 by Lemma 4.2. 1. m -1 The products Y;·Yj = p 2 y2 for i,j = 1,2, induce an associative ring structure R on G with R2 ~ 0. Therefore R = <s 1y1 + s 2y2>, s 1 ,s 2 Every element x E R has the form m2-1 kxs 1y 1 + (kxs 2 + mxp )y2 , kx,mx integers.

integers. x

=

_

_

In particular

m2-1

y 1 - k s 1y 1 , and y2 - (k s 2 + m p )y 2 • Hence if m2 > 1, yl y2 y2 m -1 k s 1 = l(mod p), and k s 2 + m p 2 : l(mod p), which implies that yl y2 y2 m -1 p k , and p s 2 • However k s? + m p 2 = O(mod p), so either yl yl "" yl

t

plky 1 or pls 2 , 44

t

a contradiction.

Therefore m2

= 1 = m1•

Theorem 4.2.4: ring groups.

There are no mixed (associative) strongly principal ideal

Proof: Let G be a mixed associative strongly principal ideal ring group. G is decomposable, Corollary 1.1.5, so by Lemma 4.2.1, G = H(i) K, H ~ 0, K ~ 0, with H and K both cyclic, or both associative nil. 1) Suppose that H and K are both associative nil. There are no mixed associative nil groups, Theorem 2.1.1, so we may assume that H is a torsion group, and that K is torsion free. Let R be an associative ring with R+ = G, and R2 1 0. Clearly H is an ideal in R, and so H = . Let lhl = m. Then mH = 0. By Theorem 2.1.1, H is divisible, and therefore not bounded, a contradiction. 2) Suppose that H = (x), and K =(e) with lxl = n < ~. and lei = ~. The products x2 = xe = ex = 0, and e 2 = ne induce an associative ring structure R on G satisfying R2 ~ 0. Therefore there exist integers s,t such that R = <sx + te>. Every y € R is of the form y = mysx + (my + uyn)te, with my and uy integers. In particular, (me+ uen)t = I. Hence t = .:t. 1. Therefore mx + uxn = 0, so that nlmx. However x = mxsx = 0, a contradiction. Theorem 4.2.5: Let G be a torsion free associative strongly principal ideal ring group. Then G is either indecomposable, or is the direct sum of two associative nil groups. Proof: By Lemma 4.2.1 it suffices to negate that G = (x1)@ (x 2 ), i = 1,2. Suppose this is so. The products for i = j ,

i = 1 ,2

for i

i ,j

~

j,

xi

~

0,

1 ,2

induce an associative ring structure R on G satisfying R2 ~ 0. Therefore there exist nonzero integers k1 ,k 2 such that R = . Every x € R is of the form x = (rx + 3sx)k 1x1 + (rx + 3tx)k 2x2 , with rx, s , t X

X

integers.

r

XI

+ 3s

Xl

=

+1,

-

so that

rx

1

= -+l(rood

3).

However

+ 3t = 0, so that r = O(mod 3), a contradiction. xl xI xl Lemma 4.2.6: Let G and H be tors1on free groups with G ~H. Then G is an (associative) strongly principal ideal ring group if and only if H is. r

45

Proof: It may be assumed, w1thout loss of generality, that G; H. There exists a positive integer n such that nG ~ H, and nH ~G. Suppose that G is an (associative) strongly principal ideal ring group. Let R = (H,·) be an (associative) ring such that R2 I 0. The products g1*g 2 = (ng 1)·(ng 2 ) for all gl'g 2 E G, induce an (associative) ring structure S = (G,*). with s2 I 0. Let I .

corollary 4.2.9: Let G be a mixed group. If G is an (associative) principal ideal ring group, then Gt is bounded, and G/Gt is an (associative) principal ideal ring group. Conversely, if Gt is bounded, and if there exists a unital (associative) principal ideal ring with additive group G/Gt' then G is an (associative) principal ideal ring group. Proof: Let R be an (associative) principal ideal ring with R+ = G. Since Gt is an ideal in R, Gt = <x>, and nGt = 0, n = 1x1. Now G = Gt®H, H ~ G/Gt' Proposition 1 .1.2. Now R =, a E Gt, 0 ~ y E H. Suppose that R2 c Gt. Let h E H. There exists an integer kn such that 2 . 2 h = khy + b, with bE R . S1nce R Gt' b = 0, and h = khy. Therefore H = (y). Clearly H is an associative principal ideal ring group. If R2 $ Gt' then R = R/Gt is an (associative) principal ideal ring with ....+-2 R ~ G/Gt' and R ~ 0.

=

Conversely, suppose that Gt is bounded, and that there exists a unital (associative) principal ideal ring T with T+ = G/Gt. Then G"" Gt G/Gt' Proposition 1.1.2. There exists a principal ideal ring S with unity, such that S+=Gt' [36,Lemmal22.3]. Let R=S(±)T, with e,f theunities of S and T respectively. Let I be an ideal in R. Then I = (I n S) (t) (I n T). Now Ins . Similarly I n T = . Clearly <X+ y> =I. However x = e(x+y) E <X+ y> and y = f(x+y) E <X + y>. Therefore I = <X + y>. Noetherian rings are rings satisfying the ascending chain condition for left or right ideals. The following is a departure from convention: Definition: A ring R will be said to be Noetherian, if every two sided ideal is finitely generated. The following trivic.l lemma will prove to be useful. Lemma 4.2.11. Let G = H(f) K, H ~ 0, K ~ 0 be a strongly Noetherian ring group. Then either G is finitely generated, or H and K are both nil. ~: Suppose that H is not nil. Let S be a non-zeroring with s+ = H, and 1et T be the zerori ng with T+ = K. The ring direct sum R = S (f) T satisfies R+ = G, and R2 ~ 0. Since T k.

Put

m = k!

(mR)+ ""(t)Q(!) @ (£) Z(p00 ) . a p a prime ap

divisible, and so boundedjandso

for all

n:;, ...

R ::J 2! R ::J 3!

R be an Artinian ring.

Clearly

(mR)+

Clearly,

R+""(±)Q+(±) (£) ~ Z(p00)(f{i~)Z(p~). 13 J a p a prime ap

p. J

is

R+/(mR)+

is

primes,

and

p~lm,

m a fixed positive integer, Proposition 1.1.9. Put ~ (+) Z(p00) . Clearly, Dc(R+)l, andso D annihilates D= p a prime ap Lemma 2.1.

The complement of

is clearly annihilated by and so every subgroup of

D.

R;

Hence

co

1

i=2

is isomorphic to (+)Q+ which a D belongs to the annihilator of R,

Di ,

R.

If

i = 1 ,2, •.• ,

D possesses infinitely tnen

00

D. :;, (f) D. :;,, • ,:;, (f) D. :;, • • • i=l

R+

D is an ideal in

many non-trivia 1 direct summands 00

in

1

of (left) ideals in

i=n R,

is. a properly descendir.g infinite chain

1

a contradiction.

Therefore

Z(p~),

®

D=

finite

primes. k

Conversely, let

k p.J m, .where

p1.

0

0

G ""® Q+ ('±) Z(p~) (f) Z(p .J) 1 a finite J

1

with

J

a,l3

are arbitrary cardinals; pi ,pj are primes; and m is a fixed positive integer. Let F be a field with F+ ""(+) Q+, S a zeroring with

a

S+ =

p~lm; 50

(±) Z(p~). 1 finite j

l, ... ,n;

k. Now (f) Z(pJ.J) 13 k

l, ..• ,nj.

n n. (+) ~ (£J j'=l k=l 13-

a prime, and

J

For

13j

finite, let

Tjk

be the sum of

~J·

copies of the ring of integers modulo p~. For ~- infinite, an J J associative commutative ring with unity Tjk can be constructed so that T;k =(f) Zlp~). and the only (left) ideals in Tjk are j n. n n. TJ.k' J:TJ.k•···•P JTJ.k = 0, [36, Lemma 122.3]. Put T = C±) d:> TJ.k" j=l k=l ring R = F(+) S 0 T is Artinian with R+ = G.

The

The ring R constructed above is commutative, associative, and possesses only finitely many ideals. These facts will prove to be useful later.· In this section it 111i 11 be shown that the additive groups of rings satisfying the descending chain condition for two sided ideals are precisely the additive groups of Artinian rings described above. The additive groups of rings possessing only finitely many ideals will be studied. The structure of the additive group of a ring possessing only finitely many ideals (satisfying the descending chain condition for two sided ideals) will be employed to obtain necessary and sufficient conditions for embedding the ring into a ring with unity possessing only finitely many ideals (satisfying the descending chain condition for two sided ideals). For the remainder of this section, ideal will mean two sided ideal and the "descending chain condition" will be abbreviated DCC. Theorem 4.3.2.

Let G be a torsion free group.

The following are equivalent: 1) G is the a~ditive group of an (associative) ring possessing only finitely many ideals.

2) G is the additive group of an (associative) ring satisfying the DCC for ideals. 3)

G ...
01

an arbitrary cardinal.

01

Proof: Clearly 1) ,. 2) and 3) ,. 1), the latter implication since Q+ is the additive group of a field. It therefore suffices to show that: 2) .. 3): Let R be a ring satisfying the DCC for ideals, with R+ =G. For every prime p, and positive integer n, pnR ~ pn+lR. Hence there exists · n sue h th at a pos1· t1· ve 1nteger

pnR = pn+ l R• Th ere f ore f or a e: G, th ere 51

exists b € G such that pna = pn+lb, or pn(a - pb) = 0. Since G is torsion free, a= pb, and so G is p-divisible for every prime p. Hence G is divisible, and G ~ Q+, Proposition 1.1.3. a Observe that the above theorem adds two equivalent conditions to those given in Theorem 4.1.3. Let G be a non-nil torsion free group.

Corollary 4.3.3: equivalent:

1) Every (associative) ring finitely many ideals.

R with

R+ = G,

2) Every (associative) ring DCC for ideals.

R with

R+ = G, and

3)

and

The following are

R2 ~ 0 possesses only R2 ~ 0 satisfies the

G ~ Q+ •

Proof:

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

By Theorem 4.3.2, G ~(±) Q+, a and so G ~ Q+(t)H. Let S be the zeroring with s+ =H. The ring direct sum R = Q(t)S satisfies R+ ""G, and R2 ~ 0. If H ~ 0, choose 0 I a € H. The infinite chain of ideals in R, (a} ~ (2!a} ~ (3!a} ~ ••• is properly descending, a contradiction. Hence H = 0, and G ~ Q+ • 2} .. 3}:

Suppose condition 2} is satisfied.

Lemma 4.3.4:

Let G be a torsion group.

The following are equivalent:

1} G is the additive group of an (associative} ring possessing only finitely many ideals. 2) G is the additive group of an (associative} ring satisfying the ascending chain condition for ideals. m ni

3)

G

=

9

.

€) Z(p~}.

i=l j=l a.

1

pi

a prime, m,ni

positive integers,

aj

an

J

arbitrary cardinal, Proof:

= l, ... ,m;

j = l, ...

,n;.

Clearly 1} • 2}:

2) • 3}: Suppose there exist infinite1y many distinct primes {pi}7=l for which G ~ 0. Let R be an (associative} ring with R+ = G such that R pi satisfies the ascending chain condition for ideals. The infinite chain of 52

2

ideals in

R, Gp

1

contradict1on. For every p

€

0 Gp

k

G c... is properly ascending, a i=l i i=l Pi Hence Gp ~ 0 for only a finite set of primes {pi}~=l. c

c ... c (f)

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

chain of ideals in

R,

positive integer ni

each contained in Gp. n. such that G = G [p. 1], Pi Pi 1

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

00

00

54

)

).

Proposition 1.1.10. For every positive integer n, pnGP ~ pn+lGP is an inclusion of ideals in R. Hence there exists a positive integer n, such that pnG = pn+lGP. Clearly pkG = pkH €) D for every positive integer p p p p k. Therefore pnHP = pn+lHP. Since Hp is reduced, this implies that pnH p = 0, i.e., for every 1 -< i -< m, there exists a positive integer n1.

n.

such that H pi

4

(tj j=l a.

Z(p~),

cxJ.

a cardinal number,

j = 1 , ••• ,ni.

J

Clearly G is of the form of condition 2). 2) ~ 1):

If G satisfies condition 2) then G is the additive group of an associative ring satisfying the DCC for left ideals, Proposition 4.3.1.

Corollary 4.3.8: Let G be a torsion group. G is the additive group of a ring with trivial annihilator satisfying the DCC for ideals if and only if G sati~fies one, and hence all, of the equivalent conditions in Theorem 4. 3.5. Proof: If G is the additive group of a ring with trivial annihilator, then G is reduced, Lemma 2.2.1. If in addition G is the additive group of a ring satisfying the DCC for ideals, then G is bounded by Theorem 4.3.7. Conversely, if G satisfies the equivalent conditions of Theorem 4.3.5, then the ring constructed in proving the implication 3) ~ 1) for Lemma 4.3.4, is a ring with trivial annihilator satisfying the DCC for ideals. Corollary 4.3.9: Let G be a non-nil torsion group. The following are equivalent: · 1) Every (associative) ring R with R+ = G, and R2 ~ 0 satisfies the DCC for ideals. m ni 2) G = €> (±) (£) Z(p~) (£) Z(p7), pi a prime, n, ni positive i=l J"=l ex j finite integers, cxj Proof:

1 , ••• ,m,

j = 1, .•• ,ni.

Similar to the proof of Corollary 4.3.6.

Theorem 4.3.10: 1)

a finite cardinal,

Let

G be a group.

The following are equivalent:

G is the additive group of an (associative) ring possessing only

55

finitely many ideals. m ni 2) G c.!{£)Q+ 0 (f) 0 Z(p~). pi a prime, m. ni non-negative integers. a i=l j=l aj a. aj arbitrary cardinals. i = l, .••• m; j = l, •••• ni. Proof: 1) • 2): Let R be a ring possessing only finitely many ideals. with R+ = G. Then R = R/Gt is a ring possessing only finitely many ideals. with 'It torsion free. Therefore it Co!~ Q+. a an arbitrary cardinal. Theorem4.3.2. Let I
a. aj

finite cardinals,

= 1 , ••• ,m;

j = 1 , ••• ,n 1 •

An argument almost identical to that used in proving Theorem 4.3.1 yields: Theorem 4.3.12:

Let G be a group.

The following are equivalent:

1) G is the additive group of an (associative) ring satisfying the DCC for ideals. n ni 2) G ... @ Q+ 0 (!) (3 a prime. m, n; a i=l j=l aj non-negative integers. a. aj arbitrary cardinals. i = 1, ••• ,m; 56

j = 1 , •••

,ni .

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

1) 2)

Let G be a non-nil group.

The following are equivalent:

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

non-negative integers, a, aj

finite cardinals,

i = l, ••• ,m; j = l, •.• ,ni.

Lemma 4.3.15: Let R be an (associative) ring with trivial annihilator, such that R+ = D for all n ~ m. Clearly <m!a>+ is divisible, and so <m!a>+ =D. This obviously implies that m!a = 0, or that K is a torsion group. 3) ~ 1): Let .G = D
R+I (S (t) N) + "" S~ /S +:

~

Let w

be the natural projection of R* onto S.

s+1/S+ 5 via

~(z)

= n

(z) + s+ for all

~

z e: R+.

Define

Clearly

~

is an

epimorphism, and (S ® N)+ ~ ker ~. Let z e: R, z = x+y, x e: S, y e: N. Suppose that z e: ker ~. Then x e: S, and so y = z- xe:R n N = N, i.e., r.'\+ r.'\+ + + ++ z E (S 1+1 N) , and so ker ~ = (S ':r' N) , or R /(S + N) ""S 1/S • Lemma 5.2.3:

S*

= -S, and N* = -~.

Proof: S* is the unique minimal Q-algebra containing S. It therefore suffices to show that S is a full subring of s. Let x e: S. Then x e: R* and so there exists a positive integer n such that nx e: R. Hence nx e: R n S = S and so s+ts+ is a torsion group. The same argument shows that N* = -N. Lemma 5.2.4: There is a Q-basis {x 1 , ••• ,xm} for s such that the free subgroup F of S generated by this basis is a subring of S. Proof:

Since s+ is a full subgroup of

z1 , ••• ,zm

in S is a Q-basis for

~.

s, a maximal independent set m

z.·zJ. =

~

a .. kzk with k=l lJ a.1J"k e: Q; i ,j,k = l, •.• ,m. Let n be a positive integer such that naijk e: Z for all i,j,k = 1 , ••• ,m. Then {X; = nz; , i = 1 , ••• ,m} satisfies the conditions of the lemma. Observation 5.2.5:

Hence

1

S~/F is a torsion group.

Proof: Clearly S+/F is a torsion group. It therefore suffices to show that S~/S+ is a torsion group, or by L;mma 5.2.2 that R+/(S N)+ is a torsion group. Let x e: R. Then x e: R , and so x = y+z, y e: S, z e: N. By Lemma 5.2.3 there exists a positive integer n such that ny e: S, and nz e: N. Hence nx = ny + nz e: S(+)N, and R+/(Sq)NJ+ is a torsion group. Lemma 5.2.6:

For all but finitely many primes

p,

(S~/F)p, and

(S+/F)p 91

are divisible and equal. The proof of Lemma 5.2.6 involves many steps, and will be broken down into a series of claims. First we introduce some notation. The degree of nilpotence of W will be denoted by t. For p a prime, k > 0 an integer, Ik = {x € F I p-k x € S}, and Jk = {x € Fl p-k x € s1}. Claim 5.2.7: {1) F = I 0 2 I 1 2 I 2 2 ... ,

=Jl =J2 =···•

(2)

F = Jo

{3)

(4)

I k !: Jk' Ik and Jk are two sided ideals in F,

{5)

Ik. It ~ IkH '

{6)

Jk.Jt!: Jk+t' and Jkt ~ I k for a 11 k ,t

(7)

Proof:

(1) - (4)

~

0.

are obviously true.

Let x1 € Ik, x2 € I 1 . Then x1 = pky 1 , and x2 = p1y2 , y1 ,y 2 € Hence x1x2 = pk+ty1y2 , with y1y2 € S, i.e., x1x2 € Ik+t . {6): follows from the same argument used to prove (5). (5):

s.

(7): Let x € Jk. Then x = pky, y € s 1 , and there exists z € N such that y-z € R. Let x1, ••• ,xt € Jk, and let z1, ••• ,zt € N such that p-kxi - zi € R, i = 1, ••• ,t. Suppose, inductively, that p-kx. x. 11 12 for l~i 1 ~a.

-

1

+ Yk.1

Yk+~,i ~~a;

for all

k

~

for all

k

k0 . Since a 1.

for all

k ~ k0 •

~

k0 and so

;

0,

so by (2') Yki ~ ~ as k ~ ~. and by Claim 5.2.20(1), This clearly implies Claim 5.2.21.

~a,. ~ ~

as

~ ~

as

oki

An immediate consequence of Claim 5.2.21 is Claim 5.2.22: For every non-negative integer ~. ~1'5" 2K(~) such that for k ~ K(~). Ik +Po~ n Jk.

there exists an integer

Proof of Lemma 5.2.18: Let p be a prime. Since r{TP) ~ r(T 1p) < ~ it suffices to show that d(Tp) = d(T 1P), or since TP .= Tlp' that d(Tlp) ~ d(TP). By Claim 5.2.21 there exists an integer K(~) such that 97

for k

~

K{R.),

4

n Jk

~

5.2.19(1), and n~ ~F.

4-.-

n Jk

~

2-

R.2

n Ik + p n B.

2-

However n Ik

{A) n4Jk ~ Ik + pR.F for all

Hence

~

Ik,

Claim

k ~ K(R.).

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

{pk-R.Tlp)[pR.]

by Claim 5.2.9(2).

eR.{I~) ~ {pk-R.Tp)[pR.].

Let n4 = pjm, m a positive integer such that

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

{p,m) = 1.

subgroup of (pk-R.Tp)[pR.]. (pk-lTlpJ[p] r {pk-1 Tlp ) pk-R. Tp

=

Similarly, employing Claim 5.2.9{1),

~

Put

R. =

j+l

is isomorphic to a

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

Then

is isomorphic to a subgroup of {pk-R.TP)[pR.], and so r {pk-R. Tp ) • For

d{TP ) • Hence

d{Tlp )

k sufficiently large ~

pk-1 T1P = d{Tlp ) , and

d ( TP ) and the claim is proved.

Finally, we can prove Theorem 5.2.1. Proof of Theorem 5.2.1: All that remains to be shown is that S (+) N is of finite index in R, Lemma 5.2.3. By Lemma 5.2.2 it suffices to show that S~/S+ is finite. S~/S+ ~ (S~/F)/(S+/F) = T1/T eo~ {t). Tlp/Tp' p a prune Observation 5.2.5 and Proposition 1.1.1. Now T1p/Tp is finite for every prime p, Lemma 5.2.18, and 0 for all but finitely primes, Lemma 5.2.6. Hence S~/S+ is finite; thus concluding the proof of Theorem 5.2.1. Translating Theorem 5.2.1 to groups, we have: Corollary 5.2.23: Let G be a finite rank torsion free group. Then H$ K, where H admits a multiplication of semisimple type, and K is the additive group of a nilpotent ring N satisfying N2 ~ 0, unless K = 0, or G is nil. G c>.

§3.

Torsion free rings with semisimple algebra type.

The results of the previous section show that, up to quasi-isomorphism, the classification of the additive groups of finite rank torsion free rings reduces to two cases; the additive groups of rings with semisimple algebra type, and the additive groups of nilpotent rings, which are not zerorings. In this section a further reduction will be obtained by showing that a finite 98

rank torsion free group admitting a multiplication of semisimple algebra type is quasi-isomorphic to a direct sum of groups admitting multiplications of simple algebra type. It will further be shown that a finite rank torsion free group admitting a multiplication of simple algebra type is quasiisomorphic to a direct sum of groups admitting multiplications of field type. Therefore, upto quasi-isomorphism, the classification of the additive groups of finite rank torsion free rings with semisimple algebra type is settled by determining the additive groups of full subrings of algebraic number fie 1 ds. Theorem 5.3.1: Let R be a finite rank torsion free ring with semisimple algebra type. Then R contains a subring S = s 1 (+) .•• fi:)Sm such that S; has simple algebra type, i = 1, ... ,m, and R+ ;s+ is finite. To prove Theorem 5.3.1 we first need: Lemma 5.3.2: Let R be a torsion free ring with Q-algebra with unity e. Let R; =RnA;. Then ..... + is bounded. and R+/(Rl C-:+"J. .@ f\n)

R*

= A1