Group and Semigroup Rings (North-Holland Mathematics Studies)

GROUP AND SEMIGROUP RINGS

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (1I1 )

Editor: Leopoldo Nachbin Centro de Brasileiro de Pesquisas Fisicas Rio de Janeiro and University of Rochester

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

126

GROUP AND SEMIGROUP RINGS Proceedings of the International Conference on Group and Semigroup Rings University of the Witwatersrand,Johannesburg, South Africa 7-13July, 1985

Edited by

Gregory KARPlLOVSKY Department of Mathematics University of the Witwatersrand Johannesburg, South Africa

1986

NORTH-HOLLAND -AMSTERDAM

0

NEW YORK

OXFORD .TOKYO

@

Elsevier Science Publishers B.V., 1986

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 or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 70043 9

Publishers:

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue New York, N.Y. 10017 U.S.A.

Library of Congress Catalogingin-PublicationData

Inttrnatiocal Conference on Group and Semigroup Rings (1935 : University of t h e Witwetersrand) Group m d semigroup rings.

(North-Holland mathematics studies ; 1 2 6 ) (Notes

ec mtem6tica ; 111)

1. Group i-incs--Congresses. 2. Semigroups-Congresses. I . Karpilovsky, Greeory, 194011. T i t l e . 111. T i t l e : Sendgroup rings. I V . Series. V . Series: Notas de matem6tica (Rio de Janeiro, Brazil) ; no. 111.

W.N86 CW713

ISBN

no. 1l.l 510

8

C512'.227

.

0-444-70043-9 ( U . S )

PRINTED IN THE NETHERLANDS

86-11545

PREFACE

The International Conference on Group and Semigroup Rings took place amidst a number of important developments in the subject. The meeting was held in the Department of Mathematics of the University of the Witwatersrand, Johannesburg, South Africa from 7

-

13 July,

1985.

Visitors from various institutions contributed lectures or series of lectures in which they covered recent advances in their own field of speciality.

Much of the life of the Conference pulsated

in the informal gatherings in the various seminar rooms on the campus, where participants from outside Johannesburg were accommodated. There were recreational activities of various kinds: A cocktail party, a one-day tour to Mabula Lodge Game Reserve, a day tour around Johannesburg, Pretoria (a beautiful city during Jacaranda time) where we saw the Government Buildings which were designed by Sir Herbert Baker. The value of such a conference cannot be assessed.

The con-

sensus of opinion of the delegates rated the complete week very high.

G. Karpilovsky Chairman International Conference on Group and Semigroup Rings Department of Mathematics University of the Witwatersrand Johannesburg, South Africa.

as

being

vi

ACKNOWLEDGEMENTS I wish to acknowledge, with very many thanks, the following for

their generosity.

Without this assistance, the Conference could not

have taken place:Professor C. Cresswell, Dean, Faculty of Science. The Cormnittee of the Deputy Registrar (Research). I.B.M., South Africa (Pty) Limited, Johannesburg. The Chairman's Fund, Anglo American Corporation, Limited. Two anonymous donors. Also:Barclays Bank for providing the folders, etc., for the delegates. South African Airways who willingly supplied the unprinted shells for the registration form, and which showed facets of South Africa. International Travels (1980) (Pty) Ltd., Johannesburg.

In

particular, Mr M Massel, who diligently and efficiently organized the routings and air tickets for the delegates. Mrs Shirley Irish, who unstintingly gave of her time at Sunnyside Residence, where the delegates resided. Mrs Mavis Nash, Catering Division, for organizing the cocktail party

so

well, and also the meals for the delegates.

The Division of Public Affairs for organizing a tour of the University Campus for the delegates. To all other University departments who assisted with equipment, etc. Last, but not least, my thanks to my colleague, Professor John Knopfmacher (Head of the Department of Mathematics), and Mrs Nan Alexander for their support throughout the Conference. Gregory Karpilovsky

vii

LIST OF CONTENTS

P

. Cherenack.

. R . Gilmer. R Gilmer.

Fundamental groups of elliptic curves internal to locally ringed spaces

................ Property E in commutative monoid rings ..............

13

Conditions concerning prime and maximal ideals of commutative monoid rings

19

..................

.

R Gilmer. Commutative monoid rings with finite maximal or prime spectrum

...................................

.

E Jespers. The group of units of a commutative semigroup ring of a torsion-free semigroup

.

...................

.........................................

. Karpilovsky. Blocks and vertices algebras

.

of twisted group

.......................................

G Karpilovsky. Defect groups of blocks of twisted group algebras G

27 35

.

E Jespers and P Wauters. A description of the Jacobson radical of semigroup rings of commutative semigroups G

1

.......................................

. Karpilovsky. Extending indecomposable modules over twisted group algebras

.........................

43

91 99 109

. ................ 117 D . Lantz. On the Picard group of an abelian group ring ......... 139 F . Levin and G . Rosenberger. Lie metabelian group rings ........ 153 W . May. Unit groups and isomorphism theorems for commuta163 tive group algebras .................................... B Kulshamer. Blocks of modular group algebras

C.P. Milies. Torsion units in group rings and a conjecture of H . J . Zassenhaus

................................

.

179

K Motose. On the nilpotency index of the radical of a group algebra IX W.D.

M.F.

.

.......................................... 193 Munn. Inverse semigroup algebras .......................... 197 O'Reilly. Sections and irreducible modules ................225

G Richter. Noetherian semigroup rings with several objects

............................................

.

W Ullery. A conjecture relating to the isomorphism problem for commutative group algebras

.

......................

P Wauters. Rings graded by a semilattice to semigroup rings

-

231 247

Applications

.................................

253

viii

LIST OF REGISTERED PARTICIPANTS

P. Cherenack, Dept. of Mathematics, University of Cape Town, Rondebosch, 7700, South Africa. V. Dlab, Dept. of Mathematics, Carleton University, Ottawa, Canada. E.W. Formanek, Pa. St. Univ., Dept. of Mathematics, University Park, Pa. 16802, U.S.A. R. Gilmer, Dept. of Mathematics, The Florida State University, Tallahassee, Florida 32306-3027, U.S.A. E.F.J.M. Jespers, Dept. of Mathematics, University of Cape Town, Rondebosch, 7700, South Africa. G. Karpilovsky, Dept. of Mathematics, University of the Witwatersrand, Jan Smuts Avenue, Johannesburg2001, South Africa.

B. K~lshammer,Fachbereich Mathematik, Universitgt Dortmund, Postfach 50 05 00, 4600 Dortmund 50, Bundersrepublik Deutschland. D. Lantz, Dept. of Mathematics, Colgate University, Hamilton, New York, 13346, U.S.A. W. May, Dept. of Mathematics, University of Arizona, Tucson, Arizona, 85721, U.S.A. C.P. Milies, Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Caixa Postal 20.570 Ag. Iguatemi, 01000 Sao Paulo S.P., Brazil.

-

K. Motose, Dept. of Mathematics, Faculty of Science, Hirosaki University, Hirosaki, Japan. W.D. Munn, Dept. of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, United Kingdom. M.F. O'Reilly, Dept. of Mathematics, Rhodes University, Grahamstown, 6140, South Africa. G. Richter, UniversitBt Bielefeld, Facultgt fur Mathematik, Universitgtstrasse 25, D-4800 Bielefeld, 1, West Germany. K.W. Roggenkamp, Mathematiche Institut der Universitgt, Pfaffenwaldring 57, 7 Stuttgart 80, West Germany. G. Rosenberger, Abteilung Mathematik, Universitgt Dortmund, 4600Dortmund 50, Vogelpothsweg 87, Dortmund, West Germany. W. Ullery, Dept. of Mathematics, University of Kansas, Lawrence, Kansas 66045, U.S.A. P. Wauters, Katholieke Universiteit Leuven, Departement Wiskunde, Celestijlaan 200B, B-3030 Leuven, Belgium.

Group and Semigroup Rings

1

~ E ~ ~ ~ ! ~ % ~ b I i s hB.V. e .(North-Holland), rs 1986

FUNDAMENTAL GROUPS OF ELLIPTIC CURVES INTERNAL TO LOCALLY RINGED SPACES Paul CHERENACK U n i v e r s i t y o f Cape Town* A formalism due t o Huber i s considered w i t h a view t o i n t r o d u c i n g a n o t i o n o f r e l a t i v e homotopy group t o a subcategory o f l o c a l l y r i n g e d To spaces. For X an e l l i p t i c curve over D t h e formalism leads t o a t r i v i a l groupa but w i t h a s u i t a b l y a l t e r e d of homotopy r e l a t i o n i t leads t o t h e group (End(X,Q),t) endomorphisms o f X keeping 0 E X fixed. Viewed as p a r t i a l submonoids o f End(X,CJ the frjndamental groups o f a f f i n e models o f X are calculated i n several cases. Some o t h e r c a l c u l a t i o n s o f homotopy groups i n t h e context of l o c a l l y ringed spaces a r e described. INTRODUCTION * F i r s t we choose f o r our model o f the u n i t i n t e r v a l a monoid X with

0.

zero i n algebraic v a r i e t i e s . A method found i n Huber [6] f o r c o n s t r u c t i n g 2) w i t h respect t o X* r e l a t i v e homotopy groups ( a t l e a s t i n dimension w i t h i n the category o f l o c a l l y ringed spaces i s described i n section 1. To This method i s r e a l i z e d i n o n l y c e r t a i n cases i n t h i s paper. Thus i f X i s an algebraic group defined over the complex numbers

I Iand 0 E X then denotes t h e d i s j o i n t union can be given t h e s t r u c t u r e o f a monoid w i t h zero 8 i n algebraic v a r i e t i e s (a t B = B f o r a l l a E X*). X

*

= X fi

{ @ I where

II

This i s n o t the most i d e a l such monoid t h a t one can choose since 0 i s n o t r e l a t e d t o the other p o i n t s o f X* i n the same way t h a t 0 i s r e l a t e d t o the other p o i n t s o f the u n i t i n t e r v a l . We attend t o t h i s d i f f i c u l t y a t with various stages. Next we s p e c i a l i z e X t o be an e l l i p t i c curve ( X , t ) i d e n t i t y 0. In s e c t i o n 3 we f i r s t compute

*

where

So = t 0 , l ) c C

*

, which

i s t h e r e l a t i v e homotopy group provided by t h e

c o n s t r u c t i o n due t o Huber [6]. To compute nlx*(Xa~) we show i n s e c t i o n 2 t h a t C(So) i s isomorphic t o (X*,B) and t h a t t h e cone C2(So) = C(C(So)) i s isomorphic t o

'pesent address : Mathematics Department, Massachusetts I n s t i t u t e o f hnology. Grants t o the Topology Research Group from the U n i v e r s i t y o f Town and t h e South A f r i c a n Council f o r S c i e n t i f i c and I n d u s t r i a l r c h are acknowledged.

2

P. Cherenack

( ( X t X)*, a ' ) ( w i t h e v i d e n t g e n e r a l i z a t i o n s t o Cr(X) ( r 2 3 ) ) . For an a r b i t r a r y monoid X* i n a l g e b r a i c v a r i e t i e s one might expect $(So) t o be o n l y a l o c a l l y r i n g e d space. We a l s o show t h a t t h e augmented 0-dimensional simplex d e f i n i n g t h e fundamental group illX*(X,Q) i s End(X,Q) , t h e s e t o f endomorphisms o f of

X

0

keeping

I n s e c t i o n 3 we see t h a t t o o t h e r p o i n t s o f X*

@

fixed. X* i s t r i v i a l , due t o t h e unrelatedness rIl (X,Q) Thus we replace @ by another p o i n t o f X

.

o b t a i n i n g a homotopy r e l a t i o n (X-homotopy) which i s seen t o be t h e diagonal r e l a t i o n . A s u i t a b l e m u l t i p l i c a t i o n on End(X,CJ turning it i n t o a fundamental group ring

End(X,Q)

.

ili(X,E)

See [5; p. 3231.

For an a f f i n e model replacing

X

.

by

i s shown i n s e c t i o n 4 t o be t h e a d d i t i o n o f t h e Y

of

X

containing

t h e above c o n s t r u c t i o n s ,

throughout, y i e l d a p a r t i a l submonoid

Y

when

X

p l i c a t i o n o r t h e complex m u l t i p l i c a t i o n a r i s e s from

T

End(X,g)

We c a l c u l a t e

nl*(Y,0-)

End(Y,Q)

nl*(Y,Q)

of

has no complex m u l t i = i ,

I n s e c t i o n 6 we

g i v e some r e s u l t s and comments on homotopy groups i n t e r n a l t o a l g e b r a i c varieties

.

As i n Popp [S] one would l i k e t o be a b l e t o compute o r d i n a r y fundamental groups by methods common t o a l g e b r a i c geometry. 1.

The

X*-fundamental

group o f an a l g e b r a i c v a r i e t y

One avenue f o r i n t r o d u c i n g homotopy groups i n t e r n a l l y t o a1 gebraic v a r i e t i e s i s i n v e s t i g a t e d . Grothendieck [4] pursues s i m i l a r s t u d i e s employing more general methods.

P o s s i b l y o n l y by c o n s i d e r i n g v a r y i n g p o s s i b i l i t i e s f o r

such homotopy groups ( d e s p i t e t h e t r i v i a l reductions which occur) can one a r r i v e a t an e f f e c t i v e homotopy theory, u s e f u l i n determining t h e s t r u c t u r e of algebraic varieties. We w i l l discuss t h e fundamental group o f an e l l i p t i c curve over E t a k i n g t h e e l l i p t i c curve i t s e l f as a model o f t h e c i r c l e . It would take considerable space t o describe t h e general homotopy background and thus we o f t e n r e s t r i c t t o s p e c i a l cases. I n t h e beginning we f o l l o w t h e formalisms found i n Huber [6] i n h i s s e c t i o n on t h e t o p o l o g i c a l cone c o n s t r u c t i o n . L e t Var denote t h e category o f a l g e b r a i c v a r i e t i e s o f f i n i t e t y p e over

t (see Hartshorne [5]) and X* a monoid i n Var . Elements o f & w i l l be c a l l e d v a r i e t i e s . We suppose t h a t X* has t h e o p e r a t i o n + , t h e i d e n t i t y 0 f o r t and t h e zero @ where Q t x = x t @ = Q f o r a l l x E X ( o p p o s i t e t o normal conventions). Suppose t h a t Z i s i n a subcategory

*E

Z

.

The p a i r

based o b j e c t s i n

(Z,*)

5

5

o f r i n g e d spaces (see [5])

i s c a l l e d a based o b j e c t i n

i s a morphism i n

C

5

and

and a morphism of

sending base p o i n t t o base p o i n t .

We l e t &* denote t h e category o f based o b j e c t s i n c o l l e t i o n o f based v a r i e t i e s .

C

.

Thus

m* i s t h e

Fundamental groups of elliptic curves

Let

Var

N

3

be t h e c a t e g o r y of (commutative) r i n g e d spaces.

N

i s a subcategory o f

(non-full).

The c a t e g o r y

See H a r t s h o r n e [5].

One can show

w i t h o u t much d i f f i c u l t y : 1.1.

Lemma:

-

F o r any two maps

f

(1x1

(ZYG)

~

9

5

in

( o r f o r added b a s e p o i n t s i n Q*)

c o e q u a l i z e r of

f

and

g

.

Let for

let

Q

open i n

U

Z

q:

-+

Q

be t h e t o p o l o g i c a l

H(U)

the r i n g

be t h e

e q u a l i z e r o f t h e maps fU

aq-l

-

1 -F ( f - l ( q - l ( u ) ) )

(U)) gU

Then

(Z,G)

q:

(a,!)

-+

f

coequalizer o f

and

X*

base p o i n t o f X*

x

X*

(Z,*)

m

the p o i n t

base p o i n t o f a based r i n g e d space. (0 x

Z ) u (x*

(0 x

Z) u (X*

x

*

F o r s i m p l i c i t y o f n o t a t i o n we o f t e n w r i t e

i:

i s the

m

i n t h e usual way (see [ 6 ] ) we l e t

and f o r a based v a r i e t y

.

Z

G(q-l(U))

N.

in

To c o n s t r u c t a cone on point o f

H(U) -+

d e f i n e d by t h e i n c l u s i o n s g

*

be t h e t h e base

f o r the

Let x

*)

-+

X* x

z

x

*)

-+

x*

z

*

.

be t h e i n c l u s i o n and

*:

be t h e c o n s t a n t map w i t h image

1.2. in

m*i s

Definition:

i,*: ( ( 0

x

0 x

x

The c o e q u a l i z e r o f Z) u

(x*

x

*).

0 x

(x*

*)

x

Z, 0

x

*)

a map

(C(Z),*) q: ( X * x z, 0 x *) i s c a l l e d t h e cone o f Z on -+

where

C(Z)

Note t h a t t h e assignment o f

(C(Z),*)

X*

.

to

(Z,*)

defines a factor

m*-+a* w h i c h may t a k e us o u t s i d e k* . We i n c l u d e t h e phrase X*" s i n c e X* p l a y s t h e r o l e o f t h e u n i t i n t e r v a l . Suppose t h a t m* i s t h e c a t e g o r y o f based commutative l o c a l l y r i n g e d To spaces. One s h o u l d be a b l e t o e x t e n d =* t o a s u b c a t e g o r y *! o f m* on w h i c h C C:

NO"

i s a monoid o r e q u i v a l e n t l y a t r i p l e (see MacLane [7]). an o b j e c t of

*!

a c o - s i m p l i c a l s e t ( t h e upper

*

I n t h a t case f o r

Wo

t o be r e p l a c e d by an

integer)

F* No = (CntlWo, X* (11 as i n Huber [6] i s obtained. R(D)

Fi,Wo

.

Let

Zo

diWo,

siWo)

Note t h a t we sometimes w r i t e

be a n o b j e c t o f

*!

.

Applying t h e f u n c t o r

i n ( 1 ) one o b t a i n s a s i m p l i c a l s e t ( t h e l o w e r

integer)

*

RD

instead o f HomD*(-,Zo)

to

t o be r e p r a c e d by an

4

P. Chemnack

* o o HomD*(FX*W ,z

nlx* (W 0 ,Z 0 )

The X * - r e l a t i ve homotopy group

.

i s d e f i n e d t o be

no(K, x* (W0 ,Z 0 ) )

.

x* 0 0 II, (W ,Z )

i n a s p e c i a l case below when t h e above We w i l l c a l c u l a t e d e f i n i t i o n s a r e c e r t a i n t o make sense. When nlx* (W0 ,Z 0 ) becomes one p o i n t and t h i s may n o t be d e s i r a b l e , a l t e r n a t i v e approaches a r e i n d i c a t e d .

X

Suppose t h a t

0.

element

+ and z e r o

i s an a l g e b r a i c group w i t h o p e r a t i o n

Var ,

t @ > be t h e s i n g l e t o n o b j e c t i n

Let

Then

X* = X u { @ I

w i l l denote t h e coproduct o f X and { @ I i n which i s also t h e coproduct i n Q I n a d d i t i o n X* can be made i n t o a monoid i n Var by l e t t i n g f o r a E X* a + @ = o + a = @ . Then X i s an a l g e b r a i c group L e t X now be an e l l i p t i c c u r v e o v e r d where h e r e and n o r m a l l y t h e o p e r a t i o n i s + L e t So be t h e c l o s e d subset o f IT c o n s i s t i n g o f 0 and 1 , We w i l l compute X* X* fl1 ( X I = 'TI ((solo), (XYO)) However we f i r s t need t o show t h a t t h e o p e r a t i o n s needed t o d e f i n e n;*(X) exist. i n t h e absence o f *!

.

.

.

.

2. D e t e r m i n a t i o n o f C 1 So Let

X

C 2So

and

diagram

((a in

x

X* = X u { @ I

be an e l l i p t i c curve, and

i

So) u ( X *

&*

x

01, @

* (X* __f

0)

x

as seen i n Lemma 1.1. (X*

ql:

4 x

There i s a c o e q u a l i n ' n g

0) - L ( C ( X ) , * )

Let

so,

x

so,

x

.

0 x

0)

+

(X*

x

1,

@ x

1)

be d e f i n e d by

Then

q'

q'(x,l)

= (x,l)

for

q'(x,O)

= (@,l)

otherwise.

i s a map o f v a r i e t i e s s i n c e

1 i n Var r : (x* x 1, cp x 1 )

and

X*

x

.

Furthermore q '

( x , l ) E X*

X* 0

x

So

x

1

i s the coproduct o f

X*

x

0

.

q' 0 * Let be t h e morphism i n 1 *

=

Var d e f i n e d by Q x I) r ( x , l ) = (x,l) As q ' i s an and t = q 0 r , One has t 0 q ' = q o r 0 q ' = q epimorphism i n $ J ( t h e maps on r i n g s d e f i n i n g q ' a r e i n c l u s i o n s ) t i s unique, Thus we have: 2.1. Lemma: The cone (C(So),*) i s isomorphic t o

+(x*

x

so,

.

(x*

x

I,

cp x 1 )

(x*,cp)

in

m* .

-

There i s a c o e q u a l i z i n g diagram o m i t t i n g base p o i n t i n one case i ( @x X*) u. ( X * x @ ) ( X * x x*, @ x @ ) 9

*

(Ao,*) .

Fundmental groups of elliptic curves

As

X*

x

X* = (X* x o) u ( 0

Lemma:

2.2.

&*

where

Remarks:

(X

x

The cone X)* = (X

x

X*) LI ( X x X ) one can show as above: 2 (C So,*) i s isomorphic t o ((X x X)*,o') X) u { o ' }

x

5

Lemma 2.1 holds f o r any monoid

in

Var .

in

Var

in

X*

p r o o f o f Lemma 2.2 does n o t c a r r y through f o r such

w i t h zero

.

X*

o

.

The

On the o t h e r hand

2 i s i r r e d u c i b l e , then the base p o i n t o ' o f C So has as l o c a l r i n g t h e s e t o f a l l f E C ( X * x X*) , the f u n c t i o n f i e l d o f X* x X* , which are

if

X*

(o

constant on

x

X*)

U

(X*

.

o)

x

Then

x*,o x o ) q: a r i s e s from a map o f l o c a l l y ringed spaces. (X*

x

(C2S o # )

-f

From ( 2 ) i n section 1 we have X* KO ((cso,*),

(x,g))

= HomD*((X*,@),

(XsQ))

-

= HomVar*((X*,@),

(X,Q))

-

since we suppose t h a t

D*

m* and maps o f v a r i e t i e s

i s contained i n

are p r e c i s e l y those a r i s i n g from maps of l o c a l l y ringed spaces. = HomVar((X x X)*,o'),

(X,O))

Ky*((C2So,*),

-

which can c l e a r l y be i d e n t i f i e d w i t h

HomVar(X

x

-

*,.A

X,X)

Similarly

(X,O))

.

t o be augmented i n which case one wishes f i r s t t o form

We would l i k e K i " (as i n Huber 161)

ri*

g}

= { o E Ki*ldoo =

(as we see from above) can be where 0 E X and *K: = K:*((CSo,*), (X,Q)) i d e n t i f i e d w i t h End(X) = HomVar(X,X) under a map p : -+ End(X) Thus one has : 2.3.

Lemma:

One can i d e n t i f y under

the s e t o f endomorphisms o f 3.

K r

-

the set

p

roX*

.

with

End(X,Q)

keeping Q fixed.

X

Homotopy r e l a t i o n s d e f i n i n g

X*

Ill

(X,Q).

Our discussion i n i t i a l l y f o l l o w s Huber [6] i n d e f i n i n g the n a t u r a l transformations which make the cone f u n c t o r i n t o a monad.

For

(Y,*)

,

an o b j e c t o f rY:

d e f i n i n g a n a t u r a l transformation

(Y,*)

r:

which allows one t o f i l l i n t h e diagram

extended t o

C

C:

m* the morphism +

m* m* -f

(CY,*)

I -+ C

i s the morphism i n

(Rng)*

P. Cherenack

6

\ \

rY

‘ \Y 1 CY where

0

a*( t h u s

map d e f i n i n g (Y,*) E

Y t o C E X* , I i s t h e i d e n t i t y I Y i s t h e i d e n t i t y on Y ) and qY i s t h e q u o t i e n t Then t h e boundary o p e r a t o r dy i s d e f i n e d f o r

i s t h e c o n s t a n t map sending

f u n c t o r on

CY

&*

.

by s e t t i n g dyY = r C Y :

(CY,*)

-+

(C2Y,*)

.

Thus dyY = (qCY)

.

(&ICY)

On t h e o t h e r hand t h e boundary o p e r a t o r d: i s d e f i n e d f o r (Y,*) as t h e unique map C r Y f o r which one has a commutative diagram

x*

where

IX*

IX* x

CY

&*

rY

Y

i s t h e i d e n t i t y on

d e f i n i n g t h e cones

x

E

2

C Y

and

X*

and

qY, qCY

a r e t h e q u o t i e n t maps

, respectively.

an e l l i p t i c curve, X* = X Y { @ I and c o n t i n u e t h e development a t t h e end o f t h e l a s t s e c t i o n . Two maps f,9 E ro a r e t h e n

X

We s p e c i a l i z e now t o

homotopic i f t h e r e i s a

such t h a t f =

o r since

qSo

,

CrS,

0

g =

F*

0

rCSo

i s a n epimorphism i f f

(3)

F*

O

q

=

F*

0

qCSo

SO

and

0

(IX*

x

rSo)

7

Fundamental groups of elliptic curves

Viewing

f

and

g

as elements of

End(X,Q)

t h a t ( 3 ) h o l d s i f and o n l y i f f o r some

F:X

a s t r a i g h t f o r w a r d p r o o f shows x

X

x x x (&Ix):

x x

-+

x

(Ix,Q): one has

F

0

3.1.

(Ix,g) =

and

f

Proposition:

F

-+

x

x

.

(g,IX) = g

0

and t h e i n c l u s i o n s

X

-+

However one t h e n has: nlX*

W i t h t h e homotopy r e l a t i o n d e f i n e d above

(X,O)

i s trivial. Proof:

Let

F(x,y)

= f(x)

.

+ g(y)

Then

F

0

(0 x

F

0

(Ix x Q ) ( x ) = F(x,Q)

I x ) ( x ) = F(0,x) = f(Q) + g ( x ) = g ( x )

.

and

As

t o expect

End(X,g) X n,(X,g)

.

= f(x)

Q.E.D.

i s c o u n t a b l e (see H a r t s h o r n e [5]), t o be n o t t r i v i a l .

.

i t i s more r e a s o n a b l e

I n f a c t f o r t h e usual topology

n,(X,E) = z x z We m o d i f y t h e d e f i n i t i o n i n [3] i n t h e r a t i o n a l case which enabled us t o c a l c u l a t e some homotopy groups o f spheres. 3.2. amap

Definition:

F:

X x X - t X

PE X

some

3.3. Proof:

Two maps a , E~ End(X,Q) a r e X-homotopic i f t h e r e i s w i t h F 0 (Ix,Q) = 0 such t h a t F o ( & I x ) = a and f o r

.

one has

F 0 (P,Ix) = B I f a i s X-homotopic t o

Lemma:

a

Suppose t h a t

i s X-homotopic t o

then B

via

a = B

.

F and t h e r e a r e a f f i n e

.

opens U,,U2 and U3 o f X c o n t a i n i n g Q such t h a t F(U1 x U2) c U3 There i s - a c a t e g o r y = ind-aff e x t e n d i n g Var which i s C a r t e s i a n c l o s e d Thus a g a i n f r o m [l] t h e re i s a natural transformation (see [l]).

-I

a : HomI(U1

x U2,

-

a ( F ) ( P ) ( Q ) = F(P,Q) For a l l

P E U1

.

U3)

and a ( F )

however

-t

HomI(U1yU3U2) -

with

UH

induced f r o m a p o l y n o m i a l map

a(F)(P)(O) = 0 and t h u s

a(F)(P)

-+

UN

.

extends t o a map

i n End(X,Q) Hence a(F) has c o u n t a b l e image. The image o f a ( F ) s i n c e i s an i r r e d u c i b l e c u r v e must be a c u r v e w i t h u n c o u n t a b l e c a r d i n a l i t y U1 ( i m p o s s i b l e ) o r one p o i n t . Hence F 0 ( P , I x ) = f f o r a l l P E U1 and t h u s

x . I f t h i s p u r e l y a l g e b r a i c p r o o f i s n o t v a l i d , choose

.

U1

f o r t h e t o p o l o g y on X i n h e r i t e d f r o m t Since maps i n deform o v e r a continuum a s i m i l a r argument t h e n a p p l i e s . 3.4.

rt*

=

Proposition:

rE*/% =

End(X,O-)

.

Let

Q

, U2

and

End(X,Q)

be t h e X-homotopy r e l a t i o n on

roX*

U3

cannot Q.E.D.

.

Then

P. Cherenack

8

ni*

n;(X,g)

We s e t

and f o r s e c t i o n 6 we w r i t e

Note t h a t t h e above d i s c u s s i o n c o u l d be c a r r i e d o u t f o r an a r b i t r a r y e l l i p t i c curve

d e f i n e d o v e r d i n which case X* + nl (X',g) = HomVar,((X,@,

X'

Also

need n o t equal Z X Z

n;(X,g)

s t r u c t u r e on

4.

.

F . i n a l l y f u l l use o f t h e monoidal

w i l l be used i n showing t h a t

X*

The o p e r a t i o n on

.

(X',0))

-

*! .

i s a monad on

C

n;(X,Q).

Huber's approach [6] does n o t l e a d t o o p e r a t i o n s on

n;(X,g)

Let

,

.

X

An element a E End(X,g) has been be an e l l i p t i c c u r v e d e f i n e d o v e r d X* which we denote by a* i d e n t i f i e d i n s e c t i o n 2 w i t h an element o f ro

nl(X,g)

F o l l o w i n g common d e f i n i t i o n s f o r m u l t i p l i c a t i o n on a * , ~ *E

F*

one must f i n d a

x

X*,X)

x

(see [2])

for

such t h a t

.

.

.

Q E X

for all on

HomVar,(X*

E

= a* I F* o (IX*,@) = 6 a n d F * ( ( @ x X*) U ( X * x 0)) = 0 . be t h e r e s t r i c t i o n o f F* t o X x X Then F 0 (Q,IX) = and 0) = 0 C l e a r l y F d e f i n e s a X-homotopy between and F 0 (Q,IX)

F

for

F*

(Q,IX*)

0

Let F(X

ri*

.

*

Thus u s i n g t h e r e s u l t s o f t h e l a s t s e c t i o n

Q E X

.

Then

a*

B* = F*

o

(IX*,IX*)

0

.

= a*

F

(Q,IX) =

0

a

Thus m u l t i p l i c a t i o n

n;(X,O) d e f i n e d i n t h i s way i s j u s t p r o j e c t i o n on t h e f i r s t f a c t o r . M o t i v a t e d by t h e d e f i n i t i o n o f t h e homotopy r e l a t i o n i n s e c t i o n 3 one

must f i n d f o r

P E X

a

Fp E HomVar(X x X,X)

such t h a t i f

f,g

E

End(X,O) =

( a f t e r i d e n t i f i c a t i o n ) then Fp 0 (0,IX) = f and Fp 0 (IX,P) = g n;(X,@ I n t h a t case t h e m u l t i p l i c a t i o n o f f and g can be d e f i n e d t o be Fp

0

(IX,IX)

.

(f I n t h a t case t h e o p e r a t i o n turning

End(X,g)

5.

End(X,g)

0.

One c a n t h e n t a k e

Fo

so

i n which case i f +* i s t h e o p e r a t i o n a e f i n e d on

t h a t FO(Q,R) = f ( Q ) + g(R) ni(X,g)- one has

+

P =

F o r s i m p l i c i t y we l e t

.

+*

g)(M) = f(M)

+* on n;(X,g)

into a ring.

+

.

g(M)

= End(X,g)

i s j u s t the operation

See H a r t s h o r n e [5] where p r o p e r t i e s o f

a r e described.

Fundamental groups o f a f f i n e models o f an e l l i p t i c curve. One can imbed an e l l i p t i c c u r v e

X

in

IP2

by a c l o s e d immersion

i: X + lP2 where X i s d e f i n e d by a c u b i c e q u a t i o n . See H a r t s h o r n e [5; p. 3281. L e t L be t h e l i n e a t i n f i n i t y . Then Y = X - (X n L) i s an Y . Then Y a f f i n e model f o r X which we now c o n s i d e r . Suppose t h a t i s a p a r t i a l monoid under t h e same o p e r a t i o n s as

X

.

c€

Using t h e same

c o n s t r u c t i o n s a s g i v e n i n t h e p r e v i o u s s e c t i o n s one can d e f i n e

nT(Y ,0) sn

9

Fundamental groups of elliptic curves

t h a t t h e X-homotopy r e l a t i o n i s t h e d i a g o n a l r e l a t i o n ( t h e homotopy r e l a t i o n f o r P r o p o s i t i o n 3.1 m i g h t a l s o become t h e d i a g o n a l

+ on

r e l a t i o n ) and t h e p a r t i a l o p e r a t i o n induced f r o m t h e p a r t i a l o p e r a t i o n

.

Y Thus nf(Y,Q) = End(Y,Q) = Homvar*l(Y,g), (Y,g)) and f o r one has CL + B = r i f r ( P ) = a ( P ) T B ( P ) E Y f o r a l l P E Y

map Y

+

extends u n i q u e l y t o a map

Y

X

(see

X

-+

i n j e c t i o n ( a r i s i n g from t h e f u n c t o r i a l i t y o f u:

End(Y,g)

-+

15;

II?)

a , E~ n?(Y,g)

.

Now as e v e r y

p. 431) t h e r e i s an

.

End(X,Q)

One t h e n f i n d s : 5.1. monoid Proof:

Proposition:

has no complex m u l t i p l i c a t i o n , t h e p a r t i a l

X

If

n1 (Y ,Q) generates t h e group u

.

( Z,+)

i s a homomorphism s i n c e

u(f+g)(P)

= (f+g)(P) = u(f)(P) = [u(f)

= f ( P ) + g(P) +

u(g)(P)

u(g)l(P) The i d e n t i t y Iyon Y for P E Y If X has no complex m u l t i p l i c a t i o n +

.

Ixgenerates

S i n c e i n t h i s case

.

extends t o t h e i d e n t i t y Ixon X End(X,Q) Z , See [5; pp 329-3301.

End(X,Q)

u ( I y ) = Iy, we a r e done. Q.E.D.

and

We use t h e f o l l o w i n g f a c t s t o be found i n Seidenberg [9; p. 57, 59, 701 (a)

One can choose Q t o be a p o i n t o f i n f l e c t i o n .

(b)

The n i n e p o i n t s o f i n f l e c t i o n o f

then s a t i s f y t h e

X

3X = Q and e v e r y p o i n t o f

equation

satisfying this

X

equation i s a p o i n t o f i n f l e c t i o n . Consider t h e s u b l a t t i c e o f

E generated by 1 and

a n a l y t i c isomorphism (see H a r t s h o r n e [5;

JI: with

$(O)

T

.

There i s an

pp 326-331 f o r t h i s and below)

e/A +

x a curve i n

X

t h e p o i n t a t i n f i n i t y on t h e y - a x i s w i t h

P2(e)

d e f i n e d by y 2 = 4x3 and

g2,g3 E E

infinity,

$(O)

.

-

g2x

- g3

B u t one r e a d i l y sees t h a t s i n c e i s a point o f inflection.

group homomorphism.

Let

~ ( 0 ) i s the only point a t

0=

(~(0)

a point o f inflection. W i t h t h i s s e t u p we s p e c i a l i z e t o

T

= i

.

L

The map

iI

R = End(X,g) = {a 1 X = { J I ( ~ ) } . Then

+ ibla,b

JI

1

is a

J I ( T )i s

Then f r o m [5; Theorem 4.191

one has We suppose t h a t

.

We see f r o m t h e above ( i n p a r t i c u l a r ( b ) ) t h a t

En}

S = End(Y,Q)

. c o n s i s t s o f those

P. Cherenack

10

z # 0 1 zw i t n + mi (n,mEZ) 3 1 implies w = -i t j t k i f o r i,k E 22 , An easy check shows t h a t the only 3 z s a t i s f y i n g t h i s requirement i s z = 0 , l . Hence n;(Y,g) = I I Y ) U CQI and n;(Y,G) generates t h e f i r s t f a c t o r o f nf(X,g) = Z X Z 1 1 1 Let W = Y { $ ( - Ti), $ ( T ) , $ ( - $ 1 Then W i s an a f f i n e open contained i n X and one can show t h a t End(W,g) = { ? l , k i } u {Ql I n t h i s case End(W,Q) i s closed under t h e m u l t i p l i c a t i o n o f End(X,Q) and a d d i t i v e l y generates End(X,Q) 1 I n between l e t U = Y - {I)(-7)) Then End(U,Q) = C?11 U {Ql w i t h the same generated s e t as End(Y,Q) , z

E

R such t h a t f o r

-

.

.

.

.

.

6. Some other fundamental groups i n t e r n a l t o algebraic v a r i e t i e s We f i r s t s t a t e a r e s u l t whose proof (using f o r instance [5; ex. 4.10,

p . 3381) i s reasonably clear, 6.1. Proposition: L e t X and X ' be e l l i p t i c curves defined over 0 XI Then (using the Suppose t h a t t h e r e i s a f i n i t e s u r j e c t i v e map f : X n o t a t i o n o f section 3) n1x* ( x , g -t = n1x* (X',Q)+ = n y ) * ( x l , Q ) + , -f

.

X* Suppose t h a t no s u r j e c t i v e map f: X + X ' exists. Then nl (X',Q)+ = Q I n [3] we ( e f f e c t i v e l y ) took X* = 0 and showed w i t h appropriate

.

.

d e f i n i t i o n s o f homotopy and m u l t i p l i c a t i o n t h a t one could obtain: x* 1 Z , S1 t h e u n i t c i r c l e . ( a ) nl ( S ,*) X* n (b) nl ( C , I ) = e

.

(c)

n;*(sn,*)

=

e

(n > 1 )

.

(d)

n:*(S1,*)

=

e

(n > 1 )

.

O f course i f

trivial.

Also i f

trivial.

When

n{J(X))*(Y,Q) i n g case i s

X

X*

i s an e l l i p t i c curve then nl (X',Q) X* X* i s p r o j e c t i v e and X I a f f i n e then nl (X',Q) is X i s an a r b i t r a r y curve one might take n,(Y,R) = X* = B and

XI

is

.

where J(X) i s the Jacobean v a r i e t y o f X Another i n t e r e s t X* = Mn(C) , t h e c o l l e c t i o n o f n x n matrices w i t h e n t r i e s from

a. REFERENCES [l] Cherenack, P., A Cartesian cl,os$d extension o f a category o f a f f i n e schemes, Cahiers Topologie Geometrie D i f f . (1982), 291-316. [2] Cherenack, P., Basic aspects o f u n i r a t i o n a l homotopy theory, Quaestiones Mathematicae 3 (1978), 83-113. [3] Cherenack, P., Cones and comparisons i n i n d - a f f i n e homotopy theory, Quaestiones Mathematicae 6 (1983), 49-66.

Fundamental groups of elliptic curves

[4] [5] [6]

Grothendieck, A., A l g e b r a i c Stacks, i n f o r m a l notes. Hartshorne, R., A l g e b r a i c Geometry (Springer, B e r l i n , 1977). Huber, P.J., Homotopy Theory i n General Categories, Math. Annalen 144 (1 961 ) , 361 -385. [7] MacLane, S., Categories f o r t h e Working Mathematician (Springer, B e r l i n , 1971 ). [8] Popp, H., Fundamentalgruppen a l g e b r a i s c h e r Mannigfal t i g k e i t e n (Springer, B e r l i n , 1970). [9] Seidenberg, A . , Elements o f t h e Theory o f A l g e b r a i c Curves (AddisonWesley, Reading, Mass., 1968).

11

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Croup and Semigroup Rings G.Karpilovsky (ed.) 0 Ekevier Science Publishers B.V. (North-Holland),1986

PROPERTY

13

I N COMMUTATIVE MONOID RINGS

E

Robert GILMER Department o f Mathematics F l o r i d a State University Tallahassee, FL 32306-3027,

USA

There has been e x t e n s i v e work on t h e groblem o f determining equivalent conditions, f o r various ring-iheoretic properties E , i n order t h a t a commutative semigroup r i n g should have p r o p e r t y E This paper i s a survey o f some known r e s u l t s i n t h i s area. We focus on p r o p e r t i e s E t h a t a r e r z l a i d e i t h e r t o c h a i n c o n d i t i o n s o r t o unique element f a c t o r i z a t i o n .

.

1.

INTRODUCTION

R

Let

be a commutative r i n g , l e t

operation w r i t t e n additively,

.

R

S

T

and l e t

T

as "polynomials"

S

sometimes w r i t e T = RCSI

write

.

rlXSl

,

T = R[X;S]

.

S

over

unitary?

Noetherian? cancellative?

(9,)

(9,) S

, one

R[X;S]

have m q - i h e o n e t i c pnupemLg

,

we

E ?

faces t h e p r e l i m i n a r y question of t h e g e n e r a l i t y

an i n t e g r a l domain? torsion-free?

For example, a field?

a group?

is Is

R

S

assumed t o be assumed t o be a

U s u a l l y one wants t o answer

b u t i n most cases,

i s n ' t known unless some r e s t r i c t i o n s are placed on i s a monoid.

S

.

(9,)

i n as g r e a t g e n e r a l i t y as possible,

As minimal hypotheses,

and

R

our n o t a t i o n r e f l e c t s t h e f a c t t h a t i n

i n which t h e question i s t o be answered. monoid?

and

R

This paper i s concerned w i t h v a r i o u s cases o f

U n d a w h a i c o n d l t i o ~d o a

(9,)

to

semigroup r i n g s have o f t e n been viewed as g e n e r a l i -

t h e f o l l o w i n g generic q u e s t i o n

Before a t t a c k i n g

with coefficients i n

N o r t h c o t t ' s n o t a t i o n , b u t more f r e q u e n t l y we

E i t h e r way,

z a t i o n s o f polynomial r i n g s .

(9,)

rnXSn

t... t

To i n d i c a t e t h e r e l a t i o n o f T

commutative r i n g theory,

and

be t h e semigroup r i n g o f

We f o l l o w t h e n o t a t i o n of N o r t h c o t t 114, p.1281 i n w r i t i n g elements o f

exponents i n

to

be a commutative semigroup, w i t h

we assume throughout t h i s paper t h a t

R

R

t h e answer and/or

S

.

i s unitary

We discuss p r o p e r t i e s t h a t are r e l a t e d t o e i t h e r ( 1 )

c h a i n c o n d i t i o n s o r ( 2 ) unique element f a c t o r i z a t i o n ;

a t r a n s i t i o n property

i n t h i s connection i s t h e ascending c h a i n c o n d i t i o n f o r p r i n c i p a l i d e a l s , f o r i t r e l a t e s t o b o t h (1) and ( 2 ) .

R. Gilmer

14

2.

THE NOETHERIAN AND ARTINIAN PROPERTIES. Under what c o n d i t i o n s i s

Noetherian?

R[X;S]

An answer t o t h i s question

S i s a group had already been

was given by Eudach i n [l];t h e case where resolved by Connell [31. THEOREM 1. (Budach [ l ] ) The if and only if R ~2Noethenian and

in

R[X;S]

monoid

Some i m p l i c a t i o n s r e l a t e d t o Theorem 1 are easy t o see: Noetherian,

then

generated,

then

S

Noethezian

S LA m e l g $enmafed.

if

R and S are Noetherian, and i f S i s finitely i s Noetherian. The b u l k o f Budach's p r o o f i s t h e r e f o r e

devoted t o showing t h a t a Noetherian monoid i s f i n i t e l y generated. case

where

S

implication

is

R[X;Sl

is

(see [5,

cancellative,

there

are

elementary

proofs

I n the this

of

Cor. 5.41 o r [13, p.2021), b u t i n t h e general case, t h e

o n l y known p r o o f uses a non-elementary primary decomposition theorem f o r A r i n g theoretic

congruences on a Noetherian monoid due t o Drbohlav 141.

p r o o f t h a t a Noetherian monoid i s f i n i t e l y generated would be o f i n t e r e s t . i s Noetherian i f

Such a p r o o f could be obtained by showing ( i ) R[X;S] and

are Noetherian,

S

and ( i i )

i s f i n i t e l y generated i f

S

R

is

R[X;S]

Noetheri an. The r e s u l t corresponding t o Theorem 1 i n t h e case o f A r t i n i a n r i n g s i s t h e f o l l o w i n g theorem. THEOREM 2.

(Zel'manov C151)

and o d y if R

Again, that on

The

and

S

w

o r on

S

R[X;S]

and

S

Noetherian,

and

S

e

I n proving

and

R

I n fact, i t i s clear that i s finite,

S

a r e commutative,

3f

S

LA

602%

R[X;S]

R[X;S]

implies that

unitary R

R

ring i s

Hence, Theorem 2 f o l l o w s from t h e

S

in

i n b a s i c a l l y f i v e steps,

as

A ~ ~ - t i n i a n and

Noethdan,

.then

.

The p r o o f

2) 3)

is

i s Artinian

of

P r o p o s i t i o n 3 i s obtained

follows. 1)

R

i s Artinian i f

and s i n c e an A r t i n i a n

t h e Artinian property i n

PROPOSITION 3.

by Connell [31.

i s a monoid can be d e r i v e d much more d i r e c t l y than by

and S i s b o t h A r t i n i a n and Noetherian. f o l l o w i n g r e s u l t (see 15, Th. 20.51).

U

S

concerning commutativity o r existence o f i d e n t i t y elements.

o b t a i n i n g i t v i a [15]. i s Artinian

A d i n i a n it

.

i s A r t i n i a n , Zel 'manov imposes no r e s t r i c t i o n s

A p r o o f o f Theorem 2 i n t h e case where

unitary,

e

in

Ring R[X;S]

monod

in

Theorem 2 was proved f o r groups is finite if

S R

JA A / L t i n i a n

An A r t i n i a n monoid s a t i s f i e s d.c.c. on i d e a l s . A c a n c e l l a t i v e monoid s a t i s f y i n g d.c.c. on i d e a l s i s a group. P r o p o s i t i o n 3 i s t r u e f o r a group S

.

15

Property E in commutative monoid rings

4) that

- denotes

If

t h e c a n c e l l a t i v e congruence on

S

,

then (1)-(3)

show

S/- i s a f i n i t e group. It f o l l o w s f r o m t h i s t h a t S i s p e r i o d i c .

5)

i s f i n i t e since

S

S

i s f i n i t e l y generated and p e r i o d i c .

We remark t h a t e q u i v a l e n t c o n d i t i o n s f o r a commutative semigroup r i n g to

ROIX;SO]

be

Noetherian o r

Artinian,

e x i s t e n c e o f an i d e n t i t y element i n

with

or i n

Ro

no assumptions c o n c e r n i n g

, have been determined i n

So

[61. ASCENDING CHAIN CONDITION FOR PRINCIPAL IDEALS.

3.

The ascending c h a i n c o n d i t i o n f o r p r i n c i p a l i d e a l s (a.c.c.p.1 most f r e q u e n t l y domains.

i s studied

i n r e l a t i o n t o u n i q u e element f a c t o r i z a t i o n i n i n t e g r a l

D

I t s r e l e v a n c e t o t h a t t o p i c i s t h a t an i n t e g r a l domain

D i s a GCD-domain s a t i s f y i n g a.c.c.p.

f a c t o r i a l i f and o n l y i f ask about a.c.c.p.

is

Rather t h a n

i n g e n e r a l commutative monoid r i n g s , we b e g i n w i t h a more

r e s t r i c t i v e question. (9) dou

A,j.lnme fhat

The h y p o t h e s i s i n

= T

R[X;S]

hold in

a.c.c.p.

(9)

Li

M

i n t e g n a l domain. U d e n w f m i c o n d i t i o m

T ? that

T

i s a domain means, o f course, t h a t

i s torsion-free

r i n g s over

R

s e r v e as f a i r models o f what may be expected i n answer t o

lative,

(Q,)

i n t h e case where

and f o r q u e s t i o n

(Q)

p o l y n o m i a l r i n g over a domain a.c.c.p.

Also,

answers

(Q)

.

, R

S

then

i n t h e case where

T

and

R 4aki~fiw a.c.c.p.

show t h a t R

i s a group,

S

polynomial

i s t o r s i o n - f r e e and c a n c e l -

i t i s easy t o

,

THEOREM 4. ( G i l m e r and Parker [7,(7.14)]) J{ and S Li a t o m i o n - b e e gnoup, ,then R[X;S] onCy ije

I n general,

is a

S

generic questions

and c a n c e l l a t i v e .

R

domain and

T

if

is a

simultaneously s a t i s f y the following result

R i,4 M i n t e g n a l m t b f i w a.c.c.p.

S

and each nonseno d e m e n t oE

domain A {

and

Li of .tgpe

(O,O,O.. . I .

if

More g e n e r a l l y , [ 7 1 shows t h a t i f R i s a domain s a t i s f y i n g a.c.c.p. S i s a t o r s i o n - f r e e c a n c e l l a t i v e monoid w i t h q u o t i e n t group G

that

S

satisfies

(O,O,O

,... 1,

hand,

if

then

R[X;S]

a.c.c.p. R[X;SI

and each

s a t i s f i e s a.c.c.p.,

n o t be o f t y p e (O,O,O,...), a.c.c.p.,

polynomial

rings,

has

element

of

G

i s of

type

On t h e o t h e r

t h e n each nonzero element of ( Q ) remains open.

G

need

and q u e s t i o n

Work on t h e non-domain case o f satisfying

nonzero

i s a domain s a t i s f y i n g a.c.c.p.

and such

been

(Q,)

limited

, where E i s t h e p r o p e r t y o f almost

entirely t o

a case t h a t remains unresolved.

t h e case o f

G e n e r a l i z i n g a.c.c.p.,

R. Gilmer

16

Heinzer and L a n t z i n 1101 c o n s i d e r e d t h e concepts o f polynomial R

satisfies

n-acc and

.

n E 2'

each

Thus

i n d e t e r m i n a t e s over

s a t i s f i e s pan-acc

1-acc

.

R

n-acc

implication.

i n h e r i t e d by a.c.c.p.,

then

For

n > 1

from R ; i f

R[Y]

R

a.c.c.p.,

and i f t h e i d e a l (0) o f then

satisfies

R[Yl

4.

in

,

the

n-generated i d e a l s

satisfies

R

Let

, t h e y show t h a t n-acc need n o t be i s a zero-dimensional r i n g s a t i s f y i n g R

if

R

i s quasi-local,

satisfies

has o n l y f i n i t e l y many weak-Bourbaki

a.c.c.p.;

if

R

is

Noetherian,

then

Other t h a n t h e s e f o u r r e s u l t s , l i t t l e more i s known

s a t i s f i e s pan-acc.

about t h i s problem, s a t i s f y i n g a.c.c.p.

.

s a t i s f i e s a.c.c.p.;

R[Y]

primes, R[Y]

R

pan-acc n E 2'

n-acc f o r Y denote a s e t of o r pan-acc i n R [ Y ] i m p l i e s t h a t Heinzer and L a n t z i n v e s t i g a t e t h e if

i s t h e same as a.c.c.p. Clearly

and For

i f each ascending sequence o f R

t h e same c o n d i t i o n i s s a t i s f i e d i n reverse

n-acc

The d e f i n i t i o n s a r e as f o l l o w s ,

stabilizes,

ring R of

rings.

and i n p a r t i c u l a r , such t h a t

R[Yl

no example i s known o f a r i n g

R

does n o t s a t i s f y a.c.c.p.

UNIQUE ELEMENT FACTORIZATION AND RELATE0 CONDITIONS.

Gilmer and Parker determined i n

[71 e q u i v a l e n t c o n d i t i o n s f o r a commuta-

t i v e monoid r i n g t o be a f a c t o r i a l domain. characterization

of

f a c t o r i a l i f and o n l y i f original

T h i s work was based on t h e

such domains mentioned i n t h e

strategy o f

equivalent conditions,

D

171 was,

f o r a domain

separately,

domain s a t i s f y i n g a.c.c.p.

l a s t section:

i s a GCD-domain s a t i s f y i n g a.c.c.p. for

T

D

is

Thus, t h e

, t o determine

T = R[X;S]

t o be ( 1 ) a GCD-domain, o r ( 2 ) a

S e c t i o n 3 i n d i c a t e s t h a t we were unsuccessful i n

o u r a t t e m p t on (21, b u t t h e f o l l o w i n g theorem answers ( 1 ) and i t determines equivalent conditions f o r

S

T t o be f a c t o r i a l .

THEOREM 5. (Gilmer and Parker [ 7 ] ) Let R be an i n l e p a 4 domain, let b e a t o m i o n - b e e c a n c d d v e monoid WAX q o - t i e n t p o u p G , and let

.

T = R[X;Sl

( a ) T J2 a GCD-domain if and o d g ije R GC D-monoid. (b) monoid,

and 04A[

T J2 f a c t o h i d

AL

and

element of

eQChtWnpm

J2 a GCD-domain

R J2 j e a c t o n i d , G i~ of 4 p e (O,O,O

S

and S

42

a

.i~ a #lactohie(

...) .

Beyond t h e f a c t o r i a l p r o p e r t y , one may c o n s i d e r o t h e r p r o p e r t i e s

E that

a r e e i t h e r more r e s t r i c t i v e (such as t h e p r o p e r t y o f b e i n g a p r i n c i p a l i d e a l domain (PID) o r a p r i n c i p a l i d e a l r i n g ( P I R ) ) o r more general ( f o r example, t h e p r o p e r t y o f b e i n g a K r u l l domain). These q u e s t i o n s have been c o n s i d e r e d i n 171,

191, 181. I n p a r t i c u l a r , Theorem 8.4 o f 171 i n c l u d e s t h e f o l l o w i n g .

statement.

Property E in commutativemonoid rings

a

THEOREM 6. ficecd and

iA

S

cancellative,

conditions f o r

T = R[X;S]

RCX;Sl

Z

PID if and

J2 a

,

eithen Zo

iAomonphcc .to

intqm, on fo

nonn@ve

For

The monoid d q

S

17

2he additive p o u p

the

R i~

o d g Ac

additive monoid

QL

integem.

Q[

Hardy and Shores have determined i n [91 equivalent

.

PIR

t o be a

complicated than t h a t o f Theorem 6,

Statements o f t h e i r r e s u l t s are more and r a t h e r than repeat them here, we

r e f e r t h e i n t e r e s t e d reader t o C9l o r t o [5, 5191.

Glastad and Hopkins [81

have considered t h e same problem i n t h e case where

S

b u t t h i s case o f t h e problem remains open. easy t o show, w i t h no r e s t r i c t i o n s on

S

, that

t h a t i s , a P I R w i t h n i l p o t e n t maximal i d e a l p # 0 and

o f characteristic

i s not c a n c e l l a t i v e ,

On t h e other hand, is a

R[X;SI

- if and o n l y

R

-

PIR

4peci.d

if

i s a c y c l i c group o f order

S

it i s

i s a field

pn f o r some n.

The class o f K r u l l domains contains t h e classes o f f a c t o r i a l domains and Noetheri an

integrall y

closed

domains,

and

Krul 1 domains

are

general l y

considered t o c o n s t i t u t e a reasonable class t o seek extension o f r e s u l t s known f o r f a c t o r i a l domains.

Recall t h a t a domain

D

i s a K r u l l domain i f

t h e r e e x i s t s a f a m i l y V = ( V a l a A o f rank-one d i s c r e t e v a l u a t i o n overrings o f D such t h a t ( 1 ) D = ;Va , and ( 2 ) f o r each d E D , d # 0 , t h e set ( a E A 1 d i s a nonunit o f V a l i s finite. I f D i s f a c t o r i a l , then ( D ( p a ) l a E A , where { p a l i s a complete set o f nonassociate primes o f D , serves as an acceptable choice f o r V ; i f 0 i s Noetherian and i n t e g r a l l y closed,

we

primes o f

can take V = I D D.

Pa

where

l,

is

{PalacA

I n order t o determine conditions under which

the

set

o f minimal

R[X;S]

i s a K r u l l domain,

Chouinard M i n t r o d u c e d t h e n o t i o n o f a K r u l l monoid,

defined as f o l l o w s .

G i s a K n u U monoid i f t h e r e e x i s t s a f a m i l y {waIa A o f homomorphisms from G onto Z such (1) S = na{g E G I wa(g) 2 0 1 , and (2) f o r g E G , g # 0 , (a c A 1 wa(g) # 01 i s f i n i t e . Chouinard's c h a r a c t e r i z a t i o n theorem was t h e A

cancellative

monoid S w i t h

quotient

group

f o l l o w i n g statement. THEOREM 7. (Chouinard [2]) t h e monoid

S.

K n u U domain, .type

(O,O,O , . . . I

Thm

R[X;S]

Let

6e f i e p o u p oe inventi6Le elem&

iA a K n u U domain

A6

and o d g if

S J2 a KnuAA monoid, and each nongeno element oc

o+!

R iA a H b oc

.

The case o f Theorem 7 where

S

Matsuda i n [12l. As s t a t e d i n the i n t r o d u c t i o n , properties

H

i s a group had already been established by (Q,)

has been considered f o r many other

E t h a t haven't been discussed i n t h i s paper.

For a d d i t i o n a l

R. Giber

18

material

on

this

topic,

the

interested

reader

may c o n s u l t

[111,

151,

and t h e references o f [51, e s p e c i a l l y those authored by R. Matsuda. REFERENCES [11 Budach, L., Monatsb. Deutsch. Akad. Wiss. B e r l i n 6(1964), 81-85. C2l Chouinard, L., Canad. J . Math. 33(1981), 1459-1468. [31 Connell, I . , Canad. J . Math. 15(1963), 650-685. [41 Drbohlav, K., Math. Nachr. 26(1963), 233-245. C5l Gilmer, R. , Commutative Semigroup Rings (Univ. Chicago Press, Chicago, 1984). Chain Conditions i n Commutative Semigroup Rings, J. Algebra C6l (toappear 1. C71 Gilmer, R. and Parker, T., Michigan Math. J. 21(1974), 65-86. [81 Glastad, 8. and Hopkins, G., Comment. Math. Univ. Carolinae 21(1980), 371-377. C91 Hardy, 6. and Shores, T., Canad. J . Math. 32(1980), 1361-1371. [ l o 1 Heinzer, W. and Lantz, D., J. Algebra 80(1983), 261-278. G., Commutative Group Algebras (Marcel-Dekker, New [ill Karpilovsky, York, 1983). 1121 Matsuda, R . , B u l l . Fac. Sci. I b a r a k i Univ. Ser. A Math. 7(1975), 29-37. , Proc. Japan Acad. 59(1983), 199-202. [131 D. G., Lessons on Rings, Modules and M u l t i p l i c i t i e s l141 KFEkcott, (Cambridge Univ. Press, London, 1968). [151 Zel'manov, E. I . , S i b e r i a n Math. J. 18(1977),557-565.

Groupand Semigroup Rings G.Karpilovsky (ed.) 0 Elsevier SciencePublishersB.V.(North-Holland),1986

19

CONDITIONS CONCERNING PRIME AND MAXIMAL IDEALS OF COMMUTATIVE MONOID RINGS Robert GILMER Department of Mathematics Florida State University Tallahassee, FL 32306-3027, USA Prime and maximal ideals play a central role in the general theory of commutative rings. This paper contains an exposition of some known results concerning prime and maximal ideals of a commutative monoid riug T Topics covered include the (Krull) dimension of T , conditious on chains of prime ideals of T , the existence of only finitely m n y maximal or prime ideals of T , and conditions under which T is a Dedekind domain, a general ZPI-ring, or a Hilbert ring.

.

1.

INTRODUCTION For a commutative unitary ring R

, denote by

Spec(R)

the prine spectrum

-

of R that is, the set of prime ideals of R -and by MSpec(R) the subset of Spec(R) consisting of maximal ideals. Several important structure properties of commutative rings deal with the sets Spec(R) and MSpeccR) for example, the Krull dimension of R , chain conditions on the set of prime ideals of R , and the properties of being a general ZPI-ring o r a Hilbert ring. This paper is a survey of some known results concerning Spec(T) and

-

, where T is a conmutative monoid ring. All rings considered in this paper are asbumed to be commutative and unitary. All semigroups are assumed to be commutative monoids, with operation written additively. MSpec(T)

2.

I(RULL DIMENSION

Recall that a conunutative unitary ring R has write dim R n , if there exists a chain

dDnWhn n

, and

we

R , but no longer such chain; if no such integer n exists, then R is . i n 6 L n L t e - d h e ~ 4 L o d (dim R =, m ) . If T R[S] is a commutative monoid ring over R , we consider the problem of relating dim T The case of interest is where dim R is to dim R and to properties of S

of proper primes of

.

finite, eince it is easy to see that T is infinite-dimensional if R is infinite-dimensional. Much work has been done on this question in the case

R. Gilmer

20

k RIZo]

where T

=

R(k)

=

RIX1,

..., \ ]

is a polynomial ring in finitely

many variables over R , and this work has served as a model for much of what has been done for more general commutative monoid rings. Let no dim R and + One basic result, due to Seidenberg let % = dim Rck) for each k E Z ( 2 2 1 , “231, is that no + 1 5 n1 5 2n0 + 1 ; these bounds are best possible in the sense that, for any integers mo, ml with 0 6 mo 5 ml - 1 2 2m0 , there exists an integral domain D such that dim D mo and dim D(1) ml On

.

-

-

.

the other hand, if R is either Noetherian or a Prufer domain, then + = mo + k for each k E Z [171, [231. One important result in the

%

dimension theory of polynomial rings is what is known as Jaffard’s Special Chain Theorem. To state this result, we need the following definition. A chain Po < PI < of proper primes of RCk)

ideal (PinR)R(k)

... < m’

is called a & P e w chain if, for each i

, the

is also a member of the chain.

THEOREM 2.1. (Jaffard [13, Th.3, p.351) L a R b e a C C ~ f n t ~ ~ Uu v de Y L i n g and Let k be a p o b w v e i n t e g a . (1) 7 6 P LA u phime idideae 06 R(~) u6 ,3inite h d g h t t , then p can be m.aLLzed ad t h e R U W I . dement ~ ~ 0 6 a ~ p t c i a echain 06 phimu 06 R(~) 0 6

lengRh t (2) 06

.

dim R(~)

can be t&zed

ah

t h e Length oh

G

d p e c k l chein

06

pnimed

R(~) , One important consequence of Theorem 2.1 is that

where MSpec(R) = {Ma)

and h(MaR(’))

denotes the height of MaR(k)

.

This

result allows reduction to the case of quasi-local rings in considering certain questions about the dimension theory of polynomial rings. To mention s final known result from the dimension theory of polynomial m

rings, Arnold and Gilmer determined in [l] those sequences {mililO of non-negative integers that can be realized in the form {dim R, dim R (1) , dim R(2), . . . I for some commutative unitary ring R There are, of course, gaps in what is known about the dimension theory of polynomial rings. One

.

open question in the area asks whether dim D(k) * k + dim D for a factorial domain D ; this is equivalent to asking whether dim D is equal to the valuative dimension of D when D is factorial. Arnold and Gilmer in [ 2 ] generalized some of the known dimension theory of polynomial rings to the case of monoid rings R[S]

, where

S

is

Conditions concerning prime and maximal ideals

cancellative. A combination of

21

[ 2 , Th.3.31 and [ 5 , Cor.11 yields the

following statement.

There is a natural extension of the notion of a special chain of prime < Pn in ideals to the case of monoid rings. Videlicet, a chain Po
0 since s is aperiodic. Hence there exists a prime ideal P of S containing I such that Pn <s> = 0 We have R[S] R[U]$R[P] , where dim R This U = S\P is a submonoid of S , s E U , and dim R[U]

-

.

.

-

.

+

implies, however, that cs + u ds + u for some u 0 U and some c,d E 2 with c f d , a contradiction to the fact that Un I = 0 Hence S is periodic, as asserted In Theorem 2.4.

.

22

R. Gilrner

CHAIN CONDITIONS FOR PRIME IDEALS

3.

Several conditions concerning behavior of finite chains of prime ideals are studied in commutative algebra. The monograph [21] by L. J. Ratliff, Jr. gives a good exposition of much of this material. We review some of the terminology in this area. A chain ("1 Po < P1
, and hence m E (l-Xs> augmentation map, m is left fixed and 1-Xs goes to 0 Hence m = 0 and R is a field. Because of the injective, inclusion-preserving correspondence

.

.

.

.

between subgroups H of S a d congruence ideals ({l-Xh(hEHl) of R[S] , the set of subgroups of S is linearly ordered, and it is easy to see that this condition implies that S is either cyclic of prime-power order or quasicyclic ( 6 , Th.19.31. We remark that Theorem 4.4 was also proved by Ohm and Vicknair [ZO], who were interested in the relation between this result and a question of Kaplansky that asked whether each chained ring is the homomorphic image of a valuation domain. L. Fuchs [31, [4,p. 1511 has recently shown that Kaplansky's question has a negative answer, but Ohm and Vicknair proved that if R[S]

is

a

chained ring, then R[S]

is a homomorphic image of a valuation

domain. It is interesting to note that the only non-Noetherian chained rings R[S] , where F is a field of characteristic p are those of the form F[Z(pm)]

.

24

R. Gilrner

-

The Noetherian ones are special principal ideal rings. In fact, if Char F p f 0 and if S <s> i e a cyclic group of order pn , then 1-XB generates the maximal ideal of P[S] and (l-Xs)pn 0

-

.

DEDEKIND DOMAINS AND GENERAL ZPI-RINGS Let I be a proper ideal of R The problem of determining whether I can be expressed as a finite product of prime ideals has classical importance. If each ideal of R is so expressible, then R is a g W U d ZPI-hing; this choice of term comes from the German “Zerlegung Primideale“, meaning “prime ideal decomposition”. An integral domain that is a general ZPI-ring is a Dedekind domain. Gilmer and Parker [9] considered the problem of determining conditions under which R[S] is a Dedekind domain or a general ZPI-ring, and the next two results follow from [9].

5.

.

Foh s # (O), .the 60UoWing condition6 me eqqlLivdent. (1) R[S] Lb a De.deLnd d o d n . ( 2 ) R Lb a 6 i d d and S ib ibomohpkic da%a to Z o h to Zo. (3) R ib a p h i n c i p d L d d d o d n .

THEOREM 5.1.

Hardy and Shores [ l l ] extended Theorem 5.2 to the case where S is not assumed to be torsion-free. Their results break down naturally into the cases where S is, or ie not, periodic. We combine these cases into a single statement. TIEOREM 5.3.

kbume that

s

Lb n o n z m , c a n c w v e , and not

tohbion-6nee.

(a) 16 S Lb not pehiodic, then R[S] ib a genehae Z P I - h i n g i d and o d y R Lb a 6 i n i t e dihect bum 06 6id& and s = zo8 H o h s s 2 8 H whehe id H f (0) Lb a &~Lte ghoup and t h e OJuf&t 06 H Lb a u n i t 0 6 R (b) 16 S Lb pehiodic, then R[S] Lb a genehae Z P I - n i t t g i 6 and O& id the doUoUlin9 h e e conditior26 &e b a t i d b i e d : (1) R Lb a genehtte ZPZ-hing, (2) s ib a h i d e ghoup, and ( 3 ) {on each p&ime p dividing Is1 ,that 0 06 Rhe domi char(R/MX) doh bOme ma&& Ld& MI 06 R Sp Lb C Y a C

.

and

& a 6.id.d.

If the hypothesio is as in Theorem 5.3, then in case (a), it can be shown

Conditions concerning prime and maximal ideals

25

that R[S]

is a principal ideal ring if it is a general ZPI-ring, but the same statement fails in case (b).

6. R[S] AS A HUBERT RING The ring R is said to be a H i t b e & hLng (or a Jacobdon h i n g ) if each proper prime ideal of R can be written as an intersection of maximal ideals of R Th.e terminology Hilbert ring stems from the fact that Hilbert's Nullstellensatz can be interpreted as saying that the polynomial ring in finitely many indeterminates over a field has the property described in the preceding sentence. The first research problem of Karpilovsky's book [14] asked for equivalent conditions in order that a commutative group ring should be a IIilbert ring. This problem was resolved independently by Krempa and Okninski [16] and by Gilmer [7], but Gilmer's paper also answers the question for monoid rings, in the case of a cancellative monoid. To wit, the following result holds.

.

Theorem 6.1 means that for S cancellative, the problem of determining whether a fixed R[S] is a Hilbert ring has the same status as that of

-

determining dim R[S] the problem reduces to the case where R[S] = R[ {X,)] is a polynomial ring in a indeterminates over R And as in the case of Krull dimension, much is known about the question of whether R[ {X,]] is a IIilbert ring, although the general problem remains open. For example, if a is finite, then R[{X,)I and R are simultaneously Hilbert rings. If a is is a Hilbert ring if and only if infinite and R is a field, then R[IX,Il

.

(RI [la]. More generally, Heinzer in [12] showed for a infinite that if R is Noetherian (or if R has Noetherian spectrum and satisfies d.c.c. on prime ideals), then R[Sl is a Hilbert ring if and only if the following conditious (i) and (ii) are satisfied: (1) IR/MI > a for each maximal ideal M of R ; (ill for each nonmaximal prime P of R , the set of primes Q of R such that Q > P and h(Q/P) 1 has cardinality greater than a a
are pairwise relatively prime. Thus Xp2-1 has at least three nonassociate prime factors over F. The general case follows in like fashion from the equality X$-l = (X-l)g(X)g(Xp) g(XPk-').

...

COROLLARY 3 . 2 . Assume that F i s a f i e l d and G = GI@ GJ, where lGil = ni > 1 and each i. Then F[Gl has a t least 2k maximal ideals.

... $Gk

i s a direct Char F X ni f o r

sum of f i n i t e cyclic groups

Proof. By induction and by Lemma 3.1, it suffices to show that if M is a maximal ideal of FIG1 8 @Gk-ll, then there exist at least two maximal

...

ideals of F[Gl lying over M.

F[Gl/M[Gl (FIG1 B, Char F Char(FIG1 @

This follows from (3.1) since

. . . @Gk-,l/M)[Gk] , where . . . @Gk-l]/M) does not divide % .

THEOREM 3 . 3 . (Lawrence and Woods [ 5 ] , Passman [9], Okninski [7]) Assume that

F i s a f i e l d and G i s an abelian group. (1) I f Char F = 0, then F[G] is sql i f and only i f

G i s finite.

31

Commutativemonoid rings

(2) If

Char F = p # 0

, then F[G]

bq.t

4 and

udg

.id

G/Gp, w h a t

G~

denotes the p-component of G, is finite.

Proof. Assume first that F[Gl is sql. If H is a subgroup of G F[H] is an algebraic retract of F[Gl , and hence is sql.

, then

Since F[Z] z F[X, X-'1 = F[X](X) is not sql, it follows that F is a torsion group. Hence G is the direct sum of its primary components G 9 Part (2) of Lemma 3.1 shows that each G has bounded order, and hence is a 9 direct sum of cyclic groups [ 4 , Th.6, p. 171. Thus, C itself is a direct sum of cyclic groupo. Moreover, if Char F = 0 , then Corollary 3 . 2 shows that

.

F is finite, and if Char F = p # 0

, then

( 3 . 2 ) implies that G/Gp is

finite. Corollary 2.3 shows that the conditions given in (1) and ( 2 ) are sufficient to imply that T is sql. This completes the proof of Theorem 3 . 3 . 4 . PROOF OF THEOREM A .

We proceed in this section t o prove Theorem A of the introduction. We have already shown in Corollary 2 . 3 that the conditions given in Theorem A imply that T is sql. To prove that they are also necessary, we use two preliminary results. The second of these, Proposition 4 . 2 , was established by Okninski in [ 6 ] for the case of a noncomutative semigroup.

LEMMA 4.1. Asswne that

is a unitary integral domain and

D

S

is a

cancellative monoid. If s E S and d E D \ I01 , then 1 - dXs is a unit of D[S] if and only if s is an invertible element of S of finite order k and 1-dk is a unit of D. k If s is an invertible element of S of finite order k and if 1-d Proof.

, then a

is a unit of D D[S]

routine computation shows that l-dXs is a unit of k -1 k-l(dXs)i with inverse (1-d Zilo Conversely, suppose that l-dXs is a

.

unit with inverse f = E:=laiXti,

(*)

1

f

Because S

- dXsf

Z;aiXt

.

where Supp(f)

Then

- Z:aidXs+ti

is cancellative and D is a domain, Supp(dXsf)

.

{s+til:

has

cardinality n From (*) it then follows that if u f Supp(f) \ Supp(dXsf) then u 0 ; similarly, if v e Supp(dXsf) \ Supp(F), then v 0. We Supp(dXsf) and 0 e Supp(f) Therefore 0 * s+ti for conclude that Supp(f)

.

,

32

R. Gilmer

+

some i, and s is invertible. Since Supp(f) = Supp(f) + us for each n E Z and since 0 E Supp(f), it follows that ns E Supp(f) for each n , and Since D[<s>] is a retract of D[S] , it consequently, s has finite order k k-lb Xis follows that f E D[<s>] Writing f = Cim0 and using (*I, it is then k straightforward to show that bi bodi for each i and bo(l-d 1; k consequently, 1-d is a unit, as asserted.

.

.

PROPOSITION 4.2

-

(Okninskit61) Assume that the commutative monoid ring

T = R[S] is sql. (1) If s is cancellative, then S is a torsion group. (2) s is periodic. Proof. By passage to (R/M)[S] for a maximal ideal of M of R , we assume without loss of generality that P. is a field. Let R* R \ { O } and

assume that T has t maximal ideals; we wish to reduce to the case where + t < I R*l If this equality fails for R , then choose n E Z such that nt + 1 < I Rln , and let F be an n-dimensional extension field of R As an R[S]-module, F[S] has a free basis of n elements, so part (3) of Proposition 2.1 shows that F[S] is sql with at most nt maximal ideals. Since

.

-

.

-.

nt < I BI" 1 I F*I , we can therefore assume, in proving that S is periodic, t Pick s E: S and let Spec(T) = {Mi}i=l For each i, there that t < I R*l exists at most one element bi E R* such that bi - XB E lli Thus, there

.

-

.

exists b E R* such that b-Xs $ U:MI that is, b-Xs and l-b-lXs are units Therefore, by Lemma 4.1, s is invertible in S and has finite of T order. Since S is cancellative, we conclude that S 10 a torsion group.

.

(2): In the general case, pick s E S and let % be the cancellative Since R[S/%] is sql, part (1) shows that there exist congruence on S

+ m,nEZ

.

,mfn,and ~ E suchthat S ms+t-ns+t. Let I = {u E S I as + u bs + u for some a, b E Z+, a f b}; note that I is an ideal of S If I n <s> = +. , then there exists a prime ideal P of S

.

containing I such that P n <s>

Q

.

We have R[S]

R[S\P]OR[PI

where

R[S\P] is a subring of R [ S ] , R[P] is an ideal, and the sum is direct as abelian groups. Thus R[S\P] is a homomorphic image of R[S] and is sql. By what has already been proved, it follows that there exist a, b E Z+ a f b, and to E S\P such that as + to bs + to Then t E I S P , a 0 contradiction. We conclude that In<s> f Q , so as + cs bs + cs for some a, + b, c E Z with a # b. Therefore S is periodic, as asserted.

-

A.

.

With these preliminaries, we are prepared to complete the proof of Theorem To do so, it suffices to prove the following statement (#).

33

Commutativemonoid rings

($1 If @;MI,...&) finite, o r

and

(2) Char(R/M1) =

T

=

R[S]

are sql, then either

... = Char(R/Mt)

= p

(1) S / r ,

is

# 0 and SICp is finite.

.

To prove (#I, there is no loss of generality in replacing S by S/C Hence, it suffices to prove that ( 2 ) holds if S is finite and is free of asymptotic torsion. Proposition 4.2 shows that S is periodic, and heuce S is a disjoint union of torsion groups {GalaEA [ 1, Cor. 101. Because S contains only finitely many idempotents, A is finite, say A {I,Z,...,rl. We wish to show that R[G ] is sql; more generally we show that if U is any i submonoid of S , then R(U] is sql. Let u be the identity element of U. If u = 0, then R[S] is integral over R[U] , so R[U] is sql. If u f 0 , let U* = Uu{Ol. By the argument just given, R[U*l is sql. We write R[u+U*] = R[U] , it follows Since X%[U*] R[U*] X%[U*]@(l-Xu)RIU*]

.

that R[U] is sql in this case as well. In particular, each R[Gi] and each (R/Mj)[Gi] is sql. Choose i such that Gi is infinite. Theorem 3 . 3 shows is finite. There that for each j, Char(R/M = p, # 0 and Gi/(G ) j pj exists, however, at most one prime p such that Gi/(G is finite. i P p , and the argument just given shows that Consequently, p1 = pt Gj/(GjIp is finite for each j We show that this implies that S/Gp is finite, For 1 6 is r , let {h )Iy be a complete set of representatives of ij j=l (GiIp in Gi If s E S , we show that B is p-equivalent to hij for some i and j. Thus, s E G for some i , and hence 8 E h + (Gi)p for ij We some j. This means that pki pkhij for some k so that 6 5 h P ij is finite, finishing the proof of both ( # ) and of Theorem conclude that S I C P

...

.

.

-

A.

h e consequence of the proof of ( I ) seems to merit a separate statement. COROLLARY 4 . 3 .

suhonoid

u

of

If T

s

.

=

R[S]

is sql, then R[U]

is sqZ f o r each

5 . PROOF OF TREOREM B.

In view of Corollary 2.4, a proof of the following statement (##I will also yield a proof of Theorem B.

R. Gilmer

34

(##I If Spec(T) finite, or p and

of

and

Spec(R)

are finite, then either

(2) there exists a prime

s/cp

Proof of (##).

(1) S / c is p such that Char R is a prime power

is finite.

As in the proof of (b), we assume without loss of

generality that S is an infinite monoid that is free of asymptotic torsion, and we-prove that (2) i s satisfied. Since T is sql, Theorem A shows that there exists a prime p such that Char(T/M) p for each M E MSpec(T) and To show that Char R SICp is finite. Lat P be a proper prime ideal of R If not, then Char(R/P) a is a power of p , we must show that Char(R/P) p 0 In this case, let N R\ P Then TN f %[S] is sql while S / L S is infinite and Char(%/PRp) = Char(R/P) 0 , a contradiction t o Theorem A. We

.

.

conclude that R/P has characteristic p

-,

-

.

.

and this completes the proof.

REFERENCES [l] Gilmer, R. and Teply, M., Houston J. Math. 3(1977), 369-385. [2] Hardy, B. and Shores, T., Canad. J. Math. 32(1980), 1361-1371. [3] Jespers, E. and Wauters, P., A Description of the Jacobson Radical of Semigroup Rings of Commutative Semigroups, this volume. [4]Kaplansky, I., Infinite Abelian Groups (Univ. Michigan Press, Ann Arbor, 1954). [5] Lawrence, J. and Woods, S.H., Proc. her. Math. SOC. 60(1976), 8-10, 161 Okninski, J. , Commun. Algebra 10(1982), 109-114. [71 , Glasgow Math. J. 25(1984), 37-44. [8] Parker, T. and Gilnrer, R., Michigan Math. J. 22(1975), 97-108. [9] Passman, D.S., The Algebraic Structure of Group R i n g s (Interscience, New York, 1977). [ l o ] Wauters, P. and Jespers, E., When I8 a Semigroup Bing of a Commutative Semigroup Local or Semilocal?, preprint.

Group and Semigroup Rings G . Karpilovsky (ed.) 0 Elsevier Science Publishen B.V.(North-Holland), 1986

35

THE GROUP OF UNITS OF A COMMUTATIVE SEMIGROUP R I N G OF A TORSION-FREE SEMI GROUP

E r i c JESPERS U n i v e r s i t y o f Cape Town, South A f r i c a We seek necessary and s u f f i c i e n t conditions f o r the u n i t s o f a commutative semigroup r i n g R [ S l t o be determined by the n i l r a d i c a l o f R[S] and the u n i t s o f R [ G I where G i s the group o f u n i t s o f S. We assume t h a t R i s a commutative r i n g w i t h i d e n t i t y and S i s a t o r s i o n - f r e e semigroup w i t h i d e n t i t y . 1. INTRODUCTION

R . Gilmer i n 131 gives a c h a r a c t e r i z a t i o n o f the u n i t s o f a commutative semigroup r i n g R[S] ,where S i s a t o r s i o n - f r e e commutative semigroup w i t h i d e n t i t y which has no n o n - t r i v i a l idempotents (e.g. c a n c e l l a t i v e semigroups).

It i s

proved t h a t a l l u n i t s o f R [ S l are o f the form u t v, where u i s a u n i t o f In

R [ U ( S ) ] and v i s a n i l p o t e n t element; U ( S ) i s the group o f u n i t s o f S. t h i s note we g ve necessary and s u f f i c i e n t conditions, i n terms o f R and S (torsion-free)

such t h a t a l l u n i t s o f R [ S I have the above mentioned form.

I n t h i s way we also g i v e a simple proof o f Gilmer's r e s u l t . the r e s u l t s we introduce some n o t a t i o n and terminology.

Before s t a t i n g

For s t r u c t u r a l r e s u l t s

on semigroups ( r e s p e c t i v e l y semigroup r i n g s ) we r e f e r t o [ll ( r e s p e c t i v e l y

PI). A l l r i n g s and semigroups considered are commutative. t i o n i s denoted m u l t i p l i c a t i v e l y . monoid. s,u,v

The semigroup opera-

A semigroup w i t h i d e n t i t y i s c a l l e d a

A semigroup S i s c a n c e l l a t i v e i f su = sv implies u = v, where

E S.

A semigroup o f idempotents i s c a l l e d a s e m i l a t t i c e .

I n t h i s note

we r e s t r i c t our a t t e n t i o n t o t o r s i o n - f r e e semigroups S, i . e . f o r a l l s , t and f o r a l l p o s i t i v e i n t e g e r s n, sn = tn i m p l i e s s = t.

E

S,

I t i s w e l l known

...................... This work was done w h i l e the author v i s i t e d the department o f Mathematics o f the U n i v e r s i t y o f Louvain, K.U. Leuven. The author thanks the U n i v e r s i t y f o r t h e i r support, and C S I R ( P r e t o r i a ) f o r t h e i r research grant.

36

E. Jespers

t h a t a t o r s i o n - f r e e semigroup i s the d i s j o i n t union o f i t s Archimedean components Say where

r

a E

r

and

i s a semilattice.

Moreover, f o r a,B

r,

E

S a S B c S a B and i f Sa contains an i d e n t i t y , which we w i l l denote by ea, then Sa

Ooviously eaeB = eag, a,B E

i s a ( t o r s i o n - f r e e ) group. ordered s t r u c t u r e

L ,

Therefore

.

x E n (Jk(R)[Sl = JJR)[SI i=I

.

of finite

To prove t h e converse, l e t

i s i n f i n i t e t h e r e e x i s t s f o r any

W

R

C -

S

By Lemma 4.2 and the f i r s t p a r t of t h e proof we o b t a i n t h a t

a g r o u p l i k e subsemigroup

series over

c_ J(R[SI).

of

, ..., s 1

1 such t h a t

S

5 Jk(R) follows.

Jk+l(R)

SO

i s a f r e e subsemigroup

Hence J m ( R ) [ S l

B = Isk

Let

o f t h e f r e e basis o f

.

.

then from t h e above and from t h e f a c t t h a t JJR)

x E J(R[S])

R[X]

a f r e e semigroup o f rank

R"Xl1

, then

and l e t 1+ rX

i s o f t h e form

For t h i s l e t R[[X]]

R[XI

be the

be t h e formal power

i s an i n v e r t i b l e element o f d = 1

- rX t r 2 X 2 - r3X 3 + ... .

E. Jespers and f. Waurers

52

, hence

But d E R [ X ] L e t N(R)

rn = 0 f o r some n

.

€IN

This f i n i s h e s t h e proof.

a non-denumerable f i e l d Amitsur [21 has shown t h a t nothing i s known. I'

Jl(R)

= N(R)

I'

.

= N(R)

I n general

i s equivalent t o the famous Koethe conjecture [181 "If a r i n g

R[XI

L

L i s contained i n N(R)".

then

( c . f . [191):

t h e even stronger conjecture polynomial r i n g

J1(R)

For algebras over

Krempa [191 has shown t h a t the v a l i d i t y o f the statement

R contains a one-sided n i l i d e a l

'I

Under

R i s a n i l r i n g , implies t h e

i s a n i l r i n g " , i t f o l l o w s t h a t Jn(R) = N(R)

This l a s t c o n d i t i o n i s o f t e n s a t i s f i e d because J1(R) R

.

denote the upper n i l r a d i c a l o f a r i n g R

o

for n

i s a n i l i d e a l , e.g.

a

1.

if

is Noetherian [41 o r i f R s a t i s f i e s a polynomial i d e n t i t y [311. The d e s c r i p t i o n t h a t we w i l l o b t a i n o f the Jacobson r a d i c a l o r i g i n a t e d from

the f o l l o w i n g conjecture o f Krempa and Sierpinska [221: f r e e c a n c e l l a t i v e senigroup o f rank proved t h i s f o r semigroups o f rank

n

1

¶

.

then

if

S

i s a torsion-

.

J ( R [ S l ) = Jn(R)[SI

They

It i s t h i s conjecture we w i l l now

f i r s t prove.

(J. Krempa, A. Sierpinska [221)

4.4 Let syl

R

be a r i n g w i t h u n i t y and S sn

...¶

be a basis f o r

normalizing extension o f R[xl, xl,

...¶

R[xly

and l e t

..., xn]

xi = si

+

sfl

.

Then R [ S I

is a

w i t h a c e n t r a l normalizing basis; and

i s t h e r i n g o f polynomials over

We prove t h i s by i n d u c t i o n on

s1-1 = R[sly

R

n , Let

Since s12 = xlsl

x,].

-1

1 and s1 as an R[xll-module.

generated by

i n n commuting v a r i a b l e s

n = 1

.

Then

i t follows that

R[S] =

result for

is

R[SI

It i s easy t o check t h a t

s1 are 1 n e a r l y independent and obviously they are c e n t r a l i n R [ S I

.

1 and The

n = 1 follows.

For n > 1 we have R [ S l = R[sly snmYl

Let

..., xn .

Proof: R[sl,

xnl

S

a f i n i t e l y generated f r e e group.

S,,~])[S~, -1

sill

.

s1-1 ,

..., sn,

s,lI

=(R[sl,

s1-1 ,

...¶

Hence, by t h e i n d u c t i o n hypothesis, one e a s i l y proves

53

A description of the Jacobson radical

that

i s a normalizing extension o f

R[SI

normalizing basis o f o f t h e group R

over

Zn elements. Because the

i t follows that

S

4.5

Lemma (J.

Let

R

i s the r i n g o f polynomials

..., xn .

xl,

0

Krempa, A. Sierpinska 1221)

be a r i n g and l e t

S

be a f r e e group o f rank

J(R[S 1 )

Proof:

syl

..., x n l

R[xl,

i n the n commuting v a r i a b l e s

..., x,l w i t h a c e n t r a l .. . , sn a r e f r e e generators

R[x19

Once t h e statement i s known f o r f i n i t e rank

t h e same method as i n the proof o f Proposition 4.3, v a l i d f o r i n f i n i t e rank.

.

Then

.

J,(R)[Sl

=

n

n

i t follows, by using

that the r e s u l t i s also

Because o f Lemma 4.1 we may assume t h a t

has an

R

identity. So we assume t h a t

where

(sly

I = J(R[Sl)

..., sn}

n

n R

i s finite.

.

Because o f Theorem 3.1,

I t remains t o prove t h a t

be a basis f o r

S

.

I = Jn(R)

n

4.6

Lemma -

Let

R

n

.

..., xn1)

R = J(R[xl,

be a r i n g and l e t S

n R

.

Let

With t h e n o t a t i o n s as i n Lemna 4.4,

f o l l o w s from Proposition 3.2 and Proposition 4.3 t h a t

x,])

J(R[Sl) = I[Sl

..., xn1)

= (J,(R)[xly

n

I = (I[Sl

n

R

Jn(R)

it

R[xl,

...)

.

0

be a c a n c e l l a t i v e semigroup o f t o r s i o n - f r e e rank

Then Jn(R)[Sl

Proof:

.

Again by Lemna 4.1 we may assume t h a t

Suppose f i r s t t h a t subgroup

c_ J(R[SI)

F o f rank

- J(R[FI) Jn(R) c

.

i s a group.

S

n

such t h a t

and

S have an i d e n t i t y .

By t h e d e f i n i t i o n o f rank,

S/F

i s a t o r s i o n group.

So i t s u f f i c e s t o prove t h a t

w i l l show t h a t f o r each a E J(R[FI)

R

and

S

By Lemna 4.5,

J(R[Fl) c_J(R[SI)

b E R[Sl

,

has a f r e e

, i.e.

we

ab has a r i g h t quasi

E. Jespers and P. Wauters

54

.

inverse i n R t S 1

.

u (supp(b)) Hence R[H]

Let H

Then H/F

S/F i s an abelian t o r s i o n group.

i s f i n i t e since

i s a normalizing extension o f

.

R[Fl

So, by Proposition 3.2,

ab E J(R[H]) ; i n p a r t i c u l a r ab has a r i g h t quasi inverse i n also i n

r i g h t ideal o f (i)

T

and l e t

R[SI

denote

v E SxT

(ii)i f

n: R [ S l

C l e a r l y ker IT

+

c_ L

n(L) n T =

, then vs

R[Tl:

.

rss

E S\T +

, n ( L ) = (n(L)R[Q(T)l) n R [ T l R[T] , r i g h t i d e a l s o f R [ Q ( T ) l

.

Q(T)

Therefore n(L)R[Q(T)l

rss

.

, the

i s a maximal r i g h t i d e a l o f

.

Since

&:

Let

S

i s a central l o c a l i s a t i o n

are generated by t h e i r i n t e r s e c t i o n w i t h

i s a maximal r i g h t i d e a l o f

f i r s t p a r t o f t h e proof shows t h a t

0 , pi E lP

1 rL(st-tL)

with

L k

>0

pL ElP and

pLre = 0

, which

S,

.

.

= 01

piai

Then pirij

= 0

re E R

, sL,tL E H' ,

. E S for a l l

sL,tL

f i n i s h e s t h e proof.

0

THEOREM

ing procedure t o compute J(R[Sl)

where

and

we can r e w r i t e t h i s sumnation such t h a t

I n t h i s s e c t i o n we deal with a r b i t r a r y semigroups

semigroups.

Because Jn(R) = I01

J 1

This means d E N

THE M A I N

.

and h . E H" f o r a l l i,j J k k and c l e a r l y ( h . x . ) P i = (hjyi)Pi and

Therefore d =

f o r some

= t$

, where

H"

x

and thus (Proposition

rij E R

J I

.

H I = GI

k

xp' = ypi Write

R[HI

R[H'l = (RIH"l)[G1l

Proposition 4.8 i m p l i e s t h a t J(F."H"l)

k

, then

i s a t o r s i o n - f r e e semigroup o f rank

i s a normalizing extension o f

d E J(R[H'l)

3.2)

H U G1

generated by

G

59

Then we consider

.

R[Sl

S

.

F i r s t we reduce the problem t o separative as a d i r e c t sum o f semigroup r i n g s

i s a c a n c e l l a t i v e semigroup.

Since we know J(R[S,I

t h i s information t o o b t a i n the complete Jacobson r a d i c a l o f we are o n l y able t o use t h e information o f

R[S,l

= J=(R)

.

R[S]

)

, we

.

R[S,]

use

However

i n case t h e r e i s a r e l a t i o n -

ship between t h e Jacobson r a d i c a l s o f d i f f e r e n t components we need the c o n t i t i o n J1(R)

We use the f o l l o w -

R[S,l

.

For t h i s

But as already mentioned i n section 4,

t h i s c o n d i t i o n i s o f t e n s a t i s f i e d and i t i s even a conjecture t h a t i t i s always satisfied.

E. Jespen and P. Waurers

60

5.1

e:L e t

R

be a r i n g

and

a semigroup, then

S

(i)

I(R,S,c)

i s a sum o f n i l p o t e n t ideals;

(ii)

J(R[S/cI.)

IJ(R[Sl

.

)/I(RS,E)

Proof: (s,t) E 6

i.e.

s,t E S , Let

rE R

(i) Let

s t n = tntl

, then 2n

Irs

-

rt

I

(ii)

.

= 0

(R(s-t))2nt1

i=O Then

.

r E R1

snt = sntl

f o r some

(rs-rt)tn = 0 = (rs-rt)sn

(?) r2ns2n-iti

1

= (rs-rt)

(rs-rt)2nt1

and

R(s-t)

= 0.

.

n2 1

,

Hence

Denote R ( s - t ) =

i s an i d e a l o f

R[SI

and

This proves (i).

This follows from ( i ) and t h e f a c t t h a t

R[S/SI

R[Sl/I(R,S,E)

GZ

.

0

Because o f thepreviouslemma i t i s s u f f i c i e n t t o study our problem f o r I n t h e sequel we use the f o l l o w i n g n o t a t i o n f o r a

separative semigroups.

separative semigroup S : S = U S, C ~rE are the Archimedean components o f S R[SI =

@

E ,

i s a r-graded r i n g , i . e .

r

where

.

If R

c_

R[SaB1,

i s a s e m i l a t t i c e and the

Sa

i s a r i n g , then

R[S,l

r

RISalRISBl

u,8 E

.

r

NOW, i f

a E R[S1

then we can w r i t e

1

a =

SES

rsE R

,

supp(a) =

a, E R I S a l

Is

E S

I

.

We denote

r s # 0)

r s =

1

a,

,

,Er

suppr(a) = ICYE

r I

a,

# 01 and remember

.

We f i r s t prove some lemnas on r i n g s graded by a s e m i l a t t i c e . a semilattice

r

r

has a natural ordered structure.

contains a maximal element.

a support.

We know t h a t

Hence any f i n i t e subset o f

The f i r s t lemma concerns maximal elements i n

A description of the Jacobson radical

(M. L. Teply,

5.2 Let

R =

J(R)

and

i s maximal i n

B

Proof:

e =

Let

r

an i d e a l o f

{a E

r

a semilattice.

suppr(r)

r I

aG y

Let

on t h e component o f degree easy t o check t h a t

g

g: R '

.

B

y

1

crEr

B'

8

Since

.

E suppr(a)l

i s an i d e a l o f

R

+

.

B

f o r some uEe

.

If 0 # r =

r E J(RB)

then

and t h e r e f o r e R = e Ra

r E J(R) n R ' = J ( R ' )

5.3

Turman, A. Quesada r351)

E.G.

be a r-graded r i n g ,

Ra

@

gr

61

acr+ a

.

R

e

Then

is

Hence

be t h e p r o j e c t i o n map

B

e it i s

i s a maximal element i n

i s an epimorphism.

1 ra E

d r

g ( r ) = r BE J(RB)

Therefore

.

( J . Weissglass [371)

r

R = e Ra be a r-graded r i n g , uEr each a E r , then J(R) = R

Let

a semilattice.

I f J(Ra) = Ra

for

.

L e t us f i r s t consider f i n i t e s e m i l a t t i c e s .

Proof:

the statement by i n d u c t i o n on

.

n > 1

Suppose

r

i s an i d e a l o f

lel

-

= n

R/R'

R

B

r

If

Let

e

, we

obtain that

and

n = 1 the r e s u l t i s t r i v i a l .

R' = e R

r

aB

e = r

Then,

i s an i d e a l o f

J(R/R')

IR / R '

.

J(R') = R'.

R

.

{B}

Since

Let

e

r

and

be t h e subsemigroup o f

r

I 5 J(R)

.

J(R) = R

0 # r E R

generated by

. is

suppr(r)

Let let

R = e R be a r-graded r i n g , r aEr a Iabe an i d e a l of Ra contained i n

5 IaB

for a l l

.

.

r E R ' = e Ra It f o l l o w s from t h e ace i s quasi i n v e r t i b l e i n R ' and thus i n R ,

r

I,RB

But since

It f o l l o w s t h a t

i s a f i n i t e s e m i l a t t i c e and

I

.

i s i n f i n i t e , i t i s s u f f i c i e n t t o prove t h a t any

Corollary:

c Ba-

If

1 the i n d u c t i o n hypothesis y i e l d s ,

For each a E R I

.

a€ 8

f i n i t e case t h a t

5.4

= n

be a maximal element i n

and t h e r e f o r e

quasi-invertible. Then

B

Irl

I n t h i s case we prove

6

E

r .

Then

a semilattice. J(R,)

-such t h a t

I i s an i d e a l o f

R

0

E. Jespers and P. Wauters

62

Proof:

Obviously

.

Ia = J(1,)

I i s an ideal o f

R

.

Because o f t h e assumptions 0

Hence Lemma 5.3 y i e l d s t h e r e s u l t .

5.5

Let

S =

i s a p e r i o d i c group.

(i)J ( R [ S l )

=

U

Sa be a separative semigroup such t h a t each Sa

r

Then f o r each r i n g R J(R[S,I)

@

;

aEr

1

( i i ) J ( R [ S I ) = J(R)[SI t

.

I(JO,p(R).S,Sp)

PEP

Proof: -

Theorem 4.11 says t h a t f o r each J(RISal)

t

= J(R)[S,I

1

r ,

a E

.

I(JO,p(R)ySaySp)

PEP Let

D

=

@

a€ r

J(R[Sa])

.

From Corollary 5.4 we deduce,

1

Conversely, l e t 0 # a =

UE

ag E J ( R I S B l ) c _ J ( R [ S I )

r

aa E J(R[SI)

f o r any maximal

D

,

aa E R[S,I

6

i n suppr(a)

c_ J(R[Sl)

.

, Then, by Lemma 5.2,

.

An i n d u c t i o n

argument y i e l d s the r e s u l t .

5.6

Corollary

Let

R

0

(J. Okninski, P. Wauters [271)

be a r i n g and S

a p e r i o d i c semigroup.

J(R[Sl) = J(R)[Sl t I(R,S,S)

1

+

Then I(Jo,p(R),Ss~p)

*

PE

Proof:

This follows from Lemma 5.1, Lemma 5.5 and Proposition 2.1 ( i , v ) . 0

We say t h a t a semigroup

group o f t h e r i n g R

, if

sn = tn and m r = 0

0 # m,n E l N and m divides

5.7

Corollary

Let S

i s torsion d i s j o i n t with

S

n

, imply

s = t

some s,t E S or

r

0

,r

E R

R, t

and R

S

R[S]

i s a union o f t o r s i o n groups which i s Jacobson-semisimple.

,

.

be a p e r i o d i c semigroup and R a r i n g . Then

i s Jacobson-semisimple i f and o n l y i f are torsion d i s j o i n t with

that

, for

the additive

R, t,

63

A description of the Jacobson radical

Proof:

Follows immediately from C o r o l l a r y 5.6.

0

Cheng i n [51 obtained t h e same r e s u l t f o r A r t i n i a n Jacobson-semisimple

C.C.

semigroup r i n g s .

Note t h a t Zelmanov 1381 proved t h a t i f a semigroup r i n g

i s A r t i n i a n then

R

5.8

&:

is A r t i n i a n and S i s f i n i t e .

Let

be a r i n g such t h a t

R

be a semigroup.

S

1

+

c_

I(Jl,p(R)SyCp)

(R[SI)

PE Ip

Because o f Lemna 5 . 1 ( i ) ¶

I(R,S,S)

deduce t h a t i t i s s u f f i c i e n t t o prove t h a t

c_ J(R[S] )

S,

n EN

, and

let

S,

where t h e

.

. From Lemma 4.9 we

jl(R)[

S1 5 J(R[SI )

S

i s separative.

we may assume, because of Lemna 5 . l ( i i ) t h a t S = u

,

Jl(R) = J=(R)

Then

J1(R)[SI + I(R,S,E)

Proof:

R[SI

.

For t h i s ,

So, l e t

are the Archimedean components, i n p a r t i c u l a r each

aEr

i s cancellative.

S

Theorem 4.11 and t h e assumption on

.

J1(R)[Sal c_ J(R[S,l) J(R[Sl)

Hence, by C o r o l l a r y 5.4,

R yield that

J1(R)[Sl

.

c_

= e Jl(R)[S,l

acr

0

I n order t o be a b l e t o s t a t e the main theorem we need some more notations.

.

S = u Sa be a separative semigroup w i t h Archimedean components S, aEr We denote by r l t h e s e t o f t h e elements a E r such t h a t S, i s not a

Let

p e r i o d i c group.

J(R,Sp,r')

If, moreover,

1

= {a =

R

a, E J ( R I S p l )

aEr

Remember t h a t

&:

I

aa E R[(S,)pl

e6 i s the i d e n t i t y element o f

o f t e n denote J(R,Sp,r')

5.9

i s a r i n g then we denote

Let

R

by

J

Q(S6)

.

For b r e v i t y , we w i l l

.

be a r i n g such t h a t J1(R) = I01 and l e t S

be a separative semigroup.

Then

=

U

a€ r

S,

E. Jespers and P. Wauters

64

Proof:

Again we may assume t h a t

prove t h a t course J' R

.

J'= J(R,Sp,l")

1

R and S

.

(R),S,S ) i s an ideal o f R [ S I Of 1,P P i s closed under a d d i t i o n and under m u l t i p l i c a t i o n by elements o f t

I(J

Because t h e second term i n J'

s E S

1

a =

a

J'

,

n(a)

such t h a t

st

and assume s E S

>6

B

n(a)

,(

sa E J '

it i s sufficient to

.

I n f a c t we prove

s a t i s f i e s the condition

We prove

i s t h e number o f elements

.

Sp

f o r every

B E

B '

sa

R[SI

as s a t i s f i e s (**).)

then

t E supp(a)

,

s t E Sp

as E J ( R I S p l )

) given i n Lemma 5.5,

d e s c r i p t i o n o f J(R[Sp]

(ii)f o r

such t h a t

, where

, i.e.

Assume n ( a ) = 0

r'

implies

(Note t h a t if a E J'

t h i s by i n d u c t i o n on t E supp(a)

i s an i d e a l o f

aa E J

e r

the f o l l o w i n g : l e t a E J(NSp1)

then as

F i r s t we

have an i d e n t i t y .

PE Ip

prove t h a t f o r

6 E

.

R[S]

i s a Jacobson-radical i d e a l o f

.

.

Hence, by t h e Moreover, i f

r , then

(as ) e6

0

.

y > 6

Hence as E J c - J' ,

.

.

Assume s E S a y E r Take B E r such t h a t BY Y i s maximal i n {ye I B E suppr(a) and 3 t E supp(a ) such t h a t t s Sp} B L e t t E supp(a ) such t h a t t s B Sp Write B Assume

n(a) > 0

.

.

1

a =

a>Bu

where if

a

suppr(a2) f l { a E

> BY

then

ay

r I

> BY

a

.

a +a2 a

> BY] Hence

0

.

, Because y

>

BY

i t follows that

A description of the Jacobson radical

as =

1 ' BY

aas

1

t

By t h e assumption on

,all

5

aas t a2s

.

a

a

aY

65

ay = BY

elements of

1

Its

1

t E supp (

aa)l

are

a

aY > BY

in

.

Sp

We c l a i m t h a t i f

.

and since

Then t a s E Sp

, ta eY E Sp ,

tas

E suppr(a)

and

,

L e t a E suppr(a)

assume t h e contrary. ta E supp(aa)

a

then ay = By

a > By

and ay >

a > By

By.

.

Indeed

Take

.

Because t a s = (taea)(se ) = ( t e )(eas) Y a Y sea E Sp But then ts = ts e we o b t a i n t h a t

.

,a

= t s eaBy = t s eaeBy = t ( s e a ) E Sp

BY

From t h e claim we

contradiction.

obtain

1

as = Clearly J'

.

BY

E

r'

1

aas t a2s = (

and t h e r e f o r e c o n d i t i o n ( w ) implies t h a t

From Lemna 5 . 5 ( i ) we o b t a i n t h a t

-d

= ae s

.

aa)eBys t a2s

a>BY

a> 5y

s a t i s f i e s c o n d i t i o n (**) and

Y thesis yields that

.

a2s E J '

.

a2 E J(R[Sp]) n(a2)


By

aaeBys E

a2s = a e s

2 Y

The i n d u c t i o n hypo-

This f i n i s h e s t h e proof o f t h e f a c t t h a t

J'

i s an i d e a l . Secondly, we have t o prove t h a t

J'

i s a Jacobson-radical

ideal.

(R),S,E ) i s a Jacobson r a d i c a l i d e a l o f 1 I(JlYp P PE'P Therefore i t i s s u f f i c i e n t t o prove t h a t J'/D = l a t D I a E J1 i s a by Lemna 5.8,

0 =

Jacobson-radical J 5 J(R[Sp])

ideal o f

, every

R[SI/D

.

element o f t h i s i d e a l i s quasi i n v e r t i b l e .

This proves 0

We now g i v e t h e main theorem.

Let

Theorem

R

R[S1.

Obviously i t i s an i d e a l and s i n c e

the r e s u l t .

5.10

Well,

(E. Jespers [121)

be a r i n g such t h a t

J1(R) = JJR)

and l e t

S =

Sa

U

aEr separative semigroup.

Then

J(R[Sl) = J1(R)[SI +

1 PEP

I(JlYp(R),S,Ep)

+ J(R,Spsr')

*

be a

E. Jespers and P. Wauters

66

Proof:

Again by Lemma 4.1 we may assume t h a t

R and

have an i d e n t i t y .

S

So we o n l y have t o

r i g h t hand side o f t h e equation i s contained i n J(R[SI). prove t h e converse i n c l u s i o n . assume t h a t

.

J1(R) = CO)

J'

c_ J ( R [ S I )

Again, because J1(R)[Sl

1

a =

CXE r

.

1

= J(R,Spyr') +

, aa

aa E J(R[SI)

we may

As before we denote

~ ~ J l , p ~ ~ ~ S a S p ~

PEP Let

, the

= I01

Because o f Lemna 5.8, Lemma 5.9 and the f a c t t h a t J1(R/J1(R))

,

E R[S,I

a

r.

E

We have t o prove t h a t

a E J'.

We do t h i s i n 4 steps. Step 1:

Let

n(a)

be t h e number o f elements i n t h e set

supp(aa) Q (Sa)p I . RISpl) t Let

6

We prove by i n d u c t i o n on

.

1 I(JlYp(R),S,Sp) PE Ip be a maximal element i n

n(a)

If n(a) = 0

N =

{a

r I

E

a E (J(R[SI) n

that

t h i s i s clear.

Suppose n ( a ) > O .

s6 E supp(a6) , s6 4 Sp

N and l e t

.

We

obtain

1

as6 =

aas6

-t

a2s6E J(RIS1)

as6

where

suppr(a2) n

{a E

Hence, by Lemma 5.2,

r I

a

1

d =

1

i t f o l l o w s from Theorem 4.11 t h a t

as = a;

+

a:

,

a >6

where

Clearly 6

supp(a;)

i s maximal i n

1

aae6 E J ( R [ S i l ) =

1

a> 6

PEP and supp(ai) FS,

c_ ( S 6 ) p

suppr(as6).

aae6) s6 1 I(Jl,p(R)~S6sSp).

Because d = (

aas6 E J(R[S61).

a3 6

Write

.

0

> 6)

,

\(S6)p

.

Hence

1

6 E Sp

a>

Obviously, i f

(t',t")

aae6 = (

, then

1 aae6 t

n(a-a;)


6

. 0

5.14 Let

Corollary

(J. Okninski, P. Wauters [271)

S be a semigroup such t h a t each Archimedean component o f

t o r s i o n - f r e e rank a t l e a s t one.

Jp) = JJR)

Then f o r any r i n g R

such t h a t

9

J(R[Sl)

= J1(R)[Sl

+ I(R,S,S)

1

+

I(Jl,p(R),S,Sp)

PEP

Proof:

has

S/S

F i r s t l y , suppose t h a t

i s separative.

S

Because

. r

=

r' ,

.

So the r e s u l t follows from Theorem 5.10. 1 I(Jl,p(R),Sp,~) PEP The non-separative case now f o l l o w s from Lemna 5.1 and Proposition 2.1.

J(R,Sp,r')

5

EXAMPLES

6.

6.1

Example

Suppose S = S

"1

U S

"2

uS

"3'

with

t o r s i o n - f r e e semigroup w i t h i d e n t i t y

r

= {al ,a2,a31

"1

e

"2

=

{call , Sa3

= {ea3}

and of nonzero rank.

and Suppose

has ordered s t r u c t u r e :

i.e.

a1 > a2 > a3

J1(R)

= J(R ,)

.

Then, by C o r o l l a r y 5.13,

f o r any r i n g

R

such t h a t

*

J(R[Sl) = J(R)ea3 However, i f

S

s" 1

+ J1(R)[Sa21 + J(R)(eal

- ea2) .

i s a t o r s i o n - f r e e semigroup o f nonzero rank, then

a

E. Jespers and P. Wauters

70

.

J(R[SI) = J1(R)[SI t J(R)e,, I f moreover

i s also t o r s i o n - f r e e o f nonzero rank, then

S "3

J(RP1)

6.2

-

= J1(R)[Sl

Example

Suppose S = Salu group,

Sa3

u

Sa2

, Sal

Sa3 u Sa4

,

=

i s a torsion-free

Sa2

i s a t o r s i o n - f r e e semigroup without an idempotent and S

Suppose the s e m i l a t t i c e

r

0

Iea41.

has the ordered s t r u c t u r e

= {al ,a2,a3 ,a41

"2

"4

"3

"4

Then, by Corollary 5.13,

f o r any r i n g

J(R[Sl

kwever, i f

S

"4

= J1(R)[Sl

Example

Let

Ho

such t h a t + J(R)ecr4

J1(R)

J=(R)

,

,let

Hn

.

i s a t o r s i o n - f r e e semigroup, then = J1(R)[SI

J(R[SI)

6.3

R

*

be t h e c y c l i c group o f order 2 and, f o r any

i n f i n i t e c y c l i c group.

The generator o f

and i t s i d e n t i t y by

.

en

r

The set

Hn

= (0,1,2,

,

n

>

1

be t h e

n B 0

,w i l l

...1

i s a s e m i l a t t i c e for t h e

be denoted by

h,

f o l l o w i n g ordered s t r u c t u r e 0

Let

S

U Hn , the d i s j o i n t union o f the groups H, nEr

.

Consider t h e f o l l o w i n g

A description of the Jacobson radical

71

group homomorphi sms :

, if

n,m,k E r

Then, f o r a l l

sn, sm in S

two elements

'nSm where

( s n E Hn =

i n [61 that

S

then

, smE

$,,,k

0

@n,m

.

= $n,k

For any

define

Hm)

On,k(sn) $m,k(sm)

n and m i n r

i s the product of

k

k < m < n

.

I t follows from Theorem 4.11

is a separative semigroup f o r t h i s multiplication.

Let

R

be

an algebra over a f i e l d k such t h a t J(R) # I01 , J1(R) = I01 and k is of characteristic zero.

I t follows t h a t , f o r any

.

{Ol and I(Jo,p(R),Sp,Sp) = J(R)(eo-ho)

.

J(R,Sp,r') = J(R)(eo-ho)

Let

HOMOGENEOUS JACOBSON

R

be a r i n g such t h a t

E

J(R[Sl) then

I(Jl,p(R),S,~p)=

Hence, Theorem 5.10 yields

.

RADICALS

J1(R) = J=(R) and l e t S be a semigroup.

this section we investigate when

1 rss

,

I t i s then'easy t o verify t h a t

J(R[Sl) = J(R)(eo-ho)

7.

p EIP

J(R[Sl) i s

(S-) homogeneous, i.e.

rss E J(R[Sl) for a l l

s ES

Sd

.

In if

Because the r i n g

does not necessarily contain an identity we have t o consider several cases, such as

J(R) # R

,

J1(R) # R

.

All the r e s u l t s contained i n t h i s section can be found i n [131.

7.1

=:

r ER\J(R) geneous then

Let

and S

R

s ES

be a r i n g such t h a t

, rs a

i s separative.

J(R[Sl)

.

R # J(R)

.

I f , moreover,

Then, f o r any J(R[Sl)

i s homo-

R

E. Jespers and P. Waurers

72

Proof: -

Let

F

Let

r E R

\

J(R)

r

denote the element determined by

Fs E J(R/J(R))[Sl)

.

,for

r s E J(R[Sl)

and assume t h a t

i n R/J(R)

.

some s E S

.

Clearly

{Ol i t f o l l o w s from

Since J(R/J(R)) = J=(R/J(R))

Corollary 5.12 t h a t the sum o f the c o e f f i c i e n t s o f any element i n J((R/J(R))[S]) Assume exist

i s zero.

J(R[SI)

s,t E S

Therefore F =

i s homogeneous.

such t h a t

( r s - r t ) E J(R[SI)

0 , i.e. S

If

,a

contradiction.

i s n o t separative, then t h e r e

s2 = s t = t2 , s # t

(Lemma 5.1).

r E J(R)

.

Let

Hence r s E J ( R [ S I )

r E R

J(R)

, then

; but t h i s i s i n 0

c o n t r a d i c t i o n w i t h the f i r s t p a r t o f the lemma.

7.2

Proposition

(i)

if

R = J(R)

(ii)

if

R # J(R)

Let R

, then , then

be a r i n g and S

a p e r i o d i c semigroup.

Then,

J(R[SI)

i s homogeneous;

J(R[Sl)

i s homogeneous i f and o n l y i f t h e f o l l o w i n g

conditions are s a t i s f i e d

If

(a)

S

i s separative; and

(b)

S

i s torsion d i s j o i n t with i s homogeneous then

J(R[Sl)

R/J(R),t.

J(R[Sl)

= J(R)[SI

.

Proof: (i)

Under t h e assumptions Corollary 5.6 says, particular J(R[SI)

(ii)Assume t h a t separative. s

#

t,

J(R[SI)

C o r o l l a r y 5.6 y i e l d s , r s E J(R[SI)

i s homogeneous. i s homogeneous.

Then, by Lemma 7.1,

Assume t h a t ( b ) i s n o t s a t i s f i e d .

r E R\J(R),

.

p EIP, n rs

E N such t h a t

- rt E J(R[SI) .

S

is

Then there e x i s t s,t E S, n n sp tP and p r E J(R).

Hence, by the assumption,

But t h i s c o n t r a d i c t s Lema 7.1

For the converse, under t h e assumptions ( a ) and becomes

,in

J ( R [ S I ) = J(R)[Sl

J ( R [ S l ) = J(R)[Sl

.

This f i n i s h e s t h e proof

b) Corollary 5.6 0

73

A description of the Jacobson radical

From now on we r e s t r i c t our a t t e n t i o n t o non-periodic semigroups, i.e. S # P(S)

the l a r g e s t p e r i o d i c i d e a l o f

.

S

Since t h e r e s u l t s f o r non-

p e r i o d i c semigroups are based on Theorem 5.10 we w i l l i n the sequel assume

J1(R)

that

7.3

.

= Jm(R)

Let R

Proposition:

semigroup.

i n g way

be a semigroup and on

P

spt

S

and l e t

be a

S

.

= J1(R)[Sl

(s,t E

Is} w i t h s i n

.

S\I

instead o f

write S / I

I an

, called s) i f s

ideal o f

= t P

o r e l s e both

I

are

.

The i d e a l

I , i n t h e follow-

modulo s

I defines

and t

belong t o

I

.

i t s e l f and every one element set

We w i l l o f t e n denote

S/P

0

.

S

t h e Rees congruence

S mod

The equivalence classes o f

7.4

R = Jm(R)

This f o l l o w s immediately from Theorem 5.10.

a congruence

S

i s easy t o deal with.

Then

Let S

is

= JAR)

be a r i n g such t h a t

J(R[SI)

Proof:

R = J1(R)

The case where

Is) also by s

.

We s h a l l

Note t h a t if I i s the empty s e t then

S/I

.

&:

group.

R

Let r E (R

If

moreover,

J (R [S ])

Proof:

Let

\

Jl(R))

S/P(S)

be t h e element determined by

.

r E R

Assume t h a t

Because s E S \ P ( S )

J1(R/Jl(R))

R # J1(R)

and s E (S\P(S))

i s homogeneous then

element determined by

s E (S\P(S))

be a r i n g such t h a t

= I01

, we

i n R/J1(R) r s E J(R[Sl)

.

Let

then

s E S

r E (R\J1(R))

B

J(R[SI).

i n S/S

r E (R\J1(R))

If,

and

r

the

and

rz E J ( ( R / J 1 ( R ) ) [ S / C I )

we may assume s B Sp and thus

c o n t r a d i c t i o n w i t h the assumption

rs

be a semi-

S

i s separative.

, then

o b t a i n from Theorem 5.10,

and l e t

f

r = 0' .

.

(S/S)p

.

.

Since

However t h i s i s i n

E. Jespers and P. Wauters

74

For the second part, assume J ( R [ S l ) element determined by

a

Hence s

s

that

s2

B

S/P(S)

,

P(S) o r

t

P(S)

B

P(S) , t

= s t = t2

.

s,t

ES

.

, such

P(S) and thus

, but

that

rs

5

,

= s

s” = Sf

t =

.

f

- rt E J(R[Sl)

i

Let

be the

i s n o t separative, then

I f S/P(S)

So we may assume, say

Lemma 5.1 y i e l d s ,

Thus r s E J(R[Sl)

.

t E S i n S/P(S)

S,f E

there e x i s t

i s homogeneous.

= f2 and

.

s B P(S)

.

It follows

,

s # t

Therefore

, where

# f

r E RxJ1(R)

.

t h i s i s i n c o n t r a d i c t i o n w i t h t h e f i r s t p a r t o f the

lemna.

0

Let

be a semigroup and l e t R

S

ideal o f

Therefore J ( R [ P ( S ) l )

i s an ideal of

7.5

R

Let

be a semigroup.

let S

C l e a r l y R[P(S)l

R[P(S)l = 101 i f

R[Sl ; we agree t h a t

Theorem:

be a r i n g .

P(S) i s the empty set.

R E 1 and J(R[P(S)l) R # Jl(R)

be a r i n g such t h a t

i s an

,

= J(R[S])

n R[P(S)l.

J1(R) = Joo(R) and

i s homogeneous i f and o n l y i f t h e

Then J(R[Sl)

f o l l o w i n g conditions are s a t i s f i e d : ( i ) S/P(S) (ii) (iii)

If

i s t o r s i o n - d i s j o i n t w i t h the a d d i t i v e group

S/P(S)

J ( R [ S l ) n RISFl 5 J ( R ) [ P ( S ) l + J1(R)[Spl i s homogeneous then

J(R[Sl)

Assume t h a t

J(R[P(S) I)

J(R [S

Because

7.2.

To prove ( i i ) , assume t h a t

, then

f o r some we o b t a i n

s

B

If,

P(S)

,

E R L J1(R)

,

LemIa 5.8 we obtain, c o n t r a d i c t s Lemma 7.4;

t

rs

B P(S)

- rt

ES/P(S) n n

and

=

Ep

, s’ # f , and

s’ = s , f

E J(RCS1)

f o l l o w s from Proposition

i s not torsion-disjoint with

S/P(S)

sp

.

Then ( i ) f o l l o w s from Lemma

I) n R[P(S) I , ( i v )

s’,f

there e x i s t

EP, r

p

.

i s t o r s i o n d i s j o i n t w i t h R/J(R),+

i s homogeneous.

J(R[SI)

7.4.

R/J~(R),t

;

.

J(R[Sl) = J1(R)[SI t J ( R ) [ P ( S ) I

P(S) i s separative and P(S)

Proof:

R/J1(R),t

, then

R # J(R)

moreover, (iv)

i s a separative semigroup ;

.

s,t

p r E J1(R) = t

.

ES

.

that

Since

s’ # f

From Lemma 4.9 and

Hence r s E j(R[SI)

hence (ii) i s proved.

, such

.

This

A description of the Jacobson radical

Next we prove ( i i i ) . Assume ( i i i ) i s not true.

r E R

homogeneous, there exist

a

rs s

a

+

J(R)[P(S)]

.

P(S)

.

Jl(R)[Sp]

s E Sp

Then because j ( R C S 1 ) rs E J(R[SI)

such t h a t

Because of Lema 7.1,

Lennna 7.4 therefore yields,

r E J1(R)

75

r E J(R) ; i.e.

.

is

but

Hence

,a

r s E Jl(R)[Spl

contradict ion. For the converse, assume t h a t ( i ) , ( i i ) and ( i i i ) are satisfied.

a

= s&

rss E J ( R [ S l )

,

then a’ =

rs E R

s E S in

the element determined by

( i i ) and because of Theorem 5.10,

E

.

S/P(S)

Since Z = s if

Because of conditions ( i ) and

s

rs E J1(R)

a

P(S)

SB

and because s 6 Sp

for all

1

.

(S/P(S))p

And hence by ( i i i ) , b E J ( R ) [ P ( S ) l

rss E J ( R [ S l )

J1(R)[S]

+

.

+ J(R)[P(S)l

if

s d Sp

, we

Therefore Corollary 5.11 implies,

sap

C_

for every

rsE E J1(R)[S/P(S)l

b =

J(R[S])

, where s’ i s

J(R[S/P(S)l

.

E (s/P(s))+/P(s))p

obtain t h a t

1 rSz E

Let

J1(R)[Spl

.

. Hence we proved t h a t

Because the converse inclusion i s also

valid (Corollary 5.11) the result follows.

7.6 S

Corollary:

0

Let R be a ring such t h a t R #

be a semigroup. Then J ( R [ S l )

and l e t

s homogeneous

lowing

conditions are satisfied ( i ) S i s a separative semigroup; ( i i ) S i s torsion disjoint w i t h the additive group R/J(R),+

Proof: -

.

That the conditions are necessary follows from Theorem 7.5.

The

conditions are also sufficient because of Corollary 5.12.

If

S i s a semigroup, then an ideal

I of

ideal of periodic elements i f a l l ellements of more t h a n one element, i.e.

I

S

is called a non-trivial

I are periodic and i f

I

has

has an element which i s not the zero element.

E. Jespers and P. Weuters

76

7.7

Proposition:

J=(R)

.

Let

Sa ,

nents

Let R

S =

# Jl(R)

J(R)

and J1(R)

=

be a separative semigroup, w i t h Archimedean compo-

S,

U

be a r i n g such t h a t

aEr

a

r , such

E

that

i s t o r s i o n d i s j o i n t with t h e a d d i t i v e

S/P(S)

...,

.

group R/J1(R),t I f e,l en E E ( S / P ( S ) ) , n €IN , are such t h a t n ( S / P ( S ) ) / U (S/P(S))ei has a n o n - t r i v i a l p e r i o d i c i d e a l , then J(R,Sp,r') i=l contains a non-zero element 1 r s s , rs E R f o r a l l s E S , such t h a t s ES n o t a l l r s s E J(R[Sl)

.

Proof: -

Because P(S)

.

i s the largest periodic ideal i n

S

we may assume

E S, , h e r e cxi E r and e, , i i i i s the i d e n t i t y o f S, Because o f t h e assumptions there e x i s t s an i n s E (Sa)p , a E r , such t h a t s B U (S/P(S))ei and such t h a t s i s coni=l n Take r E J(R) tained i n a p e r i o d i c i d e a l o f (S/P(S)) / U (S/P(S))ei i=l J1(R) and l e t w = r s = rsea Note t h a t e, E S, because S, contains that a l l

ei f P ( S )

Hence, l e t

ei = e

.

.

.

the periodic element s denote that

a

- aeB

by

.

I f eB i s an idempotentof S

.

a(1-e,)

d E J(R,Sp,r')

Consider

d

)

w(1-e a1

.

Because o f C o r o l l a r y 5.6,

1

dB w i t h

and a E R [ S I

... (1-e,

d E J(RISpl)

.

) n

.

, we

We claim

So i t i s

s u f f i c i e n t t o prove

where we w r i t e such t h a t

d =

6 & B

B

E suppr(d)

ts

t(e,s)

.

,

dB E R [ ( S B ) p l

(tir

i s t r i v i a l l y satisfied.

O f course

6 &

Q

.

NOW, l e t

.

So, suppose t E (S6'(S6)p)

i s also a non-periodic element o f S6 n on s we o b t a i n t E U (S/P(S))e, , i.e. 6 0

Suppose R [ S I

.

i s semilocal.

i s therefore a semilocal r i n g . assume t h a t J(R)

S = S

R

Because

I01 and R # {OI

being a homomorphic image o f

(R/J(R))[Sl

.

RtSI/J(R)[SI

2

Hence R

R[SI

we may

i s semisimple A r t i n i a n and

t h e r e f o r e contains an i d e n t i t y . let To prove (ii),

s E S RtSls

Because R[S]/J(R[S]) J(R[Sl)

s.sk

RISlsktL

- dskt2

Thus f o r some

.

Consider the descending chain

2 R t S l s2 2

... - R I S l s n 2 ... . 3

i s A r t i n i a n , t h e r e e x i s t s an

+ J(R[SI)

E J(R[Sl)

.

t E supp(d)

for a l l

By Theorem 4.11,

, L €IN

.

L ElN

and

€IN

k

RISlsk t

Hence f o r some d E R [ S I

1

J(RtS1) = some

such t h a t

pE

P

IP ,

I(JO,p(R),S,S

, p)

.

81

A descr@tionof the Jacobson radical

(s

L

ktl)p

= (tskt2)P

then we o b t a i n

e

Let e

i e

e = tPsp

.

be t h e i d e n t i t y o f t h e q u o t i e n t group o f e E S and a l s o t h a t

This i m p l i e s t h a t

S

S

,

is a

group. If

S

i s f i n i t e then (iii) holds.

S

i s infinite.

V[Sl

Now, l e t V

be a simple homomorphic image of

i s a semilocal r i n g , i.e.

Proposition 8.1,

V[Sl/J(V[Sl)

# I01

J(V[Sl)

So t o prove ( i i i ) we may assume t h a t

.

i s Artinian.

Because t h e c e n t r e o f

.

V[S/S

1 i s Artinian.

P

Hence V I S / S p l

By Proposition 8.1,

. Obviously

By v i r t u e o f i s a f i e l d , we

V

t h e r e f o r e o b t a i n from Theorem 4.11 t h a t t h e c h a r a c t e r i s t i c of p > 0 and J(VtS1) = I(Jo,p(V),S,Sp)

R

V

i s a prime and

V[Sl/J(V[Sl)

i s f i n i t e and t h e r e f o r e

S/Sp

x T where T i s a f i n i t e p'-group and S i s i n f i n i t e . Because S P P P P P i s t h e unique i n f i n i t e p-component o f S and since V i s an a r b i t r a r y simple

S = S

homomorphic image o f the semisimple A r t i n i a n r i n g R characteristic

.

p > 0

2A

Since

R/J(R)

that

= J(R)[SI

.

+ I(Jo,p(R),S,Sp)

i s Artinian.

I n case R

R has

and (iii) (Theorem 4.11) Because of (ii)

i s A r t i n i a n and since

R[SI/A

obtain t h a t

This proves (iii).

Next we prove t h e converse. J(R[SI)

, we

Hence R [ S I / A

(R/J(R))[S/Spl

.

i s f i n i t e , Proposition 8.1 y i e l d s

S/Sp

Hence t h e r e s u l t f o l l o w s .

U

i s comnutative t h e previous p r o p o s i t i o n i s due t o T. Gulliksen,

P. Ribenboim and T.M. Viswanatham [9J and G. Renault [33J. 8.3

Lemma:

Let

u Sa be a separative semigroup w i t h Archimedean

S =

r

components Su and R a r i n g such t h a t

(i)f o r each a E (ii)r

Proof:

r ,

R # J(R).

i s semilocal;

R[S ]

If R[S]

i s semilocal, then

and

i s finite.

For

an i d e a l o f

u E

R[S]

r

we denote Aa = and

ra: Aa

R[SB] =

@

B a-a +

R[S,l:

1 age, +

BE r

@ R [ S B ] , Then Aa i s B

Because R [ S I / J ( R [ S I ) = Aa

...>an > ... k

n+k

exists a

b E An+k

suppr(a-b)

a

# B ,

n


a2 > a3 > ...> a

does n o t contain an i n f i n i t e descending chain

= J(R[SI) an

Hence, f o r any 0 # a E R[Sa 1 t h e r e

-bE

.

J(R[SI)

n

Because an

, Lemma 5.2 y i e l d s t h a t a E J(R[Sa 1). Thus

i s maximal i n

RISal

.

J(R[S,])

n This c o n t r a d i c t s Lemma 7.1. (b) i s proved i n a s i m i l a r way because R [ S I / J ( R [ S l ) A r t i n i a n and thus Noetherian.

r

subset o f

To prove ( c ) l e t

{ai

i s semisimple

I

i ElN 1 be an i n f i n i t e

o f incomparable elements.

Noetherian, t h e r e i s an n ElN

Again because R [ S I / J ( R [ S I ) is n ntl such t h a t 1 A, + J ( R [ S I ) = 1 A, + J(R[Sl). i=l i i=l i

1

Hence f o r any 0 # a E R[Sa

such t h a t 1 there exists a b E A, i i=l n+l b E J(R[Sl) As before i t follows t h a t a E J(R[Sa I) and thus a n+l J(R[Sa I) = REa 1 , a contradiction. Therefore ( c ) i s s a t i s f i e d . ntl ntl

.

-

r

F i n a l l y we prove t h a t (a), (b) and ( c ) imply t h a t

i s finite.

then we construct by i n d u c t i o n an i n f i n i t e descending chain

...

in

(b) l e t

r al

such t h a t

ran

nE

i s an i n f i n i t e s e t f o r a l l

be a maximal element i n

r

.

Note t h a t

r

> a2

al

IN

.

If not 7

...>an >

Because o f

= Fal U {al)

and

83

A description of the Jacobson radical

i s infinite.

Assume we constructed

therefore

Fal

infinite.

Then, by ( b ) and (c), t h e r e are o n l y f i n i t e l y many maximal elements

..., B~

say

, under

B ~ ,

k

(an}

, there

antl

= B~

ra n = u rei u

i=1 Then we define chain

r

a1 > a2 >

i s finite.

8.4

Let S

either

R[SI/I(R,S,S)

where t h e

Ta

r

8.3 t h a t 8.2,

...

in

.

that

rBi

i s infinite.

However t h i s c o n t r a d i c t s (a).

Thus 0

and R

.

R # J(R)

a r i n g such t h a t

S/S

i s f i n i t e and R/J(R)

P

p > 0

.

R[Sl

i s semilocal.

i s a semilocal r i n g .

Let

a r e t h e Archimedean components o f i s f i n i t e and each R[Tff]

i s semilocal and e i t h e r

R

r

i s f i n i t e or either

S/S

Suppose f i r s t t h a t

R[S/cI

, such

B1

Hence we have constructed an i n f i n i t e descending

be a semigroup

prime c h a r a c t e r i s t i c

Proof:

, say

B~

1 G i G k. Since

i s semilocal; and

R

(ii)

an >

is a

for

is

ran

i s semilocal i f and o n l y i f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d :

R[SI

(i)

.

B~ E ran

Note t h a t

that

This f i n i s h e s t h e proof.

Theorem:

Then

... >

.

an

such

an

T

has

Then, by Lemna 5.1,

, then

T = S/C

T

.

T =

T

U

a

~

'r

It f o l l o w s from Lemna

i s semilocal.

Hence, by Proposition

i s f i n i t e or either

Talca

is finite Y P

for all

E

a

r , c,,~

R has c h a r a c t e r i s t i c

p > 0

.

either

characteristic

p > 0

5 J(R[Sl)

I(R,S,C)

R/j(R)

.

Now,

i s semilocal.

S are periodic. t

I(R,S,S)

P

5

i s f i n i t e and

R/J(R)

i s f i n i t e i t f o l l o w s from C o r o l l a r y 5.6 t h a t R[Sl/(J(R)[Sl + I(R,S,C))

i s A r t i n i a n , Proposition 8.1 y i e l d s t h a t

Hence R [ S ]

j(R)[S]

S/C

.

Conversely, i f S/C t

and since

i s f i n i t e or either

S/S

,

= 5 /S i s t h e P P S/Sp IT/Cp s a; rT,/Sa,p we

Since, Proposition 2.1

l e a s t p-separative congruence on T obtain that

, and

denotes t h e l e a s t p-separative congruence on Ta

If

S/S

+ I(Jo,p(R),S,Sp)

.

and because o f ( i i ) , Hence R / J ( R ) [ S I

. Since

i s Artinian.

i s n o t f i n i t e , then by (ii)S / S

Again by C o r o l l a r y 5.6,

J(R)[Sl

I( R / J ( R ) ) [ S / S I

(R/J(R))[S/SI

2

has

and thus

J(R[SI)

(R/J(R))[S/Spl

=

.

E. Jespers and P. Wauters

a4

Proposition 8.1 y i e l d s again t h a t

R[Sl

i s semilocal.

0

F i n a l l y we mention t h a t J . Okninski i n [ 261 studies semigroup algebras which are semilocal. ditions f o r

k[S]

I n p a r t i c u l a r he obtained necessary and s u f f i c i e n t con-

, k

a field,

S

a commutative semigroup t o be semilocal.

Nevertheless we f e e l t h a t t h e conditions i n t h e previous theorem a r e more i n t r i n s i c than i n [261.

9.

OTHER RADICALS

I n 1321 i t i s shown t h a t t h e prime r a d i c a l

Because, i n

Levitzki radical

L

8 s a t i s f y the analogues of the t w o basic r e s u l t s

and the Brown-McCoy r a d i c a l i n section 3.

, the

P

the previous sections, we only made use o f these

t w o r e s u l t s and o f some general r a d i c a l theory, i t follows t h a t a l l the r e s u l t s i n sections 3 t o 7 remain v a l i d i f we replace J

[12, 13, 271). P(R) = P(R ,)

f o r a l l rings

L

or

B

(cf.

( c f . [11 o r 1213) t h a t

We note t h a t i t i s easy t o show and L ( R ) = L,(R)

P ,

by

R

.

We also mention t h a t t o

prove Proposition 4.8 f o r t h e Brown-McCoy r a d i c a l one has t o replace maximal l e f t ideals by maximal i d e a l s and q u a s i - r e g u l a r i t y by G-regularity. That t h e analogue o f Proposition 3.2 i s s a t i s f i e d f o r the s t r o n g l y prime r a d i c a l S i s shown i n 1291. I f one assumes some a n n i h i l a t o r c o n d i t i o n on the graded r i n g then also Theorem 3.1 remains v a l i d f o r

( c f . 1111).

S

the r e s u l t s i n sections 3 t o 7 are also v a l i d f o r S ( c f . t h i s radical

S(R ,)

S(R)

For t h e upper n i l r a d i c a l

Hence

[ill). Also f o r

( c f . [81 o r [lll). N t h e s i t u a t i o n i s more complicated.

It i s

shown i n [141 t h a t the analogue o f Theorem 4.11 f o r t o r s i o n - f r e e semigroups i s again t r u e f o r

N

.

Hence i t i s easy t o check t h a t a l l the r e s u l t s on t o r s i o n -

f r e e semigroups i n sections 4 t o 7 are again v a l i d f o r

N

.

However i f one

wants t o extend these r e s u l t s t o a r b i t r a r y semigroups, problems r e l a t e d t o t h e unsolved Koethe problem w i l l appear.

A description of the Jacobson radical

B(R[S])

As an a p p l i c a t i o n on the d e s c r i p t i o n o f

85

we consider t h e problem

o f when a semigroup r i n g i s (quasi-) l o c a l .

8.5

A r i n g w i t h i d e n t i t y i s said t o be a quasi-Zocd ring i f

Definition:

has a unique maximal i d e a l

R

M

.

A ring

R

w i t h i d e n t i t y i s s a i d t o be a

R/J(R)

ZocaZ ( r e s p e c t i v e l y seaZar ZocaZ) r i n g i f

i s a simple A r t i n i a n r i n g

(respectively a division ring).

i s t h e unique maximal i d e a l o f a quasi-local r i n g

If M

then

8.6

M = B(R)

the Brown-McCoy r a d i c a l o f

Proposition ([361)

R[S]

Let

be a semigroup r i n g w i t h i d e n t i t y .

R[SI

has o n l y one element o r e i t h e r

S

Then

Proof:

i s a p-group and

S

R

Suppose f i r s t t h a t R[S] i s a quasi-local r i n g .

image o f

i s a quasi-local r i n g too.

R[S]

d i f f e r e n t from

R[S],

c_

B(R)[Sl

S

0

, rs E

1

B(R) = {0)

R)

S

8 (R[S]/o(R[Sl))

s E S

t h a t f o r any i s a group.

,

Since

B(R[SI)

B(R)[SI

.

Hence

B(R) = I01

quasi-local r i n g and we may assume t h a t

1 rss I 1 rs=

.

Let

be t h e augmentation i d e a l o f

, we

obtain that

.

R[Sls = R [ S I

R/B(R

.

p > 0

has prime c h a r a c t e r i s t i c

S

.

i s a quasi-local r i n g ;

R

(ii)e i t h e r

{

with identity,

i s quasi-local i f and o n l y i f t h e f o l l o w i n g conditions are s a t i s f i e d :

(i)

R[S],

R

R

Hence S

being a homomorph i s an i d e a l o f (R/B(R))[Sl

is a

o(R[SI) =

R[SI.

o(R[SI) = $R[Sl)

Since

.

It f o l l o w s

i s a simple semigroup, i.e.

Theorem 4.11 f o r t h e Brown-McCoy r a d i c a l implies,

w(R[Sl) =

.

1 I(%,p(R),S,~p) I f S i s a s i n g l e t o n ( i i ) i s s a t i s f i e d . So PE Since we may assume t h a t S has a t l e a s t two elements, say s1 and s2

B (R[Sl) =

.

s1

-

s2 E o ( R [ S l )

n.1 = 0

p > 0

prime that

in R

S

.

.

we o b t a i n t h a t

n(sl-s2)

But t h e center o f

R

This y i e l d s o(R[SI)

i s a p-group.

= 0

f o r some 0 # n

i s a f i e l d , thus I(B (R),S,Ep) 0,P

.

C

p.1 = 0

EN

. Hence

f o r some

It i s then c l e a r

E. Jespen and P. Wauten

86

Conversely, suppose t h a t (i) and ( i i ) are s a t i s f i e d . there i s nothing t o prove. S

So we may assume t h a t

i s a p-group and R/B(R)

R[SI/B(R[Sl)

8.7 R[SI

y

Corollary

R/B(R)

i s a singleton

i s a quasi-local r i n g ,

has prime c h a r a c t e r i s t i c

.

B(R[SI) = B(R)[S] t I(BO,p(R),S,Sp)

Theorem 4.11,

R

If S

p > 0

.

Again by

This implies t h a t

vhich i s a simple r i n g by hypothesis.

(1361)

Let R[SI

be a semigroup r i n g w i t h i d e n t i t y .

i s l o c a l ( r e s p e c t i v e l y scalar l o c a l ) i f and o n l y i f R

( r e s p e c t i v e l y scalar l o c a l ) and e i t h e r i s a p-group and R/B(R)

0

S

i s local

has o n l y one element o r e i t h e r

has prime c h a r a c t e r i s t i c

p > 0

Then

S

.

0

Note t h a t i n t h e previous p r o p o s i t i o n and c o r o l l a r y we assumed t h a t the semigroup r i n g R[S1

has an i d e n t i t y .

but n o t necessarily t h a t t i v e semigroup with identity

S = Sa

ei

.

S

This implies t h a t

has an i d e n t i t y .

R

has an i d e n t i t y ,

For example consider a separa-

i s an abelian group U Sa2 U Sag such t h a t each Sa 1 i Suppose the s e m i l a t t i c e r = {al, a2, a31 has t h e

ordered s t r u c t u r e

Let

R

be a r i n g w i t h i d e n t i t y 1, then i t i s easy t o v e r i f y t h a t

i s the i d e n t i t y element o f

RlSI

.

Clearly

S

el t e2 - e 3

i t s e l f has no i d e n t i t y element.

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This Page intentionally left blank

91

BLOCKS AND VERTICES OF TWISTED GROUP ALGEBRAS G , Karpilovsky

University of t h e witwatersrand South Africa

F be an a l g e b r a i c a l l y closed f i e l d of characteristic p > 0 and l e t Q be a vertex of an i r r e d u c i b l e FG-module V belonging t o a block B AbstMct.

Then a fundamental theorem of Knorr 131 a s s e r t s t h a t one can choose

FG.

of

Let

D of

t h e d e f e c t group

B

CD(QI C _ Q

such t h a t

SD.

The a i m of t h i s paper

i s t o extend this r e s u l t t o p r o j e c t i v e r e p r e s e n t a t i o n s , i.e. to blocks of twisted group algebras.

Introduction. L e t G be a f i n i t e group, l e t F be a f i e l d and l e t c1 G + F* b e a cocycle. Denote by F G the corresponding t w i s t e d group algebra of G over F, The a l g e b r a f G h a s an F-basis I g E GI and t h e 0.

c1 :

G x

{s

multiplication i n F G i s determined by

--

zy = u(z,y)G

for a l l

P G is isomorphic t o t h e ordinary group a l g e b r a FG

Observe t h a t

r,y E G i f and only

G over F (or simply F i f (Z is not p e r t i n e n t to the d i s c u s s i o n ) one understands any m a p p t G * GL(n , B ( f o r some n 1) with is a coboundary.

a

if

By an a-repreesntation of

projective r e p r e s e n t a t i o n o f p(1) = 1

and

:

G

Over

p(z)p(y) = a(s,y)p(Zy)

GL(n,F) , i = 1 , 2 , M E GL(n,F) such t h a t pi

G

+

P,(S)

for a l l z,y

PG,

G.

Two a - r e p r e s e n t a t i o n s

for a l l g e G

= M"Pl (g)M

G

The modular a - r e p r e s e n t a t i o n theory of

blocks of

E

are s a i d t o be ZinearZy equivalent i f t h e r e e x i s t s

i s concerned with t h e study of

a

f o r the isomorphism classes of F G-modules correspond b i j e c -

t i v e l y , i n a well-known manner, with the l i n e a r equivalence classes o f ar e p r e s e n t a t i o n s of

G over F.

The i n t e n t i o n o f the p r e s e n t paper i s t o prove t h e following r e s u l t which i s

an extension t o p r o j e c t i v e r e p r e s e n t a t i o n s o f a fundamental theorem of Knorr [31. Theorem. and l e t

B

of

F be an a l g e b r a i c a l l y c l o s e d f i e l d of characteristic p > 0 V belonging t o a block Then one can choose t h e d e f e c t group D of B such t h a t

Let

@ be a v e r t e x of an i r r e d u c i b l e PaGmodule

PG.

CD(Q)

5Q

D

.

G. Karpilovsky

92 In particular

M

(i) For any i r r e d u c i b l e module

i n a block

B

of

PG,

contains, up t o conjugation, t h e centre of a d e f e c t group of

D

(ii) I f

D

i s a b e l i a n , then

M

a v e r t e x of

B.

i s a v e r t e x of any i r r e d u c i b l e PG-module

.

in B

1. Notation and t e d n o t o g y . Let

F

a(x,g)

= a(g,r)

x

=

.

Z2(G,F*)

conjugate o f Given

g

1X

x E CG(g)

for a l l

t h a t Supp e = C, U

Let

... U Ct

(or of t h e block

x

= {g E G

PG

e

x

g

.

I t can be shown (see Karpilovsky 121)

1


9

whence since

G. Karpilovsky

112 we have

(4)

a

-1

we now set

so t h a t

cr-lB

V =

v(n,g) = v(g,g

(5)

r

Let

V(6,y) = 1 f o r a l l

-1

6,I/

E

N and, by (4)

for all

W) G

be t h e V-representation of

n

E

N,g E G

induced by t h e i d e n t i t y r e p r e s e n t a t i o n

m e n , by Lemma 1 and (5) , w e have

N.

of

( n E N,g E G)

w , g ) B ( n , g ) = a-l(g,g-llZg)B(g,g-lv)

r(n) = 1 for a l l n E N

and

therefore r ( g ) = v(n,g)r(ng)

(6) Let

T denote a t r a n s v e r s a l f o r

let

C(g) E Z!'

Then, s e t t i n g

be such t h a t

N

for a l l

G

in

g E c(g)N.

-1 X(g) = v - l ( C ( g ) g ,g)

n E N,g E G

containing 1 and, f o r any L(g) = r ( C ( g ) )

Let

g E G,

for a l l

$7 E G.

w e have by ( 6 )

L(g) = r ( c ( g ) g - l g ) = x ( g ) r ( g ) Hence

L

is a y-representation of

G

where

a-l(B6X)

y = (6X)V =

(71

Since some

L i s constant on t h e c o s e t s of w E z~(Q,F*).

replacing

f

9

by

N, it follows t h a t y = i n f W f o r X(n) = 1 f o r a l l n E N. nus, X(g)fg we conclude from ( 7 ) t h a t B can always be chosen Note a l s o that

of t h e d e s i r e d form. F i n a l l y , assume that

W

i s a coboundary, say

w (oN,YN) = t(6N)t (yN)t WyN) f o r some

t :Q

X(g) = 6 (gN) F-bases of

-+

F*

with

and l e t

P G

= 1.

B = a i n f (w) FBG,

and

t(N)

.

a-1@ = i n f

(w)

,

X

: G

-+

F*

be defined by

E GI

Denote by

r e s p e c t i v e l y , such t h a t f o r a l l

G i = a(Z,$f)s , a a Since

Let

=

-

(Z,g)Zg

and

n

and

E

6,y E G,n E

N,

= 5

w e have

( a - h (6,Y)

= w(sW,?"

=

X(6)A(y)X(6y)-1

( X ,Y

E G)

113

Extending indecomposable modules

and t h e r e f o r e

is an PG-mdoule.

V

L v

n E N,XW

Since f o r a l l

= X(n)-'iiv

=

= 1 w e have

Ev

t h u s completing the proof.

rn

W e close this s e c t i o n by providing the following p i e c e of information.

y = infce)

Let

f o r some

p : G

Then there e x i s t s

-P

8 E Z2(6?,F*)

and suppose that

Y is

a coboundary.

such that f o r all z,y E G

F*

Y(Cc,Y) = ll(Z)P(Y)ll(zY)-l E G,n E N,y(n,g) = y ( g , n ) = 1, w e have

Because f o r a l l

~ ( m = PW) which shows that the r e s t r i c t i o n

=

(n E N.g E G)

u(g)v(n)

p ( N is a l i n e a r character of

Further-

N.

more, t a k i n g i n t o account t h a t N,g E G)

P(g(g-lng) = u ( g ) u ( g - l w = P ( V ) = P ( n ) P ( g ) w e have

u!g-lw)

n E N,g

for all

= I.l(fi)

)JIN i s i n f a c t a G-invariant l i n e a r c h a r a c t e r o f N. t o P I N a s the t i n e a r c h a r a c t e r a s s o c i a t e d with inf(f3). Thus

E

G

W e shall refer

3. PROOF OF THE THEOREM.

Assume that

W

i s an PG-module such that

a - r e p r e s e n t a t i o n of X#Y

G

W.

a f f o r d e d by

WN

V

Then d i m #

and l e t

be an

p

n

= dimJ=

and, for a l l

E G P(zc)P(Y) =

a(z,Y)P(xY) an = 66,

Taking the determinants o f b o t h s i d e s y i e l d s defined by all

t ( g ) = d e t p ( g ) ,g E G.

E G,5dN is a submodule of

w e have

WN,

t

where

n a i s a coboundary.

Thus

iWN

= WN.

:

G

-+

F*

is

Because f o r

Hence, f o r a l l

9 E G, V

-I

WN

1

Wig'

1

v(g)

proving that V is G-invariant. From now on we assume t h a t V is G-invariant and Because 'L that

v

is a coboundmy. W such p r o j e c t i v e cover of W. For the an

i s p r o j e c t i v e , t h e r e e x i s t s an i r r e d u c i b l e PN-module

= P(W), where

P(W)

denotes the

sake of c l a r i t y , we d i v i d e the rest of the proof i n t o a number of s t e p s .

Step 1.Here we show that W is G - ~ n v a r k n t by proving t h a t for a l l g E G. P(W(9))= P(W)(9) $ :

A s s u m e that

submodule

W'

of

P(w)

-f

w

i s an e s s e n t i a l epimorphism (i.e. f o r every proper

P(w) ,@(W') # JI(P(w)).

Then

JI can a l s o be regarded as

G. Karpilovsky

114

W"). Because 6/' is a submodule of P(W) i f and only i f it i s a submodule of P(M)('I, t h e l a t t e r epimrphism i s e s s e n t i a l . I t t h e r e f o r e s u f f i c e s t o prove t h a t i f W is p r o j e c t i v e , then so is W(') So assume t h a t W i s p r o j e c t i v e and w r i t e an epimorphism P ( w )

-+

.

( f N )()'

and

i s obviously a f r e e flN-module.

W(')

Hence

and t h e a s s e r t i o n follows.

(see Karpilovsky [ 4 ] ) .

i s s o l v a b l e , dim W d i v i d e s t h e o r d e r of N F m = dim W , a g e n e r a l i z e d Fong's dimension

G

We f i r s t observe t h a t s i n c e

.

W is extendible t o P G

Step 2 . Here we prove that

is projective

Hence f o r

F

formula (see Karpilovsky 141) t e l l s us t h a t

n n

Thus w e must have

=

p

k

IN1 rn

=

P P'

for some

k 2 0.

Now by hypothesis

n

cx

is a

coboundary and hence (see Curtis and Reiner [ZI), t h e o r d e r of the cohomology

a

c l a s s of

P.

i s n o t d i v i s i b l e by

The conclusion i s t h a t

m

a

is also a

coboundary.

G = N.H

Write

m

Because

i n Lemma 2 (with where

6

where

H

= a i n f w.

On t h e o t h e r hand,

i s a coboundary.

Since Us

G such t h a t N n H = 1. ( m , [ H ( ) = 1. L e t w E Z2(&,F*) be a s Then W can be extended t o an F 8G-module

i s a subgroup of

IN1 w e must r e s p e c t t o W).

divides

have

rn is a coboundary, i n f ( w )

$"

i s a coboundary where

Because

6

= [HI.

t h e l i n e a r c h a r a c t e r of

As has been observed earlier,

e N,h E H ,

X is

and p u t

A

i s G-invariant.

X(g) = X ( n ) .

a linear c h a r a c t e r of

6

(m,s) = 1, we deduce t h a t i n f ( w ) i s a coboundary.

In view of Lemma 2 , w e are l e f t t o v e r i f y t h a t t h i s end, we denote by

is a l s o a coboundary. Hence i n f (w) =inf (Us)

w

To

i s a coboundary.

N Fix

a s s o c i a t e d with i n f ( w ) .

g E G,

say

g = n..hh,

Then it is s t r a i g h t f o r w a r d t o v e r i f y

G which extends

A.

From t h e d e f i n i t i o n of

X

it follows t h a t t h e r e e x i s t s a map

p : G+F*

L e t t : G * P* be defined by t ( r ) = ~ ( z ) x ( z ) - ' . Then, f o r a l l g n E N , we o b t a i n

E

G

and

Extending indecomposable modules

115

and hence

t (4t (Y)t ( Z Y )

(**I

= I.r(Z)X(") - l u ( Y ) X ( Y ) -lll(ZY)

-lX(rY)

= i n f (w) (Z,Y) =

for a l l

t r y E G.

defined and by (**I

BY

(*I,

w

t

t h e function

:

G/N

+

F*,gN t+ t ( g )

is w e l l

w e have

w(ZN,YN) Thus

w(xN,t/N)

tl (YN)t l ( W N )-l

tl

=

(Z,Y

E G)

is a coboundary, a s required.

Step 3 . We now p a r e that

V

By S t e p 2 , t h e r e i s an PG-module

v I P(LN).

is extendible t o P G . L such that LN I W

and t h e r e f o r e

Thus w e need only v e r i f y t h a t

P(LN) = P ( Q N Because

G/N

i s a p'-group,

w e have

J(PG) = PG.J(PN) = J(PN)PG where

JCPG)

P G

i s t h e . Jacobson r a d i c a l of

( P G ) N/J Write

(PN) CPG)

=

(see Karpilovsky [ 4 ] ) . (fG/J

Hence

(PG))

PG/J(PG) = y divi

vi.

with nonisomorphic i r r e d u c i b l e PG-modules

Then,

P G = @ diP(Vi) 2

and t h e r e f o r e

a (F G)N =

9 diP(ViIN 2

Taking i n t o account t h a t

P G

i s a f r e e (hence, p r o j e c t i v e ) P'N-module, we

a l s o have

(PG)N

=

P( (J'%N)

=

P ( P G )N/J (fi") (PG)

=

P(PG/J(PG) )N)

=

9 diP( (Vi"' z

Thus P((v;)N)= P ( v i ) N and t h e r e q u i r e d a s s e r t i o n follows.

G. Karpilovsky

116

Here we c a n p l e t e t h e proof bg examining t h e c a m a = 1. A s s u m e t h a t F i s an a r b i t r a r y f i e l d of c h a r a c t e r i s t i c p > 0 and G an step 4 .

a r b i t r a r y f i n i t e group.

Then a fundamental theorem of I s a a c s [ 3 ] a s s e r t s

t h a t every G-invariant i r r e d u c i b l e FN-module is e x t e n d i b l e t o

FG.

Hence t h e

d e s i r e d conclusion follows by applying t h e argument employed i n S t e p 3. References.

[l] Conlon, S B., Twisted group a l g e b r a s and t h e i r r e p r e s e n t a t i o n s , J Austral. Math. SOC. 4(1964) ,152-173. [21 C u r t i s , C W., and Reiner, I., Representation theory of f i n i t e groups and a s s o c i a t i v e algebras, New York, I n t e r s c i e n c e , 1962. [31 I s a a c s , I M., Extensions of group r e p r e s e n t a t i o n s over a r b i t r a r y f i e l d s , J Algebra (1981), 54-74, 141 Karpilovsky, G . , P r o j e c t i v e r e p r e s e n t a t i o n s of f i n i t e groups, Marcel Dekker, Pure and Applied Mathematics. A series of monographs and Textbooks, New York. (1985)

.

Group and Semigroup Rings G. Karpilovsky (4.) 0 Elsevier Science Publishers B.V. (North-Holland), 1986

117

BLOCKS OF MODULAR GROUP ALGEBRAS Burkhard Kulshammer Fachbereich Mathematik, Universitat Dortmund, Postfach 50 05 00, 4600 Dortmund 50, Bundesrepublik Deutschland Let G be a finite (multiplicative) group and F an algebraically closed field of characteristic p. If p = 0 or if p is a prime not dividing the order I G I of G then, by Maschke's Theorem [16,V.2.7], the group algebra FG does not contain nilpotent ideals. It then follows from Wedderburn's Theorem [16,V.4.5] that FG is a direct sum of complete matrix algebras over F,

.

FG = @ i=l Mat (ni,F)

(11

In this article we describe some of the known results as well as some of the open problems in the case where p is a prime dividing I G I.

1. THE CENTER

Two elements g r h E G are called conjugate in G if and only if g = hX for some x E G. This gives an equivalence relation on G, the equivalence classes being called conjugacy classes. For any subset X of

GI

set X + := C x E X x

E

FG. Then the elements C+ where C ranges over

the set of all conjugacy classes of G form a basis of the center ZFG of FG, as is easily shown. (Recall that, for any F-algebra A, the center ZA consists of all elements z a E A.

E A

such that za = az for

In particular, the dimension of ZFG coincides with the

number of conjugacy classes of

G.

Obviously the center of a direct sum of algebras is the direct sum of their centers, and the center of a complete matrix algebra consists just of the scalar matrices. So in the case where the characteristic of F is zero or a prime not dividing I G I , formula

1i a

B. Kiilsharnrner

( 1 ) i m p l i e s t h a t t h e dimension of ZFG i s a l s o k ( G ) , t h e number of

summands a p p e a r i n g i n ( 1 ) . More e x p l i c i t l y

k(G) i s t h e number of

conjugacy c l a s s e s of G. T h i s g i v e s a v e r y s a t i s f a c t o r y d e s c r i p t i o n of t h e c e n t e r ( a r i n g - t h e o r e t i c i n v a r i a n t ) i n terms o f t h e group structure.

2 . THE RADICAL

Suppose now t h a t F h a s prime c h a r a c t e r i s t i c p. Then t h e group a l g e b r a FG c o n t a i n s non-zero n i l p o t e n t i d e a l s , f o r example t h e 1dimensional F-subspace g e n e r a t e d by G

+,

t h e sum of a l l e l e m e n t s i n

G , a s i s e a s y t o check. The sum of a l l n i l p o t e n t i d e a l s of FG i s

t h e maximal n i l p o t e n t i d e a l of FG, c a l l e d t h e r a d i c a l JFG of FG. One would o f c o u r s e a p p r e c i a t e a g r o u p - t h e o r e t i c d e s c r i p t i o n of t h e r a d i c a l , s i m i l a r t o t h a t of t h e c e n t e r above. However, our knowledge a b o u t JFG i s very incomplete. For example, t h e r e i s no d e s c r i p t i o n of JFG i n terms of t h e n a t u r a l b a s i s G of FG. Even worse, t h e r e i s n o t even an e x p l i c i t formula f o r i t s dimension i n g r o u p - t h e o r e t i c terms. The same a p p l i e s t o t h e Loewy l e n g t h t ( F G ) of FG, t h e s m a l l e s t p o s i t i v e i n t e g e r t such t h a t (JFG)t = 0. Not even r e a s o n a b l e bounds f o r t ( F G ) a r e known i n g e n e r a l . I f G i s ps o l v a b l e , i. e. i f G h a s a series of normal subgroups whose succ e s s i v e q u o t i e n t s a r e e i t h e r p-groups o r p'-groups

(of o r d e r n o t

d i v i s i b l e by p ) then a(P

-

1 ) + 1 S t ( F G ) 5 pa

where a Sylow p-subgroup of G has o r d e r pa (Wallace [38], Tsushima

[35]). There a r e examples of non-solvable g r o u p s where t (FG) > pa (Erdmann

[lo]).

I t i s known what groups can a r i s e i f t ( F G ) i s

s m a l l ; i n c a s e t(FG) = 2 , f o r example, p = 2 , and a Sylow p-subgroup of G h a s o r d e r 2 (Wallace [37]). I t must be s t r e s s e d a g a i n , however, t h a t more g e n e r a l informa-

t i o n on t h e r a d i c a l would be of g r e a t v a l u e .

Blocks of modular group algebras

119

3 . THE RADICAL FACTOR

By Wedderburn's Theorem, FG/JFG is a direct sum of complete matrix algebras over F, (2)

FG/JFG =

@

i=l (FG) Mat (ni,F)

.

The number 1(FG) has been computed by Brauer in the following way [17,VII.31. For any F-algebra A , K(A) denotes the F-subspace of A generated by all Lie commutators ab

-

ba, a,b

E

A. One easily

computes that for a complete matrix algebra A over F, K(A) is just the kernel of the trace map A

-P

F; in particular, K(A) has codi-

mension 1 in A. From this it. is not difficult to see that 1(FG) = dimF (FG/JFG)/K(FG/JFG) = = dimF (FG/JFG)/(K(FG) + JFG/JFG) = di?

FG/K(FG) + JFG.

One easily computes that K(FG) consists of all elements EgEG agg in FG such that

gEC a9 Using the fact that (3)

= 0 for all conjugacy classes C of G.

(a + b)p = ap + bP(mod K(A))

for elements a,b in an arbitrary F-algebra A over a field of characteristic p it is possible to see that n K ( A ) + JA = {a E A: ap E K(A) for some n EN). From this we can get a description of K(FG) + JFG in terms of group elements, again using (3). To do this we have to introduce some new notation. For any element g E G there are a unique p-element g (i. e. P g has order a power of p) and a unique p'-element g (i. e. P 9P' P' = ,g ; gp is called has order prime to p) such that g = q g P P' QP P the p-component and g the p'-component of g. We say that two P' elements g,h E G lie in the same pl-section of G if their p'-components are conjugate in G, This induces an equivalence relation on G, and the corresponding equivalence classes are called p'sections of G. Their number coincides with the number of p-regu-

120

B. Kiilshammer

l a r conjugacy classes of G where a conjugacy c l a s s i s c a l l e d pr e g u l a r i f i t s elements have o r d e r prime t o p . I t i s now e a s y t o check t h a t K(FG)

+

JFG c o n s i s t s e x a c t l y of t h o s e elements

C

= 0 f o r a l l p ' - s e c t i o n s S of G; i n a g such t h a t C gEG g gES ag p a r t i c u l a r , t h e number 1 ( F G ) c o i n c i d e s w i t h t h e number of p-regu-

l a r conjugacy c l a s s e s of G. I n case where p does n o t d i v i d e I G I , a l l conjugacy c l a s s e s a r e p - r e g u l a r ,

and J F G = 0, so w e g e t back

t h e formula d e r i v e d i n s e c t i o n 2 .

4 . THE SOCLE

For any F-algebra A , t h e s o c l e SOC A of A c o n s i s t s of a l l elements x E A such t h a t xJA = 0. I n g e n e r a l , t h i s may o r may n o t c o i n c i d e w i t h t h e set of elements x E A such t h a t JAx = 0. For group a l g e b r a s , however, both s e t s do c o i n c i d e . More g e n e r a l l y , f o r any i d e a l I of FG, {x E FG: XI = 0 ) =

(4)

{X

E FG:

I x = O}.

This i s due t o t h e f a c t t h a t group a l g e b r a s are symmetric; an Fa l g e b r a A i s c a l l e d symmetric i f t h e r e e x i s t s an F - l i n e a r map

t: A

-

F t h e k e r n e l o f which c o n t a i n s K ( A ) b u t no non-zero l e f t

o r r i g h t i d e a l . I n a symmetric F-algebra A , t h e map A

X

A * A,

(XIY)

-

t(W)t

i s a non-degenerate symmetric b i l i n e a r form. L e t us denote by X 0 t h e subspace of a l l elements i n A orthogonal t o a l l elements i n t h e s u b s e t X of A. I t i s e a s i l y s e e n t h a t , f o r an i d e a l I of A, I'

= { x E A: XI = 0). S i n c e t h e b i l i n e a r form i s symmetric, a l s o

Io = {x E A: I x = O}, which i s t h e e x p l a n a t i o n f o r ( 4 ) .

For a group a l g e b r a FG, t h e map t: FG t(CgEG

-

F i s d e f i n e d by

agg) := a , . The d i s c u s s i o n i n s e c t i o n 3 shows t h a t , f o r

any F-algebra A , J A i s t h e sum of a l l i d e a l s of A c o n t a i n e d i n K(A)

+

JA.

I f A is symmetric t h i s e a s i l y i m p l i e s t h a t

Blocks of modular group algebras J A = { a E A:

a(K(A) + JA)'

121

= 0).

One e a s i l y c h e c k s t h a t K ( A I o = ZA, so ( K ( A ) + J A ) '

= ZA rl SOC A ,

and SOC A i s g e n e r a t e d as an i d e a l by ZA rl SOC A. For a group a l g e b r a FG, w e have computed K(FG) + JFG i n s e c t i o n 3 . Using t h i s

w e e a s i l y g e t t h a t (K(FG) + JFG)'

= ZFG rl Soc FG h a s a b a s i s con-

s i s t i n g of a l l e l e m e n t s S+ where S r a n g e s o v e r t h e s e t o f a l l

PI-

s e c t i o n s o f G. Summarizing,

(5)

JFG

=

{ x E FG:

XS+ =

o

f o r a l l p l - s e c t i o n s s of GI.

T h i s f a c t had f i r s t been proved by Brauer [ 2 ] . Then h i s r e s u l t had been f o r g o t t e n , a p p a r e n t l y . I n [ 3 3 ] , Reynolds proved t h a t t h e F-subspace o f ZFG spanned by a l l e l e m e n t s S + where S r a n g e s o v e r t h e s e t o f a l l p ' - s e c t i o n s o f G i s indeed an i d e a l of FG, w i t h o u t g i v i n g a p r o o f of

(51, however. Then i n 1363, Tsushima r e p r o v e d

B r a u e r ' s r e s u l t . The proof s k e t c h e d above i s t a k e n from [ 2 3 , I ] .

5. THE RADICAL OF THE CENTER For any F - a l g e b r a A, t h e r a d i c a l J Z A o f t h e c e n t e r ZA o f A c o i n c i d e s w i t h J A rl ZA. There i s no d e s c r i p t i o n o f JZFG i n terms o f t h e group b a s i s G o f FG. However, i t f o l l o w s from (5), o f course, t h a t JZFG

= { z E ZFG:

zs+ =

o

f o r a l l p l - s e c t i o n s s of G I .

A s was shown i n [ 1 8 ] , t h i s can be improved t o

JZFG = {z E ZFG: zG+ = 0) P where G d e n o t e s t h e s e t of p-elements i n G , t h e p ' - s e c t i o n of G P c o n t a i n i n g t h e i d e n t i t y e l e m e n t of G. Obviously, ZFG = AZFG$ JZK

(6)

where AZFG d e n o t e s t h e F-subalgebra of ZFG spanned by a l l idemp o t e n t s . S t i p u l a t e d by r e s u l t s of G. R. Robinson i n [341, M. Brou6 [31 and t h e a u t h o r [23,IIII i n d e p e n d e n t l y found a formula f o r t h e p r o j e c t i o n map 6 : ZFG

+

AZFG w i t h k e r n e l JZFG: 6 ( z ) =

= n ( z G + ) f o r z E ZFG, where t h e F - l i n e a r map n: FG

P

-D

FG i s d e f i -

122

B. Kiilshammer

ned by n ( g ) = g i f g i s of o r d e r prime t o p and T ( g ) = 0 o t h e r -

wise. No

handy d e s c r i p t i o n of JZFG i s known. I n [34], G. R. Robinson

d e s c r i b e d i t a s t h e r a n k of a c e r t a i n m a t r i x w i t h e n t r i e s i n t h e f i e l d w i t h p elements. We w i l l come back t o h i s r e s u l t when w e t a l k a b o u t b l o c k s . Nothing seems t o be known a b o u t t h e s o c l e of ZFG. Of c o u r s e , ZFG n SOC FG 5 SOC ZFG. When does e q u a l i t y h o l d ?

6 . p-GROUPS Most of t h e r e s u l t s mentioned so f a r a r e t r i v i a l f o r p-groups. I f G i s a p-group t h e n JFG i s t h e k e r n e l of t h e augmentation map FG

+

F, CglIG

agg

+

CgEG

a g ; i n p a r t i c u l a r , FG/JFG

=

F. Much i s

known a b o u t t h e dimensions of powers of JFG, due t o t h e work of J e n n i n g s [ 2 0 ] . H i s r e s u l t s can be found i n [ 1 7 , V I I I . 2 ] , f o r examp l e . However, some more s t r u c t u r a l r e s u l t s would be h e l p f u l .

7. THE UNIT GROUP For any F-algebra A , A = S isomorphic t o A / J A ;

(b

J A where S i s a s u b a l g e b r a of A

f u r t h e r m o r e , S i s unique up t o c o n j u g a t i o n

w i t h e l e m e n t s i n t h e subgroup 1 + J A of t h e u n i t group UA of A. T h i s f o l l o w s from t h e Wedderburn-Malcev Theorem [ 7 , 7 2 . 1 9 ] .

It i s

an immediate consequence t h a t UA i s t h e s e m i d i r e c t p r o d u c t of US and 1 + J A ; h e r e o b v i o u s l y US i s a d i r e c t p r o d u c t of g e n e r a l l i n e a r groups over F, and 1 + J A i s a p-group due t o t h e f a c t t h a t JA i s n i l p o t e n t . T h i s o b s e r v a t i o n i s sometimes u s e f u l f o r cohomo-

logical questions. Now l e t G be a f i n i t e group. There a r e s e v e r a l r e s u l t s r e l a t e d t o t h e q u e s t i o n how G i s embedded i n t o t h e u n i t group UFG of FG. Here w e would j u s t l i k e t o mention t h e f o l l o w i n g f a c t due essent i a l l y t o Coleman [61. For any p-subgroup P of G , t h e n o r m a l i z e r

Blocks of modular group algebras

123

NUFG(P)o f P i n t h e u n i t g r o u p o f FG c o i n c i d e s w i t h NG.(P)UCFG(P) where CFG(P) d e n o t e s t h e c e n t r a l i z e r o f P i n FG, i . e. t h e set of

a l l e l e m e n t s a E FG s u c h t h a t a u = ua f o r u E P. NUFP(P)

= P

-

In p a r t i c u l a r ,

UZFP f o r any f i n i t e p-group P which i s t h e r e s u l t

s t a t e d i n [6].

8. BLOCKS The i n v e s t i g a t i o n o f b l o c k s i s one of t h e most i m p o r t a n t t a s k s i n t h e subject of modular group a l g e b r a s . A number of r e l e v a n t r e s u l t s have been o b t a i n e d o v e r t h e l a s t few y e a r s , some o f t h e most p o w e r f u l o n e s have n o t y e t a p p e a r e d i n p r i n t . L e t A be a f i n i t e - d i m e n s i o n a l a l g e b r a o v e r a f i e l d F , and w r i t e

lA = el

+ e2 +

... +

er. w i t h p a i r w i s e o r t h o g o n a l p r i m i t i v e idem-

p o t e n t s ei i n t h e c e n t e r of A; o b v i o u s l y r i s t h e dimension of ZA/JZA.

Then w e have a d e c o m p o s i t i o n A = B1 @ B2 @

w i t h i d e a l s Bi

...

= e i A o f A;

@

Br

e a c h Bi

i s i t s e l f an F - a l g e b r a w i t h

= e i . The i d e a l s Bi a r e u n i q u e l y d e t e r m i n e d by A and Bi are c a l l e d t h e b l o c k s of A. The problem w e w i l l be c o n c e r n e d w i t h identity 1

i n t h e s u b s e q u e n t s e c t i o n s i s t o d e s c r i b e t h e s t r u c t u r e of a b l o c k of t h e g r o u p a l g e b r a FG, a t l e a s t i n some i n t e r e s t i n g s p e -

c i a l c a s e s . By a r e s u l t of O s i m a [ 1 7 , V I I . 1 2 . 8 ]

t h e i d e m p o t e n t s ei

are l i n e a r c o m b i n a t i o n s o f p - r e g u l a r e l e m e n t s i n C (0 ( G ) ) where G P

0 ( G ) d e n o t e s t h e maximal normal p-subgroup

P

of G.

9. DEFECT GROUPS

The d e f e c t g r o u p s c o n s t i t u t e t h e most i m p o r t a n t i n v a r i a n t o f a b l o c k . I n b l o c k t h e o r y t h e y p l a y a r o l e s i m i l a r t o t h a t p l a y e d by Sylow s u b g r o u p s i n f i n i t e g r o u p t h e o r y . I n o r d e r t o d e f i n e a d e f e c t group l e t u s f i r s t i n t r o d u c e an i m p o r t a n t map, t h e r e l a t i v e t r a c e map

124

B. Kiilsharnrner

TrH K , d e f i n e d f o r subgroups K and H of G such t h a t K i s c o n t a i n e d H i n H. T r K : C F G ( K )

over a t r a n s v e r s a l

-B

CFG(H) maps c E CFG(K) o n t o Zcg where g r a n g e s

f o r t h e r i g h t c o s e t s of K i n H ; i t i s obvious-

l y independent of t h e c h o i c e of t h i s t r a n s v e r s a l . For any subgroup K of G , t h e image (FG); i s an i d e a l i n CFG(G)

= ZFG which can be d e s c r i b e d i n terms of t h e group

a s follows.

basis

G (FG)K h a s an F- b a s i s c o n s i s t i n g o f t h e c l a s s sums C+

of G w i t h t h e p r o p e r t y t h a t K c o n t a i n s a Sylow p-subgroup of CG(g) f o r some g E C. A subgroup D of G i s c a l ed a d e f e c t group of t h e b l o c k B of

FG i f D i s minimal i n t h e s e t of a l l subgroups K of G such t h a t ZB

=

G (FG)K. A b a s i c p r o p e r t y i s t h a t t h e d e f e c t groups o f a b l o c k

form a unique conjugacy c l a s s of p-subgroups of G c o n t a i n i n g 0 ( G ) ; more p r e c i s e l y ,

P

t h e y a r e i n t e r s e c t i o n s of two Sylow p-sub-

groups of G [ 151. An i n t e r e s t i n g q u e s t i o n i s whether, f o r a given p-group D, t h e r e a r e o n l y a f i n i t e number of Morita e q u i v a l e n c e c l a s s e s of b l o c k s of group a l g e b r a s w i t h d e f e c t group D. A more p r e c i s e q u e s t i o n h a s been r a i s e d by L. Puig i n [31]. S e v e r a l r e s u l t s i n t h e subsequent s e c t i o n s c o n t r i b u t e t o t h i s q u e s t i o n . R e c a l l t h a t two F - a l g e b r a s A , B a r e c a l l e d M o r i t a e q u i v a l e n t i f t h e i r module c a t e g o r i e s a r e e q u i v a l e n t . An e q u i v a l e n t c o n d i t i o n i s t h a t A

e M a t ( n , B ) e and B

fMat(m,A)f f o r some p o s i t i v e i n t e g e r s m,n

and some idempotents e E M a t ( n , B ) , f E Mat(m,A). Y e t a n o t h e r e q u i v a l e n t c o n d i t i o n i s t h a t A and B have isomorphic b a s i c suba l g e b r a s ; by a b a s i c s u b a l g e b r a o f A w e mean a s u b a l g e b r a

A'

=

i A i of A f o r some idempotent i of A such t h a t A A ' A = A and A ' / J A '

i s commutative. A b a s i c s u b a l g e b r a of A i s u n i q u e l y determined up t o c o n j u g a t i o n w i t h u n i t s i n A.

Blocks of modular group algebras

125

1 0 . THE CENTER OF A BLOCK

Many o f t h e problems, c o n s i d e r e d f o r group a l g e b r a s so f a r , have a n a l o g u e s f o r b l o c k s . As an example f o r t h e i n f l u e n c e o f t h e d e f e c t g r o u p s on t h e s t r u c t u r e of a b l o c k w e would l i k e t o mention t h e f o l l o w i n g r e s u l t , due t o Okuyama [271. For any b l o c k B w i t h d e f e c t group D, (JZB)IDI = 0 ; even ( J Z B ) I D I

-

-c

SOC B.

In par-

titular, J Z B = 0 i n c a s e )Dl = 1 . I t f o l l o w s from t h e f a c t t h a t

each b l o c k , j u s t l i k e t h e group a l g e b r a i t s e l f , i s a symmetric a l g e b r a t h a t J Z B = 0 i m p l i e s J B = 0. T h i s shows t h a t b l o c k s of i. e . b l o c k s w i t h d e f e c t group of o r d e r 1 ) a r e com-

defect zero

p l e t e m a t r i x a l g e b r a s o v e r F, a r e s u l t f i r s t observed by R. Brau-

e r [12,IV.4.

9 1 . I f p d o e s n o t d i v i d e t h e o r d e r of G t h e n e v e r y

b l o c k of FG h a s d e f e c t z e r o , so w e g e t back t h e Maschke-Wedderburn r e s u l t s . An i m p o r t a n t c o n j e c t u r e due t o R.

Brauer [12,IV.5]

states that

t h e dimension of t h e c e n t e r of B i s bounded above by t h e o r d e r of

i t s d e f e c t group. The more o r less r i n g - t h e o r e t i c methods w e have d i s c u s s e d so f a r have n o t y e t c o n t r i b u t e d t o t h i s c o n j e c t u r e . Using d i f f e r e n t methods, t o be d i s c u s s e d l a t e r , t h e c o n j e c t u r e has been confirmed i n some s p e c i a l c a s e s . For g r o u p s of odd o r d e r , t h i s i s due t o Knorr [ 2 1 ] and Gluck [ 1 4 ] . For symmetric, g e n e r a l l i n e a r and u n i t a r y groups t h i s i s due t o Olsson [ 2 9 1 ( f o r most 2

p r i m e s ) . I n g e n e r a l , i t i s t r u e t h a t ID1 / 4

+ 1 i s an upper bound

f o r t h e dimension o f ZB. Reasonable lower bounds f o r t h e dimens i o n of ZB

would a l s o be d e s i r a b l e .

11. THE RADICAL OF A BLOCK Not much i s known a b o u t t h e r a d i c a l of a b l o c k . I n [ 2 3 , I I ] t h e a u t h o r proved t h a t ( J B ) q / p

# 0 where q i s t h e exponent o f a d e f e c t

group D ( # 1 ) o f t h e b l o c k B of FG, i. e. t h e s m a l l e s t p o s i t i v e i r b

126

B. Kiilshammer

t e g e r n s u c h t h a t dn = 1 f o r d E D. non-trivial,

In p a r t i c u l a r , J B # 0 i f D is

a r e s u l t due t o B r a u c r [12,IV.4.19].

Blocks B w i t h ( J B ) 2 = 0 have a d e f e c t g r o u p o f o r d e r 2, by a r e s u l t o f W a l l a c e [ 3 7 1 . More r e c e n t l y , Okuyama [26] proved t h a t b l o c k s B w i t h ( J B ) 3 = 0 have a d e f e c t g r o u p o f o r d e r e i t h e r f o u r o r p , g i v i n g some a d d i t i o n a l i n f o r m a t i o n as w e l l .

1 2 . THE NUMBER OF BLOCKS

L e t D b e a p-subgroup

o f t h e f i n i t e group G and P a Sylow p-

subgroup o f G c o n t a i n i n g D. I n [ 3 4 1 , G. R. Robinson h a s found an i m p o r t a n t f o r m u l a f o r t h e number o f b l o c k s o f FG w i t h d e f e c t group D.

H i s d e s c r i p t i o n i s g i v e n i n terms of t h e r a n k of a c e r t a i n

m a t r i x . To e x p l a i n it w e have t o i n t r o d u c e t h e f o l l o w i n g n o t i o n . The Sylow p-subgroups of t h e c e n t r a l i z e r s o f t h e e l e m e n t s i n a c o n j u g a c y class C of G a r e c a l l e d t h e d e f e c t g r o u p s of C . F o r any e l e m e n t g E G and any c o n j u g a c y c l a s s C o f G s e t n ( g , C ) :=

I I E ~gp n c: P n gpg-l E

sylp(cG(c))j~

where S y l ( G ) d e n o t e s t h e c o l l e c t i o n of a l l Sylow p-subgroups of P G. I t i s n o t d i f f i c u l t t o show t h a t n ( g , C ) i s c o n s t a n t on e a c h d o u b l e c o s e t PgP o f P i n G. L e t us d e n o t e by N t h e m a t r i x w i t h c o e f f i c i e n t s n ( g , C ) , t h e rows b e i n g i n d e x e d w i t h t h e d o u b l e c o s e t s

PgP o f P i n G I t h e columns b e i n g indexed by t h e p - r e g u l a r c o n j u g a c y classes o f G w i t h d e f e c t g r o u p D. Then t h e number o f b l o c k s o f FG w i t h d e f e c t group D c o i n c i d e s w i t h t h e r a n k of t h e m a t r i x N TN considered a s a matrix over t h e f i e l d with p elements. A s i m p l e p a r t i c u l a r case, a l r e a d y known t o B r a u e r and N e s b i t t

[7,86.10]

i s t h e f o l l o w i n g . The number o f b l o c k s w i t h d e f e c t group

P I a Sylow p-subgroup of G I c o i n c i d e s w i t h t h e number o f p - r e g u l a r c o n j u g a c y classes w i t h d e f e c t g r o u p P i n G .

Blocks of modular group algebras

127

13. REDUCTION THEOREMS

In this section we report on two reduction techniques usually attributed to P. Fong [131. Similar techniques had, however, before been used by Morita [25]. For both techniques one has to assume the presence of a (non-trivial) normal subgroup N of the finite group G. In this situation G permutes the blocks of FN under the conjugation action. Let us denote by T the stabilizer of a fixed block 8 of FN. If B is a block of FG with 1 1 # 0 then l B 1 8 B 8 is the identity of a block b of FT, and we have an isomorphism of F-algebras Mat(lG:Tl,b)

+

B, x

+

Cg-'xh,

where g ranges over a transversal for the right cosets of T in G. This shows that in order to compute the isomorphism type of B one only has to compute the isomorphism type of b. It is not difficult to prove that the blocks B and b have a common defect group. After the reduction described above has been carried out one can usually assume that the block B of FN is stable under conjugation with elements in G. In certain cases it is possible to obtain a further reduction. The most familiar situation is the one where N is a p'-group, i. e. its order is not divisible by p. The reduction also works well if, slightly more general, the block f3 of FN has defect zero. In these cases 8 is a complete matrix alge-

bra over F, and the identity 1

8

is a central idempotent in FG. The

reduction applies to the algebra 18FG, usually a sum of several blocks of FG. Since the simple F-algebra 8 is a subalgebra of 1 FG, a well-known result in ring theory implies that we have a

B tensor product decomposition 1 FG = 88, CFG(B)18. It turns out now B that CFG(B)l8 is a twisted group algebra of G/N over F. (Recall that a twisted group algebra of G over F is an F-algebra A with fixed decomposition tisfying A A = g h

A

gh

A into 1-dimensional subspaces sagEG g for g,h E G . ) This shows that blocks of FG

A = @

128

B. Kulshammer

contained in 1 FG are in bijection with blocks of this twisted B group algebra, and corresponding blocks are Morita equivalent. Furthermore, the definition of a defect group can easily be generlized to twisted group algebras, and then corresponding blocks have isomorphic defect groups. It is possible to return from twisted group algebras to (ordinary) group algebras. Let G be a finite group and A =

@

gEG Ag a twisted group algebra over F. Then there are a central extension l + Z + G + G + l

where Z is a p'-group and a block B of FZ such that the blocks of A are in bijection with blocks B of FG satisfying I B l B f 0. Furthermore, corresponding blocks are isomorphic and have isomorphic defect groups. After performing these reductions one can usually assume that Opl(G), the largest normal p'-subgroup of G, is central in G.

1 4 . TWISTED GROUP ALGEBRAS AND THE SCHUR MULTIPLIER

Let A =

A be a twisted group algebra of G over F. Then 9EG 9 the disjoint union GrU(A) of all subsets A n UA is a subgroup of @

9

UA. More precisely, it is an extension of groups 1

+

UF

+

GrU(A)

+

G

+

1

where UF is identified with UA1 and the map GrU(A) unit in A

9

+ G

maps any

n UA onto g . Using the well-known relationship between

group extensions and cohomology, each twisted group algebra induces an element in the second cohomology group H2 (G,UF), the Schur multiplier of G over F. It is not difficult to see that this procedure can be reversed, constructing a twisted group algebra from any element in H2 (G,UF).

15. p-NILPOTENT GROUPS AND BLOCKS WITH CENTRAL DEFECT GROUPS

Let us see what the reduction theorems lead to in the case of

Blocks of modular group algebras

129

blocks of p-nilpotent groups: recall that a finite group G is called p-nilpotent if G/O

(G) is a p-group. After applying the P' techniques explained in section 13 to the normal subgroup 0 (GI P' it is easy to see that each block of FG is isomorphic to a com-

plete matrix algebra over the twisted group algebra of a finite p-group. Now each twisted group algebra of a finite p-group P is isomorphic to an ordinary group algebra, due to the fact that the Schur multiplier H L (P,UF) is trivial. So indeed any block of a pnilpotent group is isomorphic to a complete matrix algebra over the group algebra of a finite p-group [25]. A careful inspection of the proof then shows that the p-group is in fact a defect group of the block. There is a similar result in a slightly different situation. Suppose that B is a block of some finite group G with defect group D such that G = DCG(D). We easily see that B is mapped onto a block of defect zero in F[G/D] under the canonical epimorphism FG

+

F[G/D]. Lifting the corresponding matrix algebra to FG we get

that B

Mat(n,FD) for some n

E

N. This is obviously a generaliza-

tion of the analogous result for blocks of defect zero. In these situations the defect group carries indeed all important information about the block. The question arises whether there is a common generalization of these two results. More precisely, the question is whether one can find necessary and sufficient conditions on the block B to be a complete matrix algebra over the group algebra of its defect group. This will lead us to an interesting result due to Brout-Puig and an interesting conjecture due to Dade. To discuss it we have to introduce additional material.

1 6 . THE LOCAL STRUCTURE OF BLOCKS

Let B be a block of the group algebra FG for some finite group

B. Kulshammer

130

G . W e had remarked above t h a t t h e d e f e c t groups o f

B play a r o l e

i n b l o c k t h e o r y s i m i l a r t o t h a t p l a y e d by t h e Sylow p-subgroups i n f i n i t e group t h e o r y . T h i s analogy h a s been made m o r e p r e c i s e and extended by work due t o A l p e r i n , Broui! and P u i g [ 1 ; 4 1 . For any p-subgroup P o f G t h e F - l i n e a r map B r p : FG

+

FCG(P)

d e f i n e d by B r p ( g ) = g f o r g E C (PI and B r p ( g ) = 0 o t h e r w i s e i n G

duces a homomorphism o f F - a l g e b r a s C,,(P)

-c

F C G ( P ) , t h e Brauer

homomorphism. For a b l o c k B o f FG, a B-subpair i s a p a i r ( Q , b ) c o n s i s t i n g of some p-subgroup Q o f G and some block b of C G ( Q ) such t h a t B r ( 1 ) 1 # 0 o r , e q u i v a l e n t l y , B r Q ( l B ) l b = l b . The BQ B b s u b p a i r s s h o u l d be c o n s i d e r e d t o be an analogue of t h e p-subgroups

of a group. There i s an i n c l u s i o n r e l a t i o n among t h e B - s u b p a i r s , d e f i n e d i n t h e f o l l o w i n g way. A B-subpair B-subpair

( P , b p ) i s s a i d t o be c o n t a i n e d i n a

(Q,bQ), w r i t t e n (P,bp) L (Q,bQ), i f P L

Brp(i)l # 0 bP 0 f o r some p r i m i t i v e idempotent i i n C F G ( Q ) . I t i s QI

and B r Q ( i ) l # bQ p o s s i b l e t o show t h a t t h e r e l a t i o n t h u s d e f i n e d has indeed t h e u s u a l p r o p e r t i e s of t h e i n c l u s i o n r e l a t i o n .

I f (D,b ) is a maximal B-subpair w i t h r e s p e c t t o t h i s i n c l u s i o n

D

t h e n D is a d e f e c t group o f B. Obviously G acts on t h e set o f a l l B-subpairs by c o n j u g a t i o n , and i t t u r n s o u t t h a t G a c t s t r a n s i t i v e l y on t h e s e t of a l l maximal B-subpairs,

an analogue o f t h e

Sylow theorem a g a i n .

17. NILPOTENT BLOCKS R e c a l l t h a t w e are aiming a t a g e n e r a l i z a t i o n o f t h e s t r u c t u r e theorem f o r b l o c k s o f p - n i l p o t e n t groups. I n o r d e r t o e x p l a i n it

w e f i r s t have t o look a t p - n i l p o t e n t groups from a d i f f e r e n t p o i n t o f view. A well-known theorem due t o Frobenius [16,IV.5.8]

states

t h a t a f i n i t e group G i s p - n i l p o t e n t i f and o n l y i f N G ( Q ) / C G ( Q )

is

a p-group f o r any p-subgroup Q o f G. T h i s p r o p e r t y o f p - n i l p o t e n t

131

Blocks of modular group algebras

groups a d m i t s a t r a n s l a t i o n i n t o t h e b l o c k - t h e o r e t i c language developed i n t h e p r e c e d i n g s e c t i o n . I n 151 a b l o c k B o f t h e group a l g e b r a FG o f an a r b i t r a r y f i n i t e group G is c a l l e d n i l p o t e n t i f N ( Q , b ) / C ( Q ) i s a p-group G G

f o r a l l B-subpairs

( O r b ) where N,(Q,b)

denotes t h e s t a b i l i z e r of (Q,b) i n G. The f i r s t major p r o g r e s s towards an u n d e r s t a n d i n g o f n i l p o t e n t b l o c k s had been a c h i e v e d i n [ 5 ] .

A f t e r t h a t , L . Puig succeeded t o

prove t h a t a n i l p o t e n t b l o c k i s isomorphic t o a f u l l m a t r i x a l g e b r a over t h e group a l g e b r a o f i t s d e f e c t group [30l. I t i s e a s y t o

see t h a t h i s r e s u l t comprises t h o s e mentioned i n s e c t i o n

5.

Dade whether t h e c o n v e r s e o f P u i g ' s

I t h a s been asked by E.C.

r e s u l t i s t r u e , i . e . whether a b l o c k which is i s o m o r p h i c t o a f u l l m a t r i x a l g e b r a o v e r i t s d e f e c t g r o u p a l g e b r a i s nece s a r i l y n i l p o t e n t . I f t h e d e f e c t group o f t h e b l o c k i s a b e l i a n , h i s q u e s t i o n h a s a p o s i t i v e answer, due t o Okuyama-Tsushima

[281. The gen-

e r a l case i s s t i l l open.

18. BLOCKS WITH NORMAL DEFECT GROUPS The r e s u l t s i n t h i s s e c t i o n have been o b t a i n e d i n d e p e n d e n t l y by L. P u i g ( u n p u b l i s h e d ) and t h e a u t h o r [ 24 1

, strengthening

earlier

r e s u l t s due t o Reynolds [32]. Our r e s u l t a l s o g e n e r a l i z e s t h e res u l t on b l o c k s w i t h d e f e c t g r o u p D such t h a t t h e c o r r e s p o n d i n g group G h a s t h e form G = D C G ( D ) . L e t u s suppose now t h a t B i s a b l o c k o f some f i n i t e g r o u p G

s u c h t h a t a d e f e c t group D of B is normal i n G . Denote by b a b l o c k o f FDCG(D) such t h a t l B l b # 0. A f t e r c a r r y i n g o u t t h e reduct i o n explained i n section

3 one may assume t h a t i n t h i s s i t u a t i o n

DC ( D ) / C ( D ) i s a normal S low p-subgroup o f G / C G ( D ) . G

G

Zassenhaus theorem [16,1.181, G/CG(D),

K on D,

DCG(D)/CG(D)

By t h e Schur-

h a s a complement K i n

unique up t o c o n j u g a t i o n . There is a n obvious a c t i o n o f and t h e isomorphism t y p e o f t h e s e m i d i r e c t p r o d u c t KD does

6.Kiilshammer

132

n o t depend o n t h e c h o i c e o f t h e complement K . The main r e s u l t o f [ 2 4 ] asserts t h a t t h e b l o c k B is i s o m o r p h i c t o a f u l l m a t r i x a l g e -

b r a o v e r some t w i s t e d g r o u p a l g e b r a o f t h e s e m i d i r e c t p r o d u c t K D . S i n c e KD i s p - s o l v a b l e t h i s r e s u l t r e d u c e s many q u e s t i o n s on b l o c k s w i t h normal d e f e c t g r o u p s t o q u e s t i o n s a b o u t b l o c k s o f ps o l v a b l e g r o u p s . We remark t h a t KD need n o t be i s o m o r p h i c t o a q u o t i e n t o f some s u b g r o u p of G.

1 9 . BRAUER'S MAIN THEOREMS

B r a u e r ' s f i r s t main theorem [ 1 2 , 1 1 1 . 9 . 7 ] s t a t e s t h a t f o r any ps u b g r o u p D of t h e f i n i t e g r o u p G t h e r e i s a b i j e c t i o n between t h e

s e t of b l o c k s of FG w i t h d e f e c t g r o u p D and t h e s e t o f b l o c k s of FNG(D) w i t h d e f e c t g r o u p D. T h i s b i j e c t i o n i s i n d u c e d by t h e Brau-

e r homomorphism: B r ( 1 ) i s t h e i d e n t i t y o f t h e b l o c k of N G ( D ) D

B

c o r r e s p o n d i n g t o t h e b l o c k B of FG. However, i n g e n e r a l B r D d o e s n o t e x t e n d t o a n a l g e b r a homomorphism between c o r r e s p o n d i n g blocks. Many o f t h e open q u e s t i o n s i n t h e s u b j e c t c o n c e r n t h e e x a c t relat i o n s h i p o f t h e c o r r e s p o n d i n g b l o c k s , and i t i s t o be hoped t h a t t h e r a t h e r d e t a i l e d i n f o r m a t i o n o b t a i n e d f o r t h e b l o c k s of NG(D) i n t h e previous s e c t i o n contributes t o these questions. B r a u e r ' s second main theorem is a s t a t e m e n t a b o u t c h a r a c t e r s and h a s no good i n t e r p r e t a t i o n i n o u r s e t u p . I n o r d e r t o e x p l a i n h i s t h i r d main theorem w e have t o i n t r o d u c e t h e n o t i o n o f a p r i n c i p a l b l o c k . For any f i n i t e g r o u p G , t h e p r i n c i p a l b l o c k o f FG, o f t e n d e n o t e d by B O ( G ) r i s t h e o n l y b l o c k B of G s u c h t h a t

~ ( 1 , ) = 1 where u: FG

-.

F d e n o t e s t h e a u g m e n t a t i o n map. Then, i n

t h e l a n g u a g e i n t r o d u c e d i n s e c t i o n 1 6 , B r a u e r ' s t h i r d main theorem

a s s e r t s t h a t t h e B o ( G ) - s u b p a i r s are e x a c t l y t h e p a i r s ( Q , B O ( C G ( Q ) ) ) where Q r a n g e s o v e r t h e s e t of a l l p-subgroups of G . F o r B O ( G ) s u b p a i r s ( Q r B O ( C G ( Q )) ) and ( P , B O ( C G ( P ) ) t h e a s s e r t i o n (Q,B,(C,(Q)))

E. ( P , B o ( C G ( P ) ) ) i s e q u i v a l e n t t o Q E P; i n p a r t i c u -

Blocks of modular group algebras

133

l a r , t h e Sylow p-subgroups o f G a r e t h e d e f e c t groups of B O ( G ) . Thus t h e a n a l y s i s o f Bo(G)-subpairs i s e q u i v a l e n t t o t h e a n a l y s i s o f t h e p-subgroups o f G [ l ] .

20. EXTENSIONS OF NILPOTENT BLOCKS I n t h i s s e c t i o n I s h a l l r e p o r t on r e c e n t j o i n t work w i t h L. Puig. Our r e s u l t s g e n e r a l i z e t h o s e o b t a i n e d i n s e c t i o n s 17 and 18. L e t G be a f i n i t e group, H a normal subgroup o f G and b a n i l -

p o t e n t b l o c k o f FH w i t h d e f e c t g r o u p Q. A s u s u a l w e may assume t h a t b i s s t a b l e i n G. Among t h e b l o c k s B o f FG such t h a t l B l b # 0 choose one w i t h a d e f e c t group o f maximal o r d e r . Then Q some d e f e c t group P o f B. Denote by N , ( Q , B ) maximal b - s u b p a i r NH(Q,B)

(Q,B).

c

P for

t h e s t a b i l i z e r of a

I t t u r n s o u t t h a t G = HNG(Q,B) and

= Q C H ( Q ) . Furthermore,

P C H ( Q ) / C H ( Q ) i s a Sylow p-subgroup

f o r a s u i t a b l e c h o i c e o f P.

of N , ( Q I ~ ) / C H ( Q ) ,

I n t h i s s i t u a t i o n w e c o n s t r u c t a group extension 1

+

Z(Q)

-

N * NG(Q,B)/CH(Q)

and an e x t e n s i o n homomorphism 1 1

-.

Z(Q)

+

11

Z(Q)

- +

P

PCH(Q)/CH(Q)

N

NG(QtB)/CH(Q)

i-

1

+

+

1

1 1.

The e x t e n s i o n i s u n i q u e l y determined by t h e diagram s i n c e Z ( Q ) i s a p-group and P C H ( Q ) / C H ( Qi)s a Sylow p-subgroup o f N G ( Q , B ) / C H ( Q ) .

Our main r e s u l t t h e n shows t h a t t h e F - a l g e b r a lbFG is a complete m a t r i x a l g e b r a o v e r a t w i s t e d group a l g e b r a o f N o v e r F. Using an i m p o r t a n t r e s u l t due t o Dade [ 9 ] w e a r e a l s o a b l e t o show t h a t t h e F - a l g e b r a s lbFG and 1 FN ( Q , B ) B G

are M o r i t a e q u i v a l e n t .

21. BLOCKS OF p-SOLVABLE GROUPS For p - s o l v a b l e g r o u p s , t h e r e d u c t i o n theorems o f s e c t i o n 1 3 are v e r y u s e f u l . Using Dade's r e s u l t mentioned above i n c o n n e c t i o n

134

8. Kiilshammer

w i t h a r e s u l t o b t a i n e d by t h e a u t h o r [221 , it i s p o s s i b l e t o show t h e f o l l o w i n g : L e t G be a f i n i t e p - s o l v a b l e group, B a b l o c k of FG and ( P , b p ) a maximal B-subpair. Q o f P and a B-subpair

Then t h e r e a r e a normal subgroup

( Q l b ) c o n t a i n e d i n ( P , b p ) and s t a b l e under

Q

NG(P,bp) such t h a t B i s Morita e q u i v a l e n t t o t h e unique block A of FN ( Q , b ) such t h a t 1 1 G

A

Q

bo

#

0. Here t h e s t r u c t u r e of t h e b l o c k A of

FN ( Q l b ) i s c o m p l e t e l y d e s c r i b e d by t h e r e s u l t s o f s e c t i o n 2 0 G

Q

s i n c e t h e block b

Q

h a s d e f e c t group Z ( Q ) and i s t h u s n i l p o t e n t .

2 2 . BLOCKS W I T H CYCLIC DEFECT GROUPS

Blocks w i t h c y c l i c d e f e c t groups have been s t u d i e d i n d e t a i l by E.

C. Dade

[a].

The s t r u c t u r e o f t h e c o r r e s p o n d i n g b l o c k i d e a l has

been determined from h i s r e s u l t s by Janusz i n [ 1 9 ] . There is a c e r t a i n graph, c a l l e d t h e Brauer tree, a t t a c h e d t o each block w i t h c y c l i c d e f e c t group, s u c h t h a t , up t o Morita e q u i v a l e n c e , a b a s i s and t h e c o r r e s p o n d i n g m u l t i p l i c a t i o n t a b l e a r e determined by t h i s graph. We are n o t going i n t o t h e d e t a i l s o f t h e s e r e s u l t s . For o t h e r t y p e s of d e f e c t groups such d e t a i l e d r e s u l t s a r e n o t a v a i l a b l e . Blocks w i t h K l e i n f o u r d e f e c t groups a r e handled by K . Erdmann i n [ l l ] . She h a s r e c e n t l y a l s o o b t a i n e d g e n e r a l r e s u l t s on t h e s t r u c t u r e o f b l o c k s w i t h d i h e d r a l d e f e c t groups.

23. CONNECTIONS WITH REPRESENTATION THEORY

I n t h i s s u r v e y w e have s t r e s s e d t h e r i n g - t h e o r e t i c approach t o t h e s u b j e c t . I t must be p o i n t e d o u t , however, t h a t t h e r e are a t l e a s t two o t h e r approaches, t h e m o d u l e - t h e o r e t i c and t h e c h a r a c t e r - t h e o r e t i c one, b o t h f i r m l y e s t a b l i s h e d by now and u s u a l l y s t r e s s e d i n t e x t b o o k s . T h i s is one o f t h e r e a s o n s why i n t h i s s u r v e y w e have c o n c e n t r a t e d on t h e t h i r d approach. I t s h o u l d be c l e a r t o e v e r y mathematician, however, t h a t i n p r a c t i c e a l l t h r e e approaches have t o be used t o g e t h e r t o o b t a i n t h e b e s t r e s u l t s .

Blocks of modular group algebras

135

This also applies to some of the results presented here. Several of our results require the use of character theory, for example. Some are proved completely within that framework. Where it was possible we have translated them into the language of this paper. I hope that this will shed some new light on these results and

give some new inspiration. Let us outline now how character theory can be applied to the things we have discussed. So let F again be an algebraically closed field of prime characteristic p. Then it is possible to find a complete discrete valuation ring R of characteristic 0 such that F is the residue class field of R. The canonical map R * F induces an epimorphism of rings RG

-

FG. Blocks of RG are

defined analogously to blocks of FG. The epimorphism RG * FG then induces a bijection between blocks of RG and blocks of FG. Many of the results mentioned in this paper have analogues for RG. More importantly, if K denotes the quotient field of R then RG is canonically embedded into KG. Since this is a field of characteristic 0 all the elaborate tools of representation theory are available now.

CONCLUDING REMARKS It is clear that in any finite survey not every important result can be mentioned. We apologize to those authors whose contributions had to omitted and for any misplaced or missing reference.

ACKNOWLEDGEMENTS The author is grateful to the Deutsche Forschungsgemeinschaft for its support and the University of the Witwatersrand for its hospitality during this conference.

136

B. Kulshammer

REFERENCES 1. J. L. ALPERIN and M. B R O a , Local methods i n b l o c k t h e o r y , Ann. Math. 1 1 0 ( 1 9 7 9 ) , 1 4 3 - 1 5 7 . 2. R. BRAUER, Number t h e o r e t i c a l i n v e s t i g a t i o n s on groups of f i n i t e o r d e r , Proc. I n t e r n . Symposium on A l g e b r a i c Number Theory, Tokyo and Nikko, 1 9 5 5 , S c i e n c e Council of J a p a n , Tokyo 1 9 5 6 , 55-62. 3. M. BROW?, On a theorem of G. Robinson, J. London Math. SOC. 2 9 ( 1 9 8 4 ) , 425-434. 4. M. BROU$ and L. PUIG, C h a r a c t e r s and l o c a l s t r u c t u r e i n G-alg e b r a s , J. Algebra 6 3 ( 1 9 8 0 ) , 3 0 6 - 3 1 7 . 5. M. BROUE and L. P U I G , A F r o b e n i u s theorem f o r b l o c k s , I n v e n t . Math. 5 6 ( 1 9 8 0 ) , 1 1 7 - 1 2 8 . 6. D. B. COLEMAN, On t h e modular group r i n g of a p-group, Proc. AMS 1 5 ( 1 9 6 4 ) , 5 1 1 - 5 1 4 . 7. C. W. CURTIS and I . REINER, " R e p r e s e n t a t i o n t h e o r y of f i n i t e groups and a s s o c i a t i v e a l g e b r a s " , Wiley, N e w York 1 9 6 2 . 8 . E. C. DADE, Blocks w i t h c y c l i c d e f e c t g r o u p s , Ann. of Math. 8 4 ( 1 9 6 6 ) , 20-48. 9. E . C. DADE, A correspondence of c h a r a c t e r s , Proc. Symp. P u r e Math. 37, A M S , Providence, 1 9 8 0 . 1 0 . K. E R D M A " , P r i n c i p a l b l o c k s o f groups w i t h d i h e d r a l Sylow 2subgroups, Comm. i n Alg. 5 ( 1 9 7 7 ) , 6 6 5 - 6 9 4 . 1 1 . K. ERDMANN, Blocks whose d e f e c t groups a r e K l e i n f o u r groups: a c o r r e c t i o n , J. Algebra 7 6 ( 1 9 8 2 ) , 5 0 5 - 5 1 8 . 12. W. FEIT, "The r e p r e s e n t a t i o n t h e o r y of f i n i t e groups'' , NorthHolland, Amsterdam, 1 9 8 2 . 1 3 . P. FONG, On t h e c h a r a c t e r s of p - s o l v a b l e g r o u p s , T r a n s . AMS 9 8 ( 1 9 6 1 1 , 263-284. 1 4 . D. GLUCK, On t h e k(GV)-problem, J. Algebra 8 9 ( 1 9 8 4 1 , 46-55. 1 5 . J. A. GREEN, Blocks of modular r e p r e s e n t a t i o n s , Math. 2 . 7 9 ( 1 9 6 2 ) , 100-115. 1 6 . B. HUPPERT, " E n d l i c h e Gruppen I " , S p r i n g e r - V e r l a g , B e r l i n , 1 9 6 7 1 7 . B. HUPPERT and N . BLACKBURN, " F i n i t e g r o u p s I I " , Springer-Verlag, Berlin 1982. 18. K. I I Z U K A and A. WATANABE, On t h e number of b l o c k s of i r r e d u c i b l e c h a r a c t e r s of a f i n i t e group w i t h a g i v e n d e f e c t group, Kumamoto J. S c i . (Math.) 9 ( 1 9 7 3 ) 5 5 - 6 1 . 19. G. J. JANUSZ, Indecomposable modules f o r f i n i t e g r o u p s , Ann. of Math. 8 9 ( 1 9 6 9 ) , 209-241. 20. S. A. JENNINGS, The s t r u c t u r e of t h e group r i n g of a p-group over a modular f i e l d , T r a n s . AMS 5 0 ( 1 9 4 1 1 , 1 7 5 - 1 8 5 . 21. R. KNBRR, On t h e number of c h a r a c t e r s i n a p-block of a ps o l v a b l e group, I l l . J. Math. 2 8 ( 1 9 8 4 ) , 181-210. 22. B. KULSHAMMER, On p-blocks of p - s o l v a b l e groups. Comm. Algebra 9 ( 1 9 8 1 ) I 1763-1785. 23. B. KULSHAMMER, Bemerkungen liber d i e Gruppenalgebra a l s symmet r i s c h e Algebra I - I V , J. Algebra 7 2 _ 1 1 9 8 1 ) , 1-7; 7 5 ( 1 9 8 2 1 , 59-69; 8 8 ( 1 9 8 4 ) , 2 7 9 - 2 9 1 ; 9 3 ( 1 9 8 5 1 , 3 1 0 - 3 2 3 . 24. B. KULSHAMMER, Crossed p r o d u c t s and b l o c k s w i t h normal d e f e c t groups, Comm. Algebra 1 3 ( 1 9 8 5 1 , 147-168. 25. K. MORITA, On group r i n g s o v e r a modular f i e l d which p o s s e s s r a d i c a l s e x p r e s s i b l e as p r i n c i p l e i d e a l s , S c i . Rep. Tokyo Bunrika Daigaku (A) 4 ( 1 9 5 1 ) , 177-194. 26. T. OKWAMA, On f i n i t e group a l g e b r a s w i t h r a d i c a l cube z e r o , Proc. 1 6 t h Symp. on Ring Theory (Tokyo 1 9 6 3 ) , Okayama Math. L e c t . , Okayama Univ., Okayama, 1 9 8 3 , 1 0 5 - 1 1 1 . 27. T. OKUYAMA, On t h e r a d i c a l of t h e c e n t e r of a group r i n g , preprint.

,

Blocks of modular group algebras 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

137

T. OKUYAMA and Y . TSUSHIMA, Local p r o p e r t i e s of p-block a l g e b r a s of f i n i t e g r o u p s , Osaka J. Math. 2 0 ( 1 9 8 3 1 , 33-41. J. OLSSON, On t h e number of c h a r a c t e r s i n b l o c k s o f f i n i t e g e n e r a l l i n e a r , u n i t a r y and symmetric g r o u p s , Math. Z . 1 8 6 ( 1 9 8 4 ) , 41-47. L. P U I G , The s o u r c e a l g e b r a of a n i l p o t e n t b l o c k , p r e p r i n t . L. P U I G , Une c o n j e c t u r e de f i n i t u d e s u r l e s b l o c s , p r e p r i n t . W. F. REYNOLDS, Blocks and normal subgroups of f i n i t e g r o u p s , Nagoya Math. J. 2 2 ( 1 9 6 3 ) , 1 5 - 3 2 . W. F. REYNOLDS, S e c t i o n s and i d e a l s of c e n t e r s of group a l g e b r a s , J. Algebra 2 0 ( 1 9 7 2 1 , 1 7 6 - 1 8 1 . G. R. ROBINSON, The number of b l o c k s w i t h a g i v e n d e f e c t group, J. Algebra 84 ( 1 9 8 3 ) , 4 9 3 - 5 0 2 . Y. TSUSHIMA, R a d i c a l s of group a l g e b r a s , Osaka Math. J. 4 ( 1 9 6 7 ) , 179-182. Y . TSUSHIMA, O n t h e p ' - s e c t i o n sum i n a f i n i t e group r i n g , Math. J. Okayama Univ. 2 0 ( 1 9 7 8 ) , 8 3 - 8 6 . D. A. R. WALLACE, Group a l g e b r a s w i t h r a d i c a l s q u a r e z e r o , Proc. Glasgow Math. Assoc. 5 ( 1 9 6 2 1 , 1 5 8 - 1 5 9 . D. A. R. WALLACE, Lower bounds f o r t h e r a d i c a l of t h e group a l g e b r a of a f i n i t e p - s o l v a b l e g r o u p , Proc. Edinburgh Math. SOC. ( 2 ) 1 6 ( 1 9 6 8 1 , 1 2 7 - 1 3 4 .

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Group and Semigroup Rings G. Karpilovsky (ed.) 0Elsevier Science Publishers B.V.(North-Holland), 1986

139

ON THE PICARD GROUP OF AN ABELIAN GROUP RING David LANTZ Department of Mathematics Colgate University Hamilton, New York 13346 United States of America

1. INTRODUCTION Let

A

be a ring (commutative and with unity, as are all the

rings in this paper).

The purpose of this paper is to survey

results relating the Picard group of a ring ring

A[G]

that, if

for G

G

A

to that of the

a free abelian monoid or group.

Swan has shown

is a free abelian monoid, then the factor group A/(nil A)

is a

Pic(A[G])/Pic(A)

vanishes if and only if

seminormal ring.

(The definition of seminormality appears below.)

Bass and Murthy have used an early version of this €act and the fundamental Mayer-Vietoris sequence oE algebraic K-theory to compute this factor group under rather restrictive hypotheses.

We

will review these results and note some recent attempts to weaken or remove Bass and Murthy's hypotheses for more general results. We will also note that the state of knowledge is much less satisfactory in the case that

G

is a free abelian group.

case, for instance, the property that may be true of A

.

A

Pic(A[G] )/Pic(A)

In this

is trivial

and fail to be true for a ring of quotients of

(The example is due to Pedrini.)

Though the motivation for

many of these results comes from algebraic geometry, in order to minimize prerequisites this presentation will be phrased in terms

of rings rather than schemes.

The reader interested in a

scheme-theoretic interpretation is referred to such papers as [GT] and [LV]

2.

.

PICARD GROUPS

140

D. Lantz

The isomorphism classes of A-modules form a monoid under the

[PI + 191 = [ P a Q]

operation A-module

;

the isomorphism class of the

itself is the identity for this monoid.

A

The classes

having an inverse in this semigroup are precisely those of the rank

is a rank one projective module,

P

one projective modules; if

then the isomorphism class of the module for the isomorphism class of

P

.

is an inverse

HomA(P,A)

The "Picard group" of the ring

A is defined to be the subgroup of this monoid consisting of these invertible classes: it is denoted ring

of

K(A)

example, if Pic A

A

Pic A

.

If the total quotient

has only finitely many maximal ideals (for

is an integral domain or a Noetherian ring), then

A

can also be described as follows:

A-submodules of

K(A)

Consider the monoid of

under the operation of multiplication; i.e.,

given two such submodules

I and

,

J

the product

is the

IJ

A-submodule consisting of the set of finite sums of products of the form

xy

as

A-submodule

x A

and

y

vary over

I

and

J

respectively.

The

The set of

is itself the identity in this monoid.

invertible elements of this monoid form a group, and the set of principal invertible elements (i-e., the set of all such submodules each of which is generated by a single nonzerodivisor) form a subgroup.

The factor group of the invertible submodules by the

principal invertible submodules is isomorphic to

Pic A

.

This

second interpretation of the Picard group is more elementary, but it follows more immediately from the first that

Pic

is a functor

from the category of commutative rings with unity to the category oE abelian groups:

to another, to

Pic B

B

,

Given a ring homomorphism from one ring, A

the corresponding group homomorphism from

Pic A

is given by associating to the isomorphism class of the

rank one projective A-module Bemodule

,

B

.

P

the isomorphism class of the

(A general reference for the Picard group is

[Bo], Chapter 11, section 5.7.)

On the Pica& group

141

It is an important fact that the functor direct limits. of

We outline a proof:

Pic

commutes with

An element of the Picard group

is the isomorphism class of a finitely generated projective

A

module over

,

A

that is, the image of an idempotent endomorphism

of a finitely generated free A-module; and this endomorphism is in turn represented by an idempotent matrix with entries from columns of which generate the projective module.

,

A

the

This module has

rank one if and only if its second and higher exterior powers Thus, any element of

vanish.

finitely many elements of

A

Pic A

,

is described in terms of

namely the entries in the matrix and

the elements appearing in the equations showing that the generators of the exterior powers vanish.

It follows that, if

limit of the directed system of rings its Picard group does appear in some

[Ai] Pic Ai

that a rank one projective module over some cyclic when tensored with

some other Pic A

Ai

A

,

A

then each element of

. Ai

It is easy to see that becsmes

is already cyclic when tensored with

further along in the system.

So we conclude that

!Pic Ail.

is the limit of the directed system of groups

(The result that

Pic

is the

commutes with direct limits is Theorem 1.3

of [GHJ. I We denote by nil A

.

Ared

the factor ring of

A

by its nilradical

Since idempotents can be lifted modulo a nil ideal in a

noncommutative ring [Lm, page 72, Proposition 11

--

for instance,

the ideal of matrices with nilpotent entries in a ring of square matrices over Pic Ared

A

--

the matrices representing the elements of

can be lifted to idempotent matrices over

A

this fact i t can be shown that the Picard groups of

A

.

Using

and

Ared

are isomorphic. 3 . THE PICARD GROUP OF A MONOID RING OR GROUP RING

Let

G

group ring

be a multiplicative abelian monoid. A[G]

The monoid ring or

is the free A-module with basis the set G

and

142

0. Lantz

m u l t i p l i c a t i o n d e f i n e d by e x t e n d i n g t h e m u l t i p l i c a t i o n s of and assuming e l e m e n t s o f

G

let

G

and

.

Now

d e n o t e a f i n i t e l y g e n e r a t e d f r e e a b e l i a n monoid o r g r o u p ,

G

and d e n o t e by A

commute w i t h e l e m e n t s of

A

A

: t h a t is,

basis for

A[G]

,

G

(Here,

homomorphism from unity i n

.

A

is a

T

i s a f r e e a b e l i a n monoid o c a Eree

i s a r e t r a c t of

A

to

A[G]

because t h e

A[G],

sending a l l elements of

A

Thus, t h e f a c t t h a t

complementary summand

, where

A[T,T-l]

d e n o t e s t h e s e t of i n v e r s e s o f

T-'

i s a d i r e c t summand o f

Pic A

or

A[G] is a r i g h t i n v e r s e f o r t h e r i n g

into

A

A[T] G

The r i n g

T .)

i n c l u s i o n of

is e i t h e r

according as

a b e l i a n group. e l e m e n t s of

t h e a s s o c i a t e d monoid r i n g o r g r o u p r i n g o v e r

A[G]

Pic(A[G] ) / P i c ( A )

to the

is a f u n c t o r shows t h a t

Pic

Pic A[G]

G

. by

W e denote t h e NPic(A,G)

.

In t h e i r f o u n d i n g p a p e r [EM], Bass and Murthy d e s c r i b e d NPic(A,G)

in the case that

is a N o e t h e r i a n r e d u c e d r i n g of

A

K r u l l d i m e n s i o n o n e o f which t h e i n t e g r a l c l o s u r e g e n e r a t e d as a module o v e r

A

.

AC

is f i n i t e l y

T h e i r o b j e c t i v e was t o d e s c r i b e

t h e Picard group of t h e group r i n g o v e r t h e r a t i o n a l i n t e g e r s of a f i n i t e l y g e n e r a t e d a b e l i a n g r o u p . i n t h e form

GOxG where

Go

r i n g " i n t h i s case,

ZIGo]

Such a g r o u p c a n b e w r i t t e n

is f i n i t e and

t h e i n t e g r a l g r o u p r i n g h a s t h e form

, does

2

G

ZIGOl[GI

i s f r e e , so t h a t

.

The " c o e f f i c i e n t

s a t i s f y a l l t h e h y p o t h e s e s on

A

m e n t i o n e d a b o v e , so t h i s r e s u l t was s u f f i c i e n t f o r t h e i r p u r p o s e s . B u t f o r more g e n e r a l a p p l i c a t i o n s i t i s f o r t u n a t e t h a t some o f

t h e s e hypotheses can be relaxed:

The f a c t t h a t

Pic

commutes w i t h

d i r e c t l i m i t s allows us, a t l e a s t i n some cases, t o remove t h e hypotheses t h a t

for,

A

A

i s N o e t h e r i a n and h a s f i n i t e i n t e g r a l c l o s u r e ;

is t h e d i r e c t l i m i t o f i t s f i n i t e l y g e n e r a t e d s u b r i n g s ,

and t h e s e s u b r i n g s , a s a l g e b r a s of f i n i t e t y p e o v e r t h e i n t e g e r s Z

, are

N o e t h e r i a n and have f i n i t e i n t e g r a l c l o s u r e s [Ma, page 2 4 0 ,

Theorem 721.

The h y p o t h e s i s t h a t

A

is r e d u c e d c a n o f t e n b e

On the Picard group

removed by using the fact that requirement that

A

143

.

Pic A = Pic Ared

But the

have (Krull) dimension one is very

restrictive, and there are few results that avoid it.

It would be

of great interest to find a way to avoid this hypothesis.

4.

SEMINORMALITY For the case of a monoid ring over a ring

for the case of a polynomial ring when the complementary summand

,

A[T]

NPic(A,G)

hard to see that, for the coordinate ring (over a field),

.

Pic A and

Y

Pic A[T]

A

and let

2

(A[Zl/(Z -aIz3-b))red

Then

B

by

a

a

includes

and

element of

A

,

and

is an inverse for

.

vanishes. A

It is not

of the cusp

X3 = Y2

denote the images of

b

and by

c

of

Pic A[T]

(It is easy to check that

aA[T] + (l+cT)A[T]

Denote by

the image of

and the A[T]-submodule

l+cT gives an element of Pic A

it is now known precisely

be an indeterminate.

2

ring

or in other words

has elements not represented in

In particular, let in

,

A

2

X

B

in

the

.

B

B[Tl generated that is not an

aA[T] + (l-cT)A[T]

.)

In [Tr] , Traverso related this phenomenon to the behavior of the spectrum under passage to the integral closure, at least for irreducible curves: He defined the "seminormalization" of

A

the smallest extension of

AC

A

within its integral closure

to be

having the property that, in every properly larger extension within AC

, some

prime ideal experiences either proper extension of its

residue field or splitting. trivial if and only if the cusp

X3 = Y2

,

A

He then showed that NPic(A,G) is is equal to its seminormalization.

For

of course, the prime ideal corresponding to the

origin does not split in the integral closure, nor does its residue Eield grow larger; the integral closure is the seminormalization. (Later work on the geometric meaning of seminormality is summarized in [Da] . I

0.Lantz

144

For more general rings, however, the vanishing of

NPic(A,G)

seems to be more easily described in terms of the equation defining the cusp than in terms of ,the spectrum.

In [Hal, Hamann gave a a3

definition of seminormality based on the equation

=

b2

and

showed that, for pseudogeometric rings, it was equivalent to the vanishing of

.

NPic

An example in [GH] showed that Hamann's

result does not hold Eor general rings, but Swan [Sw] refined Hamann's definition to provide the desired characterization: Pic A[T]

=

Pic A

if and only, for any elements

a

and

b

of

Ared satisfying a3 = b2 , there is an element c of Ared for which c2 = a and c3 = b The necessity of this condition is

.

easily seen by mimicking the argument above for the cusp.

5.

THE FUNDAMENTAL SEQUENCE As mentioned above, Bass and Murthy computed

case of

A

NPic(A,G)

a Noetherian reduced ring of Krull dimension one, with

integral closure

AC

had proved that

a finitely generated A-module.

NPic(AC,G)

.

Since they

trivial (the first version of

s

Swan's result), they needed a method of relating NPic(AC,G)

in the

NPic(A,G)

to

They found that method in a special case of the

Mayer-Vietoris sequence of algebraic K-theory [Ba, Chapter IX, (5.311, and several authors since have also found their sequence

very useful.

To describe this sequence, we will use the following notation: The multiplicative group of units (invertible elements) in the ring

A

will be denoted

U(A)

.

Then

U

is also a functor from

the category of commutative rings with unity to the category of abelian groups, so that, as in the case of direct summand of by

NU(A,G)

of elements

.

b

U(A[G] A

,

U(A)

is also a

: we denote the complementary summand

Next, the conductor of in

Pic

for which

bAC

A

in

Ac

(that is, the s e t

is included in

A

)

is an

On the Picard group

ideal of both Ac

A

and

.

AC

145

We denote the factor rings of

with respect to this ideal by

B

and

B'

A

respectively.

and The

sequence used by Bass and Murthy now can be stated as follows:

NU(A,G) -> NU(A~,G)XNU(B,G) -> NU(B',G)

Let us suppose that

is a free abelian monoid.

G

a group is considered in the next section.)

(The case of

Then the first terms

in this sequence admit an explicit representation: The ring homomorphism from of

G

A[G]

to

to the unity in

the form

t-1

as

t

A

defined by sending every element

A

has kernel generated by the elements of

varies over a basis

of this kernel with the nilradical of the set of sums isomorphic to

l+f

t

where

NU(A,G)

1 + (T-l)(nil A)[T] as

,

.

, where

T-1

A

and hence

T

AC

.

G

.

The product

is a nil ideal of

AIG];

varies over this nil ideal., is

In symbols,

varies over a basis

hypotheses,

f

A

T of

NU(A,G) =

means the set of elements

t-1

Under Bass and Murthy's

are reduced, so that

NU(A,G)

and

NU(A~,G) are trivial. Next we turn OUK attention to the other end of the sequence. The conductor of because AC

A

in

AC

does not consist only of zerodivisors,

is a finite A-module; since the dimension of

one, this means that

B

and

B'

have dimension zero.

also Noetherian, so they are Artinian; so finite products of fields.

It follows that

NPic(B',G)

are trivial.

NPic(AC,G)

is also trivial.

shows that

Bred

And, since

AC

and

NPic(B,G)

A

is

They are Blred

are

and

is seminormal,

Thus, the Mayer-Vietoris sequence

146

D. Lantz

+ (T-l)(nil 1 + (T-l)((nil

NPic(A,G) = (1 =

The o p e r a t i o n o n t h e g r o u p t h e m u l t i p l i c a t i o n on

A[T]

though t h e d e s c r i p t i o n o f e l e m e n t s , t h e set right.

1

, not

B ' ) [ T ] ) / (1 + ( T - l l ( n i . 1 B ) [ T I ) B')/(nil B))[Tl

+

( T - l ) ( n i l A)[T]

i s i n d u c e d by

by t h e a d d i t i o n o n

NPic(A,G)

( n i l B ' ) / ( n i l B)

: so,

A

is c o r r e c t a s a set o f

h a s n o o p e r a t i o n i n i t s own

The r e a d e r c a n e x p l o r e t h i s s t a t e m e n t i n t h e f o l l o w i n g

(almost g e n e r i c ) e x a m p l e , d r a w n f r o m t h e d o c t o r a l t h e s i s of M. B. M a r t i n , t h e U n i v e r s i t y o f N o r t h C a r o l i n a , 1984:

a positive integer

,

n

A = k[[an,an+',

subring k[[all

a

in

over

k

a

and a n i n d e t e r m i n a t e

...I ]

.

Take a f i e l d

, and

The i n t e g r a l c l o s u r e

AC

of

an+',

... .

is t h e

A

in

A

( A s an i d e a l i n

is n e e d e d ) . T h u s , B = k , a n d .n-l : a n d so space w i t h b a s i s 1, a , a 2 ,

o n l y t h e generator k-vector

a",

consider the

o f t h e f o r m a l power s e r i e s r i n g

e n t i r e f o r m a l power s e r i e s r i n g , a n d t h e c o n d u c t o r of

i s t h e i d e a l g e n e r a t e d by

,

k

a"

AC AC

,

is t h e

B'

...,

( n i l B ' ) / ( n i l B ) is t h e k - v e c t o r a, a2, a n-1 ,

space with generators

...,

6 . THE GROUP R I N G CASE

I n t h i s s e c t i o n we e x p l o r e t h e d i f f e r e n t c i r c u m s t a n c e s e n c o u n t e r e d b y Bass a n d Murthy i n a p p l y i n g t h e i r method t o t h e case where

G

is a f r e e a b e l i a n g r o u p ; w e a s s u m e t h r o u g h o u t t h a t

such a group.

L e t u s begin with t h e r i g h t end of t h e

Mayer-Vietoris

sequence.

integrally closed, then h y p o t h e s e s we g e t

Bass a n d M u r t h y showed t h a t i f

NPic(A,G)

NPic(AC,G) = N P i c ( B ' , G ) = N P i c ( B , G ) = 0

vanishes, b u t f i r s t l e t u s c o n s i d e r t h e l e f t end o f t h e sequence.

is

A

i s t r i v i a l ; so u n d e r t h e i r

s h a l l s a y mqre b e l o w a b o u t t h e c o n d i t i o n s u n d e r w h i c h

Mayer-Vietoris

G

.

W e

NPic(A,G)

is

On the Picard group

When

G

is a group, the summand

longer all of not in

I el, in

A

A

1 + (T-l)(nil A)[T,T-']

NU(A,G) : The new units in the group ring elgl +

are sums of the form

... , en and

147

... + engn

is no k[Gl

where

is a complete family of orthogonal idempotents

1

... , gn

gl,

are elements of

determine a partition of the spectrum of

G A

.

The idempotents

into finitely many A) [TIT-']

NU(A,G)/(l+(T-l)(nil

clopen sets; so in this case

)

may

be identified with the set of continuous functions from the spectrum of Since

A

A

into the group

is Noetherian and

G

AC

with the discrete topology. is a finite module, the spectrum

of each can be partitioned into a finite collection of connected components, so that the corresponding group of continuous functions is isomorphic to a finite product of copies of by

h(A)

ring group to

Then, with the hypothesis of reduced rings, the factor

NU(A~,G)/NU(A,G)

is a free abelian group with rank equal

.

(rank G)x(h(Ac)-h(A))

Similarly, the factor group summand

Let us denote

the number of connected components in the spectrum of the

.

A

.

G

1 + (T-l)((nil B')/(nil

NU B',G)/NU(B,G)

has, besides the

as in the monoid case,

B

a summand measuring the difference of the number of components in the spectra of

B'

and

B

1 + (T-l)((nil B')/(nil

of rank

.

Thus,

NPic(A,G)

and a free abelian group of

B))[T,T-l]

(rank G)(h(B')-h(B)-h(AC)+h(A))

condition for the vanishing of

.

NPic(A,G)

finite integral closure, is the equality h(A) = 0

is the direct sum

In particular, a necessary

,

for

h(B')

A

-

Noetherian with h(B) - h(Ac) +

*

The general problem of conditions under which vanishes, for

G

NPic(A,G)

a free abelian group, remains unsolved however.

Bass and Murthy introduced the term "quasinormality' in this context; it is not clear precisely which conditions they meant to describe by this term, and other authors have used it both in this

148

0. Lantz

context and several others, each with a slightly different meaning. A

Here we will call a ring

is reduced and

group.

NPic(A,G)

A

quasinormal if and only if

vanishes for

a free abelian

G

Using a result of Horrocks [Ho], Bass and Murthy showed

that a quasinormal ring was seminormal [BM, Corollary 6.41.

In

fact, quasinormality is strictly stronger than seminormality and One way to see this is

strictly weaker than integral closedness.

to note that both seminormality and integral closedness are preserved under passage to a ring of quotients; but an example due

to Pedrini [Pd, page 501 shows that a ring of quotients of a The example is obtained

quasinormal ring need not be quasinormal.

by "gluing" the ideals generated by the indeterminates in the polynomial ring A

be the subring of

f(X,Y)

for which

k[X,Yl

.

and

.

X2Y

A

in

AC

This ideal is prime in

ideals of

B = A/P A

and

and AC

B' = AC/P

P

component in its spectrum (indeed, B'

while the new and

AC

That is, let

has two.

A X

P

,

is

A

generated in

but in and

AC

.

Y

,

k[X,Yl A

i t is

The

are connected by the maximal P : but if these rings are

that contain

localized at the complement of

of

is the ideal

Ac

the intersection of the primes generated by spectra of

.

k

Y

consisting of those polynomials

The integral closure

and the conductor of XY

over the Eield

and

f(X,O) = f(0,X) ; or equivalently

A = k[X+Y,XY,X2Yl

by

k[X,Y]

X

in B

A

,

the new

has only one

h

has only one prime ideal),

Since the spectra of the domains

A

are connected (before or after localizing), we conclude

that, before localizing:

h(B')

-

h(B)

and in fact A

-

h(Ac)

+

h(A) = 1

-1-

1 + 1 = 0

is quasinormal; but after localizing:

I

On the Picard group

-

h(B')

h(B)

-

-

h(Ac) + h(A) = 2

so that the localized

A

149

-

1

1 + 1 = 1

I

is not quasinormal.

It would be useful to have a elementary condition for quasinormality similar to the "2,3-closed" condition €or seminormality described above.

One candidate €or such a condition

was suggested by Asunuma: A ring only if, for every element that

b2-b

and

b3-b2

A

is called "u-closed" if and

of the integral closure of

b

are in

,

A

b

is itself in

A

.

A

such

An

article by Onoda and Yoshida [OYI shows that a one-dimensional domain is quasinormal if and only if it is seminormal and u-closed.

(As with 2,3-closedness and seminormality above, it is

not hard to see the necessity of u-closedness €or quasinormality: If there were such an element AIT,T-l]-submodule of 2

b -b

in

h

ACIT,T-l]

has inverse generated by

isomorphism class is in

AC

but not in

generated by (b-1)T

Pic A[T,T-l]

,

bT

-

,

A

then the

(b-1)

and

,

so its

but i t is not in

Pic A

b

and

b3-b2

.)

As mentioned above, it would be desirable to remove the hypothesis of one-dimensionality, and there seems to be a reasonable chance of doing s 3 in this result.

To remove the hypothesis of domain,

however, it seems likely that it will be necessary to recast the definition of u-closedness in the spirit of Swan's recasting of Hamann's 2,3-closedness: only if, for any elements a3 + ab = b2 and

,

c3-c2 = b

A reduced ring

A

is "u-closed" if and

of

A

for which

a

and

there is an element

b c

of

A

for which

c2-c = a

.

7. OTHER RESULTS We close with a quick review of related results from the literature. Most closely related to the results described above are the

150

D. Lantz

papers of Greco [Gr] and Rush [Rul, in which it is shown that, for a ring

A

and a finite abelian group

seminormal,

m

ideals in both

G

is a nonzerodivisor in

A

and

AC

, and

divisors, then the group ring

mA

A[Gl

of order A

m

,

if

A

is

generating radical

has no embedded prime is also seminormal.

(Greco

asserts the converse, but Rush points out that only a partial converse holds.) quasinormality.

Both papers also contain results on In particular, Greco comments on the passage of

quasinormality to the group ring of a finite abelian group. In several papers, of which [An] is the most recent, David F. Anderson has investigated projective modules over subrings of polynomial rings over fields.

This study has developed into

investigation of seminormal graded domains and the conjecture that, if

A

is a quasinormal domain, then

torsion-free semigroup

G

NPic(A,G)

vanishes for every

.

A condition slightly weaker than seminormality and also suggested by algebraic geometry is weak normality.

In [Yal,

Yanagihara extends some of the results on seminormality to weak normality, using a result of Itoh identifying weak normality as 2,3-closedness identifies scminormality.

He develops a "weak

gluing" of a ring corresponding to Pedrini's methods in the case of seminormality. Ramella's interpretation of one-dimensional quasinormal rings in [Ra] is geometric not in the sense of algebraic geometry but sense of graph theory.

n the

She does, however, provide results on etale

coverings and examples which are complex curves. Finally, we may regard the passage from

A[T]

to

adjunction of the inverse of a single nonzerodivisor.

A[T,T-l

as

In [Ill,

Ischebeck investigates the behavior of the resulting homomorphism of Picard groups under any adjunction of an inverse of a prime element.

In

[I21, he extends Swan's work on the kernel of the map

On the Picard group

NPic A,G) -> NPic(Ac,G) from the ring extension

,

for

G

A -> AC

151

a free abelian monoid, arising to more general ring extensions

A -> B .

REFERENCES [An] Anderson, D. F., Seminormal graded domains 11, J. Pure Appl. Alg. 23 (19821, 221-226. Ba 1 Bass, H., Algebraic K-Theory (Benjamin, New York, 1968). BM 1 Bass, H., and P. Murthy, Grothendieck groups and Picard groups of abelian group rings, Ann. of Math. 86 (19671, 16-73. Bo 1 Bourbaki I N., Commutative Algebra ( Addison-Wesley, Reading, Massachusetts, 19691. Da 1 Davis, E., On the geometric interpretation of seminormality, Proc. h e r . Math. SOC. 68 (19781, 1-5. Gilmer, R., and R. Heitmann, On Pic(R[X]) for R seminormal, J. Pure Appl. Alg. 16 (19801, 251-257. Greco, S., Seminormality and quasinormality of group rings, J. Pure Appl. Alg. 18 (19801, 129-142. Greco, S., and C. Traverso, On seminormal schemes, Comp. Math. 40 (19801, 325-365. Ha 1 Hamann, E. , The R-invariance oE R[X], J. Algebra 35 (19751, 1-16. Ho 1 Horrocks, G., Projective modules over an extended local ring, Proc. London Math. SOC. 14 (19641, 714-718. 11 1 Ischebeck, F., Uber die Abbildung Pic A -> Pic Af, Math. Ann. 243 (19791, 237-245. I21 ------------- , On the Picard group of polynomial rings, J. Algebra 88 (19841, 395-404. Lml Lambek, J., Lectures on Rings and Modules (Blaisdell, Waltham, Massachusetts, 1966). LVI Leahy, J., and M. Vitulli, Seminormal rings and weakly normal varieties, Nagoya Math. J. 82 (19811, 27-56. [Ma] Matsumura, H., Commutative Algebra, second edition (Benjamin/Cummings, Reading, Massachusetts, 19801. [Pdl Pedrini, C., Incollamenti de ideali primi e gruppi di picard, Rend. Sem. Mat. Univ. Padova 48 (19731, 39-66. [Ral Ramella, L. , A geometric interpretation of one-dimensional quasinormal rings, J. Pure Appl. Alg. 35 (19851, 77-83. [Rul Rush, D., Picard groups in abelian group rings, J. Pure Appl. Alg. 26 (19821, 101-114. [ SWI Swan, R., On seminormality, J. Algebra 67 (19801, 210-229. [Trl Traverso, C., Seminormality and Picard group, Ann. Sc. Norm. Sup. Pisa XXXIV (IV) (19701, 585-595. [Yal Yanagihara, H., Some results on weakly normal ring extensions, J. Math. SOC. Japan 35 (19831, 649-661.

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Cmup and Semigroup Rings G.Karplovsky (ed.) 0 Elsevier Science Publishers B.V. (North-Holland), 1986

LIE METABELIAN GROUP

153

RINGS

F. L E V I N (Bochum) and G. ROSENBERGER (Dortmund)

Abstract We c h a r a c t e r i z e t h e nonabelian groups w i t h L i e metabelian group r i n g s . Moreover, such group r i n g s are shown t o be s t r o n g l y L i e metabelian and s t r o n g l y L i e n i l p o t e n t o f c l a s s a t most 3.

1. For any a s s o c i a t i v e rincj R we d e f i n e (x,y)

= x y - y x t o be t h e L i e

product o f x,y E R. S[21(R) t o be the a d d i t i v e group o f a l l L i e products and S(')(R)

i n R. R i s L i e solvable i f

t o be the i d e a l generated by S['](R)

6[n1(R) = 0 f o r some n, where 6[n1(R) i s defined i n d u c t i v e l y by (S[n-ll(R),S[n-ll(R))

= S[nl(R),

and s t r o n g l y L i e solvable i f S(')(R)

= 0 for

some n, where S('))(R) i s t h e i d e a l generated by (S(n-l)(R),S(n-l)(R)).

In this

note we c h a r a c t e r i z e those groups G which have a L i e metabelian group r i n g RG, t h a t i s , S["(RG)

= 0. For a general discussion o f L i e solvable group r i n g s

the reader i s r e f e r r e d t o Chapter 5 o f the book by Sehgal [31. I f R has a u n i t element 1 E R, as w i l l be assumed f o r a l l r i n g s i n the sequel, then strong L i e s o l v a b i l i t y c l e a r l y implies L i e s o l v a b i l i t y o f R. The reverse i s not t r u e i n general. For instance, as observed i n [31, i f K i s a f i e l d o f c h a r a c t e r i s t i c p # 0, then KS3 i s L i e solvable f o r a l l primes, s t r o n g l y L i e solvable f o r odd primes b u t n o t s t r o n g l y L i e solvable f o r p = 2. On the other hand, as we show i n t h i s note, a group r i n g o f a nonabelian group i s L i e metabelian if and o n l y i f i t i s s t r o n g l y L i e metabelian and, moreover, the group o f u n i t s of the group r i n g must be n i l p o t e n t o f class 5 3. I n a d d i t i o n , t h e group r i n g w i l l be L i e n i l p o t e n t o f class ( a t most) 3, t h a t i s , (rl,r2,r3,r4) w i l l be t r i v i a l f o r a l l group r i n g elements, where we d e f i n e i n d u c t i v e l y (rly.

..,rn)

=

((rly..

.,~-,-~),r~).

2. For the remainder o f t h i s note G w i l l denote a group and R an a s s o c i a t i v e r i n g w i t h u n i t element 1 E R. Lemma 1. ( a ) The f o l l o w i n g i d e n t i t i e s hold i n any a s s o c i a t i v e r i n g .

(i) (a,bc) = (a,b)c t b(a,c). (ii)

(a,b)(c,d)

t

(a,d)(c,b)

= (ac,b,d)

-

(a,b,d)c

-

a(c,b,d).

( b ) The f o l l o w i n g i d e n t i t i e s hold f o r the group o f u n i t s o f an associative ring.

154

F. Levin and G. Rosenberger

( i i i ) [a,bl

-

1 = a -1b -1(a,b) = (a-',b-')ab,

1 Ib,alc -1(a -1b -1(a,b,c) t (a-'b-',c)(a,b)), where [a,bl = a -1b -1ab and [a,b,cl = ~ [ a , b l , c l . (iv)

-

[a,b,cl

The p r o o f s f o r Lemma 1 a r e r o u t i n e and a r e l e f t t o t h e reader. Lemma 2 . The f o l l o w i n g i d e n t i t i e s h o l d i n any a s s o c i a t i v e L i e metabelian r i n g

R and any elements, a,b E R such t h a t (a,b) ( i ) (r,s,a)t(b,u) = 0 f o r any r,s,t,u E R. If, i n a d d i t i o n , ( r , s )

t,u E R. ( i i ) (a,t)(b,r)(s,u) ( i i i ) (a,r)(b,s)(t,u)

=

= 0.

= 0, then t h e f o l l o w i n g i d e n t i t i e s h o l d f o r any

0 0

-

= (r,s,a)(b,tu) (r,s,a)(b,t)u. Proofs. ( 1 ) BY Lemma l ( i ) , (r,s,a)t(b,u) i t s u f f i c e s t o show t h a t (r,s,a)(b,t) = 0. By Lemna l ( i i ) , (r,s,a)(b,t)

(at,b,(r,s)) metabel i a n .

-

a(t,b,(r,s)),

Hence, =

which i s t r i v i a l since, by hypothesis, R i s L i e

( i i ) BY Lemma l ( i i ) , (a,t)(b,r)(s,u) i s t r i v i a l by ( i ) .

= -(a,t)(ru,s,b)

t

(a,t)r(u,s,b),

-

(a,r)(bt,u,s) (a,r)b(t,u,s) ( i i i ) By Lemma l ( i i ) , (a,r)(b,s)(t,u) (a,r)(b,u)(t,s), which i s t r i v i a l by ( i ) and ( i i ) . (a,r)(b,u,s)t

-

which

-

3. I n t h i s paragraph we apply t h e preceding lemmata t o L i e metabelian group

r i n g s . Our f i r s t r e s u l t l i m i t s the n i l p o t e n c y o f t h e group o f u n i t s o f the group r i n g and t h e c h a r a c t e r i s t i c o f t h e r i n g . Theorem 1. L e t G be a group w i t h a L i e metabelian group r i n g RG o v e r some a s s o c i a t i v e r i n g R w i t h u n i t element. (i)

= 1 f o r a l l a,b,c i n G. ( i i ) I f R has c h a r a c t e r i s t i c 0 o r g r e a t e r than 4, t h e n G i s a b e l i a n .

G i s n i l p o t e n t o f c l a s s ( a t most) 2 , t h a t i s , [a,b,cl

Proofs. ( i ) By Lemma 2 ( i ) , (a,c,b)d(a,b)

= (c,b,a)d(a,b)

= 0 f o r any elements

o f RG and, hence, (a,b,c)d(a,b) = 0. Thus, by Lemma l ( i v ) , f o r any a,b,c i n G, ([a,b,cl-l)([a,bj-1) = ( [ a , b , ~ ] - l ) a - ~ b - ~ ( a , b ) = ( b,a)c -1(a -1b -1,c)(a,b)a-'b-l(a,b),

and t h i s , by Lemmas l ( i i i ) , l ( i ) and 2 ( i i ) i s t r i v i a l . Thus,

([a,b,c]-l)([a,b]-1)

-

= 0 f o r any a,b,c

i n G. I n p a r t i c u l a r ,

-

[a,b,c]-[a,b] [a,b,c] [a,b] t 1 = 0 so i f [a,b] # 1 then e i t h e r [a,b,c] = 1 o r R has c h a r a c t e r i s t i c 2 and [a,b,c][a,b] = 1, whence i t follows t h a t [a,b][a,b,c] = [a,bIc = 1. Thus, i n any event, [a,b,c] n i l p o t e n t of c l a s s 2 .

= 1, and G i s

Lie metabelian group rings

( i i ) By Lemma a-'b-'(

155

l(iii), f o r any a, b E G, ([a,bl-1) 3 = a, b)a-'b-l( a, b) , and i t f o l lows from L e n a 2 (i ) that

a ,b)a-'b-'( 3 = (a-1b-1)3(a.b)3, which, by Lemna 2 ( i i ) i s t r i v i a l . Thus, ([a,bl-1) 2 3[a,bl ([a,bl-1)3 = [a,b13 t 3[a,bl 1 = 0, so i f R has c h a r a c t e r i s t i c e i t h e r 0 o r greater than 4, then [a,bl 2 = [a,bl and [a,bl = 1.

-

-

Before l i s t i n g other p r o p e r t i e s o f G we record the f o l l o w i n g lemma f o r l a t e r use. Lemna 3. L e t G be n i l p o t e n t o f c l a s s 2. For any ai,bi (i)

(al,bl)(a2.b2)

...(a,,bk)

E G,

=

.

blalb2a2. ..bkak([a,,bl]-l)([a,,b21-1). . ( [ a k , b k l - l ) , k 2 1. I n p a r t i c u l a r , (alybl) ...(ak,bk) = 0 i f and only if ([a,,b,I-l) ...( [ a k , b k l - l ) ( i i ) f o r any k 2 2, (a1,...,ak) ci = ai-l...a2al.

= 0;

Proof. ( i ) By Lemna l ( i i i ) , (alybl) bkak( [ak,bkl-1),

=

c

k

X ([ci,ail-l),

k+l i=2

...(ak,bk)

where, f o r each i,

= blal([al,bll-l)

...

and the r e s u l t f o l l o w s since, by hypothesis, each [ai,bi

I is

central. ( i i ) I n d u c t i o n on k . For k = 2 the r e s u l t f o l l o w s from Lemna l ( i i i ) . Thus, are c e n t r a l , l e t k > 1. Since a l l [ci,ail k-1 (al, ak) = (al,,..,ak-l,ak) (ck,ak) n ( [ C i a i -11 i=2 k-1 k -1) 9 = akck( [ck.ajl -1) in=2 ([ci,ail-l) = ' k t l 2:i (['i 'ai

...,

which completes t h e induction. Theorem 2. I f R has c h a r a c t e r i s t i c 3, then e i t h e r G i s abelian o r the derived group, G ' , o f G i s c y c l i c o f order 3. Proof. I f R has c h a r a c t e r i s t i c 3, then t h e proof o f Theorem 2 ( i i ) shows t h a t = 1 f o r any a,b i n G. Also, by Lemma 2 ( i i ) , f o r any c,d i n G, 2 2 (a,b) (c,d) = 0 and, hence, by Lemna 3 ( i ) , ([a,bl-1) ( [ c , d l - 1 ) = 0. It now f o l l o w s d i r e c t l y by comparing terms t h a t i f [a,bl # 1, then [c,dl i s dependent

on [a,b],

t h a t i s , G ' i s c y c l i c o f order 3 generated by [a,bl.

Before s t a t i n g the r e s u l t s f o r the c h a r a c t e r i s t i c 2 case, we need the f o l l o w i n g observations. Lemma 4. Let RG be L i e metabelian and R have c h a r a c t e r i s t i c 2 (i)

For any elements a,b i n G, (a,b,a)

= 0.

F. Levin and G. Rosenberger

156

( i i ) For any elements a,...,e = 0. ( i i i ) (a,b,c)(a,d)

i n G, (a,b,c)(d,e)t

(a,b,e)(c,d)

= 0.

Proofs. ( i ) By Theorem 1, G i s n i l p o t e n t o f c l a s s 2 and by Lemmas 2 ( i i ) and 3 3 2 3(i), ([a,bl-l) =[a,b] t [a,bl t [a,bl t 1 =O,since R has c h a r a c t e r i s t i c 2. I t follows t h a t i n any event [a,bI2 = 1, and s i n c e G i s n i l p o t e n t o f c l a s s 2, 2 2 t h a t [ a ,b] = 1. Thus, (a ,b) = 0, and by Lemma l ( i ) , a(a,b) t (a,b)a = 2a(a,b) t (a,b,a) = (a,b,a) = 0, s i n c e R has c h a r a c t e r i s t i c 2 . (ii) By Lemma l ( i i ) , (a,b,c)(d,e) t (a,b,e)(c,d) = (ce,d,(a,b)) -c(e,d,(a,b)) (c,d,(a,b))e, which i s t r i v i a l s i n c e RG i s L i e

-

metabelian. ( i i i ) f o l l o w s immediately from ( i ) and ( i i ) . Theorem 3. I f R has c h a r a c t e r i s t i c 2, then e i t h e r G i s a b e l i a n o r G' has exponent 2 and o r d e r 2 o r 4.

...,

a r e independent f o r some elements a, f in ( [ e , f l - 1 ) # 0 i n RG, G, n o t n e c e s s a r i l y d i s t i n c t . Hence, ([a,bl-1) ([c,d]-1) # 0. From hmna 2 ( i i ) i t f o l l o w s t h a t i f and, by Lemma 3 ( i ) , (a,b)(c,d)(e,f) Proof. Suppose [a,b],

a t r i p l e {al,a2,a3}

[c,d],

[e,f]

i s composed o f elements i n d i s t i n c t p a i r s {a,b},

{c,d},

= (alya2)(alya3)(c.d) = (a13a2)(alya3)(e,f) = 0. {e,f} then (al,a2)(alya3)(a,b) Hence, by Lemma 3 ( i ) , i f (al,a2)(al,a3) # 0, then [a,bl, [c,dl, [ e , f l w i l l a l l and [al,a31 and, thus, cannot be be i n the subgroup o f G' generated by [al,a21 independent s i n c e G ' i s a b e l i a n . Hence, we may assume t h a t f o r each such

t r i p l e , (al,a2)(al,a3)

t h a t e i t h e r [al,a21 or [al,a31 i s t r i v i a l o r [ a 1 ,a 2 1 = [al,a31. Suppose, f o r instance, t h a t [a,cl = 1. By Lemna 2 ( i i ) , i f any o t h e r p a i r o f elements from such a t r i p l e commute, then (a,b)(c,d)(e,f)

= 0, and, thus, by Lemma 3 ( i i ) ,

= 0. Hence, we may assume t h a t i n t h e remaining t r i p l e s t h e

i n d i c a t e d e q u a l i t y between group commutators holds, t h a t i s [al,a21 = [al,a31 i f n e i t h e r o f these p a i r s i s a,c, t h a t i s , al-1a2-1(al,a2) = ai1ai1(al,a3) or ai1(al,a2) (df,e,c)

= ai1(al,a3).

-d(f,e,c)

By Lemna l ( i i ) , (a,b)(c,d)(e,f)

- (d,f,c)e).

(a,b).( (f,c)(e,d)

t

Since (a,c) = 0, i t f o l l o w s from Lemma 2 ( i ) t h a t

( a ,b) (c ,d)(e,f) = (a, b ) ( f ,c) ( e ,d) and , hence (a ,b) (c ,d) ( e ,f) = (a,b)ca-l(f,a)(da)-l(e,a), which, by Lemna 4 ( i i i ) and Lemma 2 ( i i ) , i s t r i v i a l . Thus, we may assume t h a t f o r any t r i p l e {a ,a2,a3} as described above, = [al,a31, whence, ai1(al,a2) a; (alya3). I n p a r t i c u l a r , we may [al,a21 1 assume t h a t c - l ( a , c ) = f - (a,f). However, i f we observe by Lemma l(i) that

;t

(a,b)(c,d)(e,f) = (a,b)(c,d)(ef , f ) f - l , e s s e n t i a l l y t h e same argument above shows t h a t we may assume t h a t c - 1(a,c) = (ef)-l(a,ef). Hence,

f - 1(a,f) (a,e)f-')

= f-'e-l(a,ef) = (a,f)

and, by Lemma l ( i ) , (a,f)

t e-l(a,e)f-l,

= e-l(a,ef)=e'l(e(a,f)

and, hence e-'(a,e)f-'

= 0. Thus,

t

( a y e ) = 0,

Lie metabelian group rings

157

and the previous argument shows t h a t ( a , b ) ( c , d ) ( e , f ) = 0. This proves the theorem. Next we consider the case of characteristic 4. Theorem 4. If RG i s Lie metabelian and R has characteristic 4 , then G ' has order a t most 2. Proof. As we have observed, f o r any a,b in G, ([a,bl-1)3 = 0 and, hence, [a,bl 2 t [a,bl t 1 = 0 since R has characteristic 4 . Hence, by an easy comparison of terms i t follows t h a t [a,bl has order a t most 2. Further, 2 by Lemma 2 ( i i i ) , f o r any elements c,d i n G, ([a,bl-1) ([c,dl -1) = 0. However, 2 2 since R has characteristic 4 , i f [ a , b l # 1, then ([a,bl-1) = [a,bl 2[a,b] + 1 = 2[a,bl t 2 # 0, and, hence, [c,dl i s dependent on [a,bl. Thus, i f [a,b] # 1, then G ' i s generated by [ a , b ] , which proves the theorem. Before turning to strong Lie properties of Lie metabelian group rings we record the following lemma.

z t

Lemma 5. Let R be a ring with characteristic p = 2,3 or 4 and G be a nonabelian nilpotent group of class 2 such t h a t ( i ) G' ( i i ) G' ( i i i ) G' Then for

has exponent 2 and order 2 or 4 i f p = 2 , i s cyclic of order 3 i f p = 3, i s cyclic of order 2 i f p = 4. any ai in G,

([al,a21-1)( [a3,a41-l)([a5,a,l-1) = 0 in RG. In particular, the above conclusion i s valid i f RG i s Lie metabelian. Proof. Let s = ([al,a21-1)([a3,a4]-1)([a5,a61-1). If p = 2, then G ' has a t most two generators and s i s divisible by ( [ a , b ] - 1 )2 f o r some a,b in G with 2 [ a , b ] = 1. Since p = 2 , ([a,b]-1)2 = 0 and s = 0. If p = 3 or 4, G' i s

3 cyclic, so f o r some a,b in G, s i s d i v i s i b l e by ([a,bl-1)3, where [a,bl = 1 i f p = 3 and [a,bI2 = 1 i f p = 4. In e i t h e r case expanding modulo p gives ([a,b]-1) 3 = 0 and s = 0. The final statement follows directly from our previous results. A ring S i s strongly Lie nilpotent of class n i f S ( n t l ) 0, where S ( n ) i s defined inductively by: S ( l ) = S and S ( n ) i s the ideal of S generated by (S("'),S).(cf. [23 and [31). The following i s stated f o r RG with R cornnutative. In the next section i t will be extended to a r b i t r a r y associative R.

F. Levin and G. Rosenberger

158

Lemma 6. The following are equivalent p r o p e r t i e s f o r a group r i n g RG o f a group G over a commutative r i n g R. (i) RG i s L i e metabelian. ( i i ) RG i s s t r o n g l y L i e metabelian. ( i i i ) RG i s s t r o n g l y L i e n i l p o t e n t o f class ( a t most) 3. (iv)

RG i s L i e n i l p o t e n t o f class ( a t most) 3.

Proof. Since R has a u n i t element, ( i i ) c l e a r l y implies ( i ) and ( i i i ) implies ( i v ) . That ( i v ) implies ( i ) f o l l o w s from the i d e n t i t y ((a,b),(c,d)) = (a,b,c,d) (a,b,d,c), which holds f o r any associative r i n g

-

and i s e a s i l y v e r i f i e d by d i r e c t expansion. Thus, t o complete the proof i t remains t o show t h a t ( i ) implies ( i i ) and ( i i i ) . By Theorem l ( i ) , i f RG i s L i e metabelian, then G i s n i l p o t e n t o f c l a s s 2, t h a t i s , G ' i s c e n t r a l . Hence, by Lemma l ( i i i ) , (a(b,c),d(e,f)) = (acb( [ b , c l - l ) , d f e ( [ e , f l - 1 ) ) = (acb,dfe)( [ b , c l - I ) ( [ e , f l - I ) = dfeacb( [acb,dfel-I)( [ b , c l - I ) ( [ e , f l - I ) f o r any = 0, by Lemma 5. elements a,...,f i n G, which implies t h a t (a(b,c),d(e,f)) Thus, ( i ) i m p l i e s ( i i ) . To see t h a t ( i ) implies ( i i i ) we apply e s s e n t i a l l y the same argument. Thus, i f RG i s L i e metabelian, then f o r any a,

...,f

i n G,

(((a,b)c,d)e,f) = ((ba([a,bl-l)c,d)e,f) = ((bac,d)e,f)([a,bl-1) = (dbac( [bac,dl-l)e,f)( [ a , b l - I ) = (dbace,f)( [bac,dl-1)( [ a , b ] - l ) = fdbace( [dbace,fl-I)( [bac,dl-1)( [a,bl-I), which i s t r i v i a l by Lemma 5 . Hence, ( i ) implies ( i i i ) , which completes the proof. Results i n Gupta and Levin [ I 1 g i v e t h e f o l l o w i n g immediate c o r o l l a r y .

Corollary. I f R i s commutative and RG i s L i e metabelian, then the group o f u n i t s o f RG i s n i l p o t e n t o f class a t most 3. 4. I f R i s n o t commutative, then f o r RG t o be L i e metabelian the conditions imposed by our previous r e s u l t s w i l l s t i l l hold, but the f a c t o f noncommutativ i t y o f R w i l l impose f u r t h e r conditions. I f G i s a b e l i a n the L i e p r o p e r t i e s o f RG w i l l depend e n t i r e l y on those o f R, so f o r t h e sequel we assume G t o be nona be1 ian. Lemma 7 . I f RG i s L i e metabelian and R n o t commutative, then f o r any ri i n R, al i n G,

Lie metabelian group rings

159

(iv)

(rlYr2,r3) = 0; (v) 2(rl.r2) = 0; (Vi) G' is cyclic of order 2. In particular, R does not have characteristic 3. Proof. By hypothesis, ((rlal,a2),(r2a3,a4))

=

0. Hence, by Lemma l(i),

((rp1 (r2a3Ya4)1 = (rl(a1 'a2 1 r2(a3 .aq)) rf2((al'a2) ( a 3 9 1 1 + (rlYr2)(alYa2)(a3Ya4) (ryr2)(a,+) (a3441 = 0 5 Y

Y

Y

= =

since RG is Lie metabelian. This proves (i). Similar expansion of ((a1,a2), (rla3,r2)) yields ( i i i ) and expanding ( (rlal,r2)y(r3a2,r4)) and ( (rl,r2), (r3al,a2)) yield, in turn, (a) (rlYr2)(r3,r4)(al.a2) = 0 and (b) (rlYr2,r3)(al.a2) = 0, respectively. If we replace (al,a2) by ([al,a21-l) in each of (a) and (b) we obtain (a') (rl.r2)(r3,r4)([al.a21-1) = 0, (b') (rlYr2,r3)( [al,a21-l) = 0. Thus, if al,a2 are chosen with [al,a2] # 1, (a') and (b') will be possible only if (r1,r2)(r3,r4) = 0 and (rl,r2,r3) = 0, that is, if (ii) and (iv) are valid. Finally, to prove (v) and (vi) we observe that since RG is metabelian, G is nilpotent of class 2 , by Theorem 1, and, hence, (i) is equivalent to ( rlyr2)( [al,a21 -1) ([a3,a41-1)

=

0,

if (rlyr2) # 0, [a3,a4] will be dependent on [al,a21 and G ' will be cyclic. In particular, (rl,r2)([al,a2]-1) 2 = 0, from which it follows that [al,a21 has order at most 2 and 2(rl,r2) = 0, which completes the proof of the lemna. For the converse to Lemna 7 we have the following result. so

Lemma 8. Let R be an associative ring with characteristic dividing 4 and satisfying the following conditions for ri in R: (i) (rl,r2)(r3.r4) = (rlYr2,r3) = 0; (ii) 2(rl,r2) = 0. If G is a nilpotent group of class 2 with G ' of order 2, then RG is Lie meta be1 ian. Proof. For the proof we must show that for any ri in R, ai in G, ((rlal,r2a2),(r3a3,r4a4)) is trivial in RG. If such a commutator is expanded by Lemna l(i), the resulting expression will be a linear combination of terms of

F. Levin and G. Rosenberger

160

the forms ( ~ ~ , ~ ~ ) ( b ~ . b (pl,p2)(bl,b2)(b3,b4), ~,b~), where both pi are i n G o r i n R and bi a r e i n G, and terms which vanish by ( i ) and ( i i ) . By Lemma 3, t o show t h a t such terms are t r i v i a l i n RG i t s u f f i c e s t o show t h a t (pl,p2)(~bl,b23-1)([b3,b~l-1)

i s t r i v i a l f o r any bi i n G. I f both pi are i n G,

the above expression i s equal t o p2p1( [p1,p21-1)( [bl,b21-l)( [b3,b41-1), which = [b3,b41 i s t r i v i a l by Lemna 5. Otherwise, both pi a r e i n R. Thus, i f [bl,b21 = [a,bl,

say, w i t h [a,bl of order 2 , i t f o l l o w s from ( i i ) t h a t

(p1,p2)( [a,bI-l)' = 0 by r o u t i n e expansion. The f o l l o w i n g theorem summarizes the above r e s u l t s . Theorem 5. L e t R be a non-comnutative r i n g and G a nonabelian group. The f o l l o w i n g two conditions a r e equivalent. ( i ) RG i s L i e metabelian. ( i i ) G i s n i l p o t e n t o f class 2 w i t h G ' o f order 2 , R has c h a r a c t e r i s t i c 2 o r 4 and f o r any ri i n R, (rlyr2)(r3,r4)

= (r1 ,r 2 ,r 3 ) = 2(rlyr2)

= 0.

An example o f a noncommutative r i n g s a t i f y i n g the above hypotheses i s provided by t h e subring R o f 3 x 3 matrices over GF(2) , the f i e l d o f two elements, o f matrices o f the forms

(; ;)

*)

o * *

1 * *

and

( 0

0

0 0 0

We leave the r o u t i n e proof o f t h i s f a c t t o the reader. I t f o l l o w s from Lemma 7 and the p r o o f o f Lemma 8 t h a t i f RG i s L i e metabe-

l i a n and R i s noncomutative, then f o r any pi i n RG, (p1,p2)(p3,p4)(p5,p6) = (p1,p2)(p3,p4,p5) = 0. Moreover, (p1,p2,p3,p4) w i l l a l s o vanish, since i n the expansion of (rlal,r2a2,r3a3,r4a4),

ri i n R, ai i n G, using Lemma l ( i ) ,

the o n l y term n o t o f the above forms and n o t t r i v i a l due t o the p r o p e r t i e s o f R w i l l be rlr2r3r4(al,a2,a3,a4), and t h i s term i s t r i v i a l by v i r t u e o f Lemmas

3 and 5 . However, i f the L i e products (pl(p2,p3),p4(p,,p6))

and

(((pl,p2)p3,p4)p5,p6), pi i n RG, are expanded using Lemma l ( i ) , the terms which w i l l occur w i l l be m u l t i p l e s o f terms having t h e above forms, and, hence, such L i e products w i l l be t r i v i a l . Thus, i f RG i s L i e metabelian, then RG i s s t r o n g l y L i e metabelian and i s s t r o n g l y L i e n i l p o t e n t o f class a t most 3. As observed i n the p r o o f o f Lemma 6, however, e i t h e r o f the l a t t e r condit i o n s implies t h a t RG i s L i e metabelian. This observation Lemma 6 y i e l d the f o l l o w i n g r e s u l t .

and the r e s u l t o f

Lie metabelian group rings

Theorem 6. The f o l l o w i n g are equivalent p r o p e r t i e s f o r a group r i n g RG o f a nonabelian group G over an a s s o c i a t i v e r i n g R. (i)

RG i s L i e metabelian.

(ii) RG i s s t r o n g l y L i e metabelian. ( i i i ) RG i s s t r o n g l y L i e n i l p o t e n t o f class ( a t most) 3. (iv)

RG i s L i e n i l p o t e n t o f c l a s s ( a t most) 3.

REFERENCES

[I]

N.D. Gupta and F. Levin: On the L i e i d e a l s o f a r i n g , Jour. o f Algebra, 83, 1983, 225-231

[21

F. Levin and S . Sehgal: On L i e n i l p o t e n t group rings, Journal of Pure and Appl. Algebra, 3, 1985, 33-39

[3]

S.K. Sehgal: Topics i n Group Rings, Dekker, New York, 1978

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Group and Semigroup Rings G. Karpiiovsky (ed.) 0 Elsevier Science Publishers B.V.(North-Holland),1986

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UNIT GROUPS AND ISOMORPHISM THEOREMS FOR COMMUTATIVE GROUP ALGEBRAS Warren May Department of Mathematics University of Arizona Tucson, Arizona 8 5 7 2 1 INTRODUCTION We shall describe methods and results which lead to various isomorphism theorems in the abelian case; henceforth, all groups will be assumed abelian and all rings commutative with identity. We take our direction from two well-known results. The first says that isomorphism of integral group algebras implies isomorphism of the groups; the second says that if we are dealing with a finite p-group, then isomorphismof the group algebras over a field of characteristic p implies the same conclusion. Now let R denote a commutative ring with identity and G an abelian group. In order to obtain similar strong isomorphism results for group algebras over R, it is natural to assume that whenever G has an element of order p, then p (more precisely, p * 1 R ) is not a unit in R. We shall see, in fact, that this assumption is necessary to ensure a strong conclusion. To describe this good situation, we shall say that G is R-favorable if whenever a prime p is a unit in R, then G has trivial p-component. We first discuss extensively the fundamental cases where R is an integral domain. Suppose that R is an integral domain of characteristic 0 , and that G is an R-favorable abelian group. We proceed by describing the structure of the group V of normalized units of RG. We prove a direct factor theorem stating that G is a direct factor of V. More precisely, we shall see that V = G x F, where F is a torsion-free group of units which is supported on the torsion subgroup of G. This splitting, together with a simple generalization of a classical result of G. Higman, yield the conclusion that if H is another group such that RG and RH are isomorphic R-algebras, then G and H are isomorphic. The modular case, as expected, has a different flavor from that of characteristic 0. Let R be an integral domain of characteristic 1.

p, and let G be an abelian group whose only torsion is p-torsion

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W. May

(i. e. G is R-favorable). We prove the well-known fact that the Ulm p-invariants of G are derivable from the group algebra. Moreover, information on the divisible part of torsion is also inherent in the algebra. Thus one would expect that if G is a totally projective p-group, and if RG is isomorphic to RH, then G is isomorphic to H. The problem is that one must be able to deduce that H is also totally projective in order to apply Ulm's theorem. We shall show that this is the case if the p-length of G is less than the first uncountable ordinal. We also demonstrate that under a similar restriction, a direct factor theorem holds for G. One would, of course, like to remove the restriction that G be totally projective, or even a torsion group. Unfortunately, knowledge about the characteristic p case is rather incomplete. Now consider an arbitrary commutative ring R. The main feature of this case is that the concept of R-favorable may not be sufficient to obtain a suitable isomorphism theorem if R contains nontrivial idempotents. Thus, one would like to characterize those R for which the hypothesis of R-favorable is sufficient. We shall discuss the isomorphism conjecture of W. Ullery that this class of rings is precisely the class of "nicely decomposing" rings. In fact, this conjecture is true if and only if isomorphism theorems hold for all fields of prime characteristic. Finally, we shall state a direct factor conjecture which underscores the close relation of isomorphism theorems and direct factor theorems in the abelian case. We wish to stress that we only discuss abelian isomorphism theorems that lie in the direction that we indicated earlier. Some examples of other possible directions would be S. D. Berman's exhaustive account 1 2 1 of algebras of countable p-groups over fields of characteristic not equal to p, or the problem of isomorphism over the field of rational numbers, or isomorphism questions in which conditions are imposed on both groups, G and H.

2.

CHARACTERISTIC 0 INTEGRAL DOMAINS In this section we shall assume that R is a characteristic 0 . integral domain and that G is an R-favorable abelian group. Most of the material presented here can be found in Karpilovsky [8] in somewhat different form. Lemma 2 . 1 . Let S be an indecomposable, reduced commutative ring with identity, and F a torsion-free abelian group. Then every unit of SF has singleton support.

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165

Proof. First we observe that the conclusion is clear if S is an integral domain by using the fact that F can be ordered. Now consider S as given, let Po be a fixed prime ideal of S, and let a be a unit of SF. Then the natural image of a in (S/P ) F has single0 ton support If}. Let I be the intersection of all prime ideals of S such that the image of a has support (f}. If these are all the primes, then I = 0 since S is reduced, hence a has support [f}. Therefore, suppose that there exist other primes, and let J be a similar intersection for supports different from If}. Then I ll J = 0 and I + J must be contained in some maximal ideal M since S is indecomposable. However, the natural image of a in (S/M)F must have support both {f) and different from [f}. This contradiction proves the lemma. / / Let V(RG) denote the group of normalized units of RG, i. e. those of augmentation 1 . Lemma 2 . 2 . Let R be an integral domain of characteristic 0, and G an R-favorable abelian group with torsion subgroup T. Then V(RG) = G * V(RT). Proof. One inclusion is obvious, therefore let a E V(RG). We may assume that G is finitely generated, hence G = T x F for some free abelian subgroup F. If S = RT, then S is reduced and indecomposable (the latter since G is R-favorable). But RG = SF, hence the previous lemma shows that a E F . V(RT) 5 G V(RT). / / The following theorem was proved by G. Higman [ 6 1 for finite abelian groups and rings of algebraic integers. Theorem 2 . 3 . Let R be an integral domain of characteristic 0, and G an R-favorable abelian group with torsion subgroup T. Then the torsion subgroup of V(RG) is T. Proof. Let a E V(RG) be of finite order. Then a E V(RT) by the previous lemma since G n V(RT) = T. We may assume that T is finite and that R is finitely generated. Thus we may regard R as a subring of the complex numbers Q. If IT1 = n, then we have an algebra isomorphism QT + Q x---x Q under which a corresponds to (cl,...,cn) for certain roots of unity (1 I i I n). Explicitly, if a = EtET att, then at = (l/n)Xi yi(t-’) for appropriate characters yi ( 1 I i 5 n), of T. Adjusting a by a factor from T, we may assume that ae # 0. Let o be an element of the Galois group of Q((,, ...,en) over ( P I let N denote the norm map for this field extension, and let d be the field degree. Then o(5,) = k (1 I i S n ) , for some integer k. Thus ae = (l/n) (5, +--- + en), hence k k = ( a I e E R. Thus o(ae) = (l/n)(X

-

ci

ci

ci

ci)

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W. May

"ae) E R n Q n (l/nd)z[G1,. . . , G I, 5 R n (l/nd 12. But this last ring is equal to 2 since T is R-favorable. Hence N(ae) is a nonzero integer, consequently IN(ae)l L 1. But la(ae)I I 1 for all 0 , hence l a e l = 1 . This can occur only if 5 = 5, =...= 5, = 5 , hence 1 at = (1/n)5Zyi(t-') = 0 for t # e. Thus a E T. / / We now need a criterion for telling when the torsion subgroup of an arbitrary abelian group is a direct factor. We state such a criterion abstractly. A general reference for abelian group theory is Fuchs [ 5 1 . Proposition 2 . 4 . Let A be an abelian group with torsion subgroup T. Then T is a direct factor of A if and only if there exists a countable chain of subgroups A1 5 A2E..., such that A = Uill Ai, and such that for every i, the torsion subgroup Ti of Ai is bounded, and A/AiT is torsion-free. Proof. If T is a direct factor of A, we leave it to the reader to give an appropriate countable chain. Therefore we shall assume and we shall show that T is a that we are given A1 5 A 2 5 direct factor of A. We shall freely use the familiar fact that if the torsion subgroup of an abelian group is bounded, then it is a direct factor. Let T : A + A/T be the natural map, and put Fi = n(Ai) for every i. Then F1 5 F2z..., and Uikl Fi = A/T. We note for later use that Fi+l/Fi Ai+l '/Ai T 5 A/Ai T, hence Fi+l/Fi is torsionfree. We must show the existence of a map cp : A/T + A such that T o cp is the identity on A/T. Note that the kernel of TI on Ai is precisely Ti. Since T1 is bounded, it is a direct factor of A 1' hence there is a map 'pl : F 1 + A1 such that TI o 'pl is the identity on F1. If we can show that 'p, may be extended to a map 'p2 : F2 + A2 such that TI o 'p2 is the identity on F2, then by induction the map q~ may be obtained as cp = Ui21 'pi. Since T is bounded, there is a 2 map : F2 + A2 such that n o is the identity on F2. We may regard 'p, as a map into A2, hence 'pl - 'pi may be regarded as a map

...,

'pi

'pi

JI : F1 A2. But then 71 o J, = 0, hence J, : F1 + T2. If K is the kernel of J,, then F1/K is a bounded torsion group. Moreover, we have noted above that F2/F1 is torsion-free. Thus F2/K is a group with bounded torsion subgroup F1/K, which thus must be a direct factor. Consequently, J, can be extended to a map J,' : F2 + T 2 . If we now put 'p2 = cp; - J , ' , then this is the desired extension of 91.

// We can now prove the direct factor theorem for characteristic 0 .

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Theorem 2 . 5 . (Direct Factor) Let R be an integral domain of characteristic 0, and G an R-favorable abelian group with torsion subgroup T. Then V(RG) = G x F, where F is a torsion-free subgroup of V(RT) such that V(RT) = T x F. Proof. Suppose we know that V(RT) = T x F. Since G fl V(RT) = T I we could conclude from Lemma 2 . 2 that V(RG) = G x F. By Theorem 2 . 3 , T is the torsion subgroup of V(RT), hence it suffices to show that T is a direct factor of V(RT). We shall apply Proposition 2 . 4 with A = V(RT). For each positive integer i, let Ti be the subgroup of all elements of T whose orders divide i!. Let Ai be the kernel of the map V(RT) -B V(R(T/Ti)) induced by the natural map T + T/Ti. Then it is immediate that A1 5 A 2 c Uitl Ai = A, and Ti is the torsion subgroup of Ai. Finally, A/Ai T 5 V(R(T/Ti))/(T/Ti), which is torsion-free by Theorem 2 . 3 . Hence the hypotheses of Proposition 2 . 4 are met. / / We remark that in the special case R = 2, R. K. Dennis [ 3 1 has given a canonical splitting of V(ZG) Before proving an isomorphism theorem for characteristic 0, we examine subgroups of V(RG) that consist of linearly independent elements. Proposition 2 . 6 . Fix F in the statement of the theorem. Then H is an R-independent subgroup of V(RG) if and only if there exists a subgroup K of G I and a homomorphism p : K -B F such that H = [ a p ( a ) l a E K) The map sending a to a p ( a ) is an isomorphism of K with H. Proof. Suppose that H is an R-independent subgroup of V(RG) and let a E H. Note that RT is integral over R. If a is of infinite order, then the powers of a are distinct, hence R-independent. Therefore a 4 RT, and consequently a 4 F. If a is of finite order, then a E T by Theorem 2 . 3 . Thus we see that H n F = 1. Let n , and n2 be the projections of G x F onto the first and second facotrs, respectively, and put K = n 1 (HI. Then n l is injective on H I and we define p = n2 o n -l 1 : K + F. It is clear that K and p express H as desired. For the converse, assume that K and p are given. We must show that the elements of H = { a p ( a ) l a E K) are R-independent. Since p maps into a torsion-free group, p is trivial on K fl T. Thus we may extend p to KT (denoting the extension also by P ) , in such a way that p is trivial on T. Hence the map sending a to a p ( a ) on KT induces an endomorphism cp of R(KT). But p ( a ) € RT, hence c p ( p ( a ) ) = = p ( a ) . Similarly, the map sending a to a p ( a ) - ’ induces an endo-

...,

.

.

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W. May

rnorphism of R(KT) satisfying @ ( p ( a ) )= p ( a ) . One easily checks that J, is the inverse of cp. Since H = q ( K ) , H is an R-independent subgroup. / / Corollary 2.7. If H is an R-independent subgroup of V(RG), then H is isomorphic to a subgroup of G. Now suppose that H is a subgroup of V(RG) that is an R-basis for the group algebra. Thus we may regard RG as RH. By Theorem 2.3, we may conclude that T is the torsion subgroup of H. If K is the subgroup of G given in Proposition 2.6, the isomorphism from K to H is the identity on K rl T I hence H 5 K V(RT) 5 RK implies that K = G. Thus H G. Theorem 2.8. (Isomorphism Theorem) Let R be an integral domain of characteristic 0, and G an R-favorable abelian group. If H is another group such that RG and RH are isomorphic R-algebras, then G and H are isomorphic. Proof. By a standard manipulation, we may assume that the isomorphism of RG with RH is normalized (augmentation preserving). We may thus regard H as a subgroup of V(RG) that is an R-basis for the group algebra. By the above, we are finished. / / We now apply the discussion above to normalized automorphisms of the group algebra. Proposition 2.9. Let R, G and F be as above, and let cp be a normalized automorphism of RG. Then there exists an automorphism 0 of GI and a homomorphism p : G F such that cp(g) = B(g)p(g) for every g E G. Proof. Put H = cp(G). Then the discussion before the theorem tells us that K = GI and there is a homomorphism p ' : G + F such that the map sending a to a p ' ( a ) is an isomorphism of G with H. -1 Let 8 be the automorphism of G such that e - ' ( a ) = cp ( a p ' ( a ) ) for every a E G. If we put p = p ' o 0 , then cp(g) = e(g)p(g) for every

-

9 E G- // In the setting that we have been discussing, if R is a finitely generated ring, then the complement F is known to be a free abelian group (see [lo], p. 503). Hence a group can have a nontrivial homomorphism into F only if it has an infinite cyclic direct factor. Corollary 2.10. Let R be a finitely generated integral domain of characteristic 0, and let G be an R-favorable abelian group which has no infinite cyclic direct factor. Then every normalized automorphism of RG is induced by an automorphism of G.

Unit groups and isomorphism theorems

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169

SOME EXAMPLES

We now give three examples which put limits on the applicability of certain types of theorems. The first and third examples are concerned with limits on direct factor theorems, while the second example deals with isomorphism theorems. A l l three depend upon a lemma which involves the notion of p-height in an abelian group. We recall the definition. Let p be a prime number, and A an abelian group. Inductively define A for every ordinal a by A = A, a 0 A p if a is a limit ordinal. This defines Aa+l = A,: and Aa = a descending chain of subgroups whose intersection we denote by We call y the There is a least ordinal y such that A = A,. A,. Y p-length of A. Clearly A, is the maximal p-divisible subgroup of A. we say that x has p-height a. If x$A,, the p-height If x E A,, of x is the largest ordinal a such that x E Aa. We now state the lemma that we need. Lemma 3.1. Let R = Z [ l / p ] , let T be a p-group, and let a E V(RT) have p-height at least o in V(RT). Then the p-height of a is in fact a. Proof. A s in the first part of the proof of Theorem 2.3, by examining how the coefficients of a are related to the values of characters, we see that the torsion subgroups of RT and QT are the same. Hence we may assume that R = Q. It suffices to show that if a E V(QT) has p-height at least O , then there exists p E V(QT) such that a = pp, and p has p-height at least w in V(QT). We may choose a finite subgroup A 5 T such that a and a p-root of a lie K1 x x Kr for certain fields Ki in (PA. We know that QA (1 I i 5 r). Regard a as lying in this product, and choose p as follows. If a has i-th coordinate 1, let 8 have i-th coordinate 1. (Note that this assures that we are dealing with normalized units.) If the coordinate of a is different from 1 , then our choice of A guarantees that we may take the coordinate of p to be a p-root. We only need show that given a positive integer k, then p has a pk-root in V(QT). Choose a finite subgroup B 2 A such that a x . . x L for certain pk+l-root of a lies in QB. Then QB L1 S fields L (1 5 j I s ) . For every coordinate of fi that is 1 , we j select 1 as the coordinate for a pk-root. Now consider a coordinate of p which is different from 1 , say in L But this comes j' from a coordinate of p in some Ki which is different from 1. Hence the coordinate of a in Ki will be different from 1 , and hence also By choice of B, this coordinate of a has a pk+'-root in L in L j' j' Thus the coordinate of p must have a pk-root in L.. Consequently,

...

.

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W. May

170

p has a pk-root. / / Example 3.2. This example will show that if the prime p is invertible in a ring R, then there exists a p-group T such that T is not a direct factor of V(RT). Thus some restriction such as R-favorable is necessary in a direct factor theorem. Let T be the abelian group with generators {tili 2 0) subject i to relations tP = 1 , and ty = to (i 2 1). Then T is a p-group such 0 that to has p-height w in T. Let p be invertible in the ring R. Then there is a ring homomorphism Z[l/p] R which induces V(Z[l/p]T) -+ V(RT). By the lemma, the p-height of to in V(Z[l/plT) is m , hence also in V(RT) since homomorphisms cannot decrease height. But it is easy to see that the height of an element in a direct factor is the same as its height in the full group. Thus T cannot be a direct factor of V(RT). We remark that this example answers in the negative two questions of R. K. Dennis 131. Take R = Q to see immediately that a direct factor theorem does not hold for the field of rational numbers. Moreover, for this example, we claim that T is actually a pure subgroup of V(QT), thus purity is not enough to guarantee that T is a direct factor. The problem is that purity does not give any information on elements of height at least o in the subgroup. The claim will follow from Lemma 3.3. Any abelian group G is a pure subgroup of V ( Q G ) . Proof. It suffices to show that G is p-pure in V(QG) for an arbitrary prime p. Let g E G I and suppose that g has a pk-root in V(QG). Then there is a subgroup of G of form TF such that T is a finite subgroup, F is free of finite rank, and a pk-root of g lies in Q(TF). We may write g = tf (t E T I f E F). The projection TF -+ F induces V(Q(TF)) + V(QF). But V(QF) = F by Lemma 2.1, hence f has a pk-root in F. Thus it suffices to show that t has a k p -root in T. The projection TF + T shows that t has a pk-root in QT. If p does not divide the order of t, then we can extract a pk-root in T; therefore assume that t has finite p-height m in T. Then there is a homomorphism of QT onto Q ( c ) , where 5 is a root of m m unity such that # 1 , and t maps to Since the image of t must have a pk-root in Q ( 6 ) , we conclude that k I m. Hence t has a pk-root in T. / / Example 3 . 4 . This example will show that if p is invertible in a ring R, then there exists an abelian group G whose only torsion is p-torsion, and such that isomorphism fails for R and G. In fact, -+

cp

cp .

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if T is the torsion subgroup of G, then RG R(T x (G/T)) , but G 7 T x (G/T) since G will not be a split abelian group. Let G be the abelian group with generators {gili 2 O} and i j {hij I i,j 2 1 1 , subject to relations gy = go, and hp. = gi 11 (i,j t 1). Note that all these generators are torsion-free elements in G. The effect of the hij is that each g . (i 2 l), has 1 p-height 0 , and consequently go has p-height 02. The torsion subgroup T contains certain differences. In particular, let ti

- gi gi+1 -p for i

t 1. Then ti ha; order pir and its p-height is has p-height w, and g& has p-height w+l). i in V(Z[l/p]T) by Lemma 3.1, thereThe p-height of each ti is fore also in V(RT) by a homomorphism argument. We now claim that go has p-height in V(RG). To start, gy = go. Now choose s1 of in V(RT) such that s y = tl. One then verifies that p-height (sl g,)' = gl. Next choose s2 of p-height m in V(RT) such that s z = s1 t2. Then (s2 g,)p = s1 g2. One can continue this divisibility chain by induction. If F is the subgroup of V(RG) generated by the successive roots, then F is a torsion-free subgroup of V(RG) which is mapped isomorphically onto G/T by the natural map V(RG) + V(R(G/T)). It follows that TF is a subgroup of V(RG) which is a linear basis for RG, thus RG R(TF). Finally, G does not split, for if it did, it would have to contain torsion-free elements of p-height m. But every torsion-free element in G has some power equal to a positive power of go, thus having p-height of form w+n for some n < w. Example 3.5. We shall be able to apply this example in Theorem 5.4 to the direct factor problem over rings that are not integral domains. Let pl,...,p n be prime numbers, let R1,...,Rn be rings such that pi is invertible in Ri (1 I i I n) , and put xRn. We shall construct a group G whose torsion inR = R1 x volves only the primes pl,...,pnr and such that G is not a direct factor of V(RG). If we take Ri = Z[l/pi], and use Theorem 2.5, it is not hard to show that if G is a torsion group, then G is a direct factor of V(RG) (assuming that n t 2 and the pi are distinct). Thus we must incorporate torsion-free elements into G in order to give our example. Here the construction in Example 3.4 is what we need, but with roots of go taken for every pi. Let G be the abelian group with generators go and

o (since g

-

...

(,sir

phij

I i,j

2

1 , p = pl,...,pn} subject to relations g0 - pgiPi

W. May

172

1 , p = pl,...,pn). As in the previous examin V(RiG). We may regard RG ple, g0 has pi-height w in G, but x RnG. Relative to this product, let ai have coordias RIG x nate g in RiG, and coordinate 1 elsewhere ( 1 I i I n). Then ai 0 has pi-height in V(RG). Assume that G is a direct factor of V(RG). Then there exists a homomorphism TI : V(RG) + G that is the identity in G. Since every torsionon G. But a(ai) must have pi-height free element in G has a power that equals a positive power of go, an implies r(ai) must be a torsion element. But then go = a l a2 that go = a(go) = TI (a,) 7~ (a,) must be a torsion element. This contradiction establishes the example. and

hp

P ij

= pgi (i,j 2

0

...

00

00

. ..

...

4.

CHARACTERISTIC p FIELDS Let p denote a fixed prime number. Since the notion of R-favorable is the same for all characteristic p domains, we shall let F denote a field of characteristic p throughout this section. We note that a group G is F-favorable if all the torsion of G is p-torsion. In contrast to the characteristic 0 case discussed earlier, in characteristic p one does not know very much about even the isomorphism of p-torsion groups. The chief feature of the characteristic p abelian case is that the p-power map is an endomorphism. Thus certain subalgebras and subgroups of units can be intrinsically identified. Assume now that F is a perfect field. Recall the definition at the beginning of section 3 of the transfinite p-powers of G. We can apply a similar process to FG. Since (FG)p = F(Gp) , we obtain a subalgebra (FG), = F(G U ) for every ordinal a (and we may write FG,). Similarly, we can define V(FG),. Since V(FG)' = V(FGp), we see that V(FG), = V(FG,) for every a . We now prove the theorem on the persistence of the Ulm p-invariants (Berman [ 2 ] , May [ 9 ] , Dubois and Sehgal [ 4 ] ) . We first state without proof a variation of a familiar result (see [ l ] , Theorem 3.1).

Lemma 4 . 1 . Let R be a ring, B a subgroup of A, M the augmentation ideal of RA, and K the kernel of the natural map RA -+ R(A/B). Then K/MK R 8 B. Theorem 4 . 2 . Let F be a field of characteristic p, and let G and H be abelian groups. If FG and FH are F-isomorphic, then G and H have equal Ulm p-invariants. Proof. Note that isomorphism over F implies isomorphism of the group algebras over the algebraic closure of F, hence we may assume

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173

without loss of generality that F is a perfect field. Let us recall what the a-th Ulm p-invariant of G is. The p-power map takes Ga into Ga+l, and Ga+l into Ga+2, thus inducing a homomorphism Ga/Ga+l + Ga+1/Ga+2 of vector spaces over the integers modulo p. Let the kernel be Ua. Then the dimension of Ua is the a-th Ulm p-invariant of G. Let I a+ 1 denote the augmentation ideal of FGa+l. Then F(Ga/Ga+l) is naturally isomorphic to F(Ga)/F(Ga)Ia+l. But we noted above that F(Ga) may be obtained intrinsically. Hence, if we are given the augmentation ideal of FG, we may construct the homomorphism F(Ga/Ga,l) + F(Ga+1/Ga+2) induced by the p-power map. Let Ka be the resulting kernel. But Ka remains the same if we look at the homomorphism induced by the p-power map on Ga and the identity on F. This is since the p-power map is injective on F. We may assume that the isomorphism of FG with FH is normalized, hence augmentation ideals correspond, and the K we have constructed are the a same up to isomorphism. Let J be the augmentation ideal of a F(Ga/Ga+l). Since K is the kernel resulting from reduction modulo a Ua, the lemma implies that the dimensions over F of Ka/JaKaand F C3 Ua are the same. But this is just the desired Ulm invariant. Hence it is the same for both G and H. / / Additional invariants can be extracted from FG. For example, the rank of the maximal divisible p-torsion subgroup of G may be gotten as in the final part of the proof above. The maximal perfect subring of FG is (FG)- = F(Gm). By looking at the kernel of the p-power map on this subalgebra, we may obtain the desired rank. Other invariants can be gotten in case G is a Warfield group or an S-group. For an account of these other possibilities, see Beers, Richman and Walker [ l ] . In view of the information on the divisible part of p-torsion, Theorem 4 . 2 suggests that we should consider p-groups that are totally projective. Of course, this assumption should only be imposed on the group G, thus in order to apply the theorem, we must answer. Question 4 . 3 . Let F be a field of characteristic p, and T a totally projective p-group. If S is a group such that FT and FS are F-isomorphic, then is S a totally projective p-group? To give a partial answer to this question, we must first prove that V(FT) is totally projective under certain hypotheses. We note some facts about V(FT) if T is a p-group. First of all, V(FT) is a p-group and consists of all elements in FT of augmentation 1.

174

W. May

Suppose further that F is perfect. We showed earlier that V(FT), = V(FTa) for every ordinal a. Thus the p-height of an element in V(FT) is the minimum of the p-heights of the elements in its support. We recall that a totally projective p-group is a direct sum of countable p-groups precisely if its p-height does not exceed the first uncountable ordinal (see [ 5 1 , Theorem 8 2 . 4 ) . Proposition 4 . 4 . Let F be a perfect field of characteristic p, and let T be a totally projective p-group of p-length not exceeding the first uncountable ordinal. Then V(FT) is a totally projective p-group of the same p-length as T. Proof. The p-lengths of T and V(FT) are equal since V(FT), = V(FTa) for every a. We first claim that if A is a finite subgroup of T and N 5 V(FA), then N is a nice subgroup of V(FT). Let a E V(FT) , and put S = (ga 1 g E (support a) , a E A). Then the p-height in V(FT) of any element in the coset aN equals the p-height of some element in the finite set S . Thus some element in the coset will have maximum p-height, proving the claim. We now prove that V(FT) is totally projective by induction on ITI. First suppose that T is at most countable, and let 0 = A c A c... be a chain of finite subgroups with Uiz0 Ai = T. 0 1 By our claim above, it follows that we can insert a nice composition series into each step from V(FAi) to V(FAi+l) (see [ 5 1 ) . Taking the union, we get a nice composition series for V(FT), which must therefore be totally projective. Now suppose that T is uncountable and let T be the least ordinal with I T I = ITI. Our assumption on p-length means that we may assume that T = up,, AP, where each A P is countable. For each a < T I put T, = uaxa AP and V, = V(FTa). (Note that the subscripts here do not refer to p-height.) The natural projection Ta+l + T, induces a projection Va+l + Val and hence a splitting V a + l = V ax Wa. Clearly V(FT) = LIa < T W,. But la1 < I T I , hence ITa+,I < ( T I . Induction implies that Va+l is totally projective. Therefore so is the direct factor War and hence the weak direct product V(FT). / / We are now ready to give a very limited isomorphism theorem for characteristic p. Let F be a field of characteristic p, and let T Theorem 4 . 5 . be a totally projective p-group of p-length less than the first uncountable ordinal. If S is a group such that FT and FS are F-isomorphic, then T and S are isomorphic. Proof. Without l o s s of generality, we may assume that F is perfect and that the isomorphism preserves augmentation. Thus

Unit groups and isomorphism theorems

175

V(FT) E V(FS), hence V(FS) is a totally projective p-group of countable p-length. By our remarks on p-heights in unit groups, we see that S is an isotype subgroup of V(FS). A theorem of P. Hill ( [ 7 ] , Theorem 1 ) implies that S is therefore totally projective. We now conclude S T by Theorem 4 . 2 . / / The restriction that the p-length of T cannot equal the first uncountable ordinal is unfortunate, but dictated by the result of Hill which is used in the proof. It would be good to remove this condition, and even better to remove any restriction on the p-length of T. Since one can prove a limited isomorphism theorem in characteristic p, Corollary 2 . 7 suggests the following question. Let G be an F-favorable abelian group. Is every Question 4 . 6 . F-independent subgroup H of V(FG) isomorphic to a subgroup of G? We end this section by giving a direct factor theorem for characteristic p which has restrictions similar to those in Theorem 4.5. We leave the proof of the following simple lemma to the reader. Let R be a commutative ring and G an abelian group. Lemma 4 . 7 . Assume that there exists an abelian group B such that V(RB) has a direct factor that is isomorphic to G. Then G is a direct factor of V(RG). Theorem 4 . 8 . Let F be a field of characteristic p, and let T be a totally projective p-group of p-length not exceeding the first uncountable ordinal. Then T is a direct factor of V(FT). Proof. Let B = Uo T. Then B is totally projective of the same p-length as T. Thus Proposition 4 . 4 implies that V(FB) is totally projective. But B is an isotype subgroup of V(FB), hence the Ulm p-invariants of V(FB) are at least equal to those of B. Consequentlyr V(FB) and T x V(FB) have the same Ulm p-invariants. Since they are both totally projective p-groups, they are isomorphic. One now applies Lemma 4 . 7 . / / ARBITRARY COMMUTATIVE RINGS We now describe recent work on the case where R is not assumed to be an integral domain. First, we shall summarize without proof some results of W. Ullery 113, 1 4 1 on the isomorphism question. Since the presence of idempotent elements tends to work against strong isomorphism results, it is not surprising that indecomposable coefficient rings are an important step after integral domains. The case of finitely general indecomposable rings of charac5.

W. May

176

teristic 0 was considered in [lo], and recently Ullery disposed of the hypothesis of finite generation by proving a version of Theorem 2 . 8 for indecomposable rings of characteristic 0. Moreover, he suggested a certain new class of commutative rings as possibly the largest class for which one has strong isomorphism results. These are the so-called ND-rings. A ring R is an ND-ring if whenever x Rn for rings R1,...,Rn, then there is an index i R 2 R x 1 ( 1 I i 5 n), such that Ri has the same set of invertible prime numbers as R. The connection with indecomposable rings is given in the following proposition. Proposition 5.1. (Ullery) A commutative ring R is an ND-ring if and only if there exists an indecomposable homomorphic image of R that has the same set of invertible prime numbers as R. Let us say that "R has the isomorphism property" if whenever G is an R-favorable group, and H is another group such that RG RH, then G H. As we have seen, integral domains (more generally, indecomposable rings) of characteristic 0 have the isomorphism property. The role of ND-rings is revealed in the following theorem and conjecture of Ullery. Theorem 5 . 2 . If R has the isomorphism property, then R is an ND-ring. The converse holds if the set of invertible primes in R is not the complement of a single prime. Isomorphism Conjecture. A ring R has the isomorphism property if and only if R is an ND-ring. The case for which the converse of the theorem is not known, namely, every prime except one inverts in R, is suggestive of characteristic p fields. In fact, Ullery has shown Theorem 5 . 3 . The following statements are equivalent. (1) The Isomorphism Conjecture is true. ( 2 ) Every field of prime characteristic has the isomorphism property. ( 3 ) Every ND-ring of characteristic 0 has the isomorphism property. The equivalence of ( 1 ) and ( 2 ) can be easily shown by using Proposition 5.1. What is surprising is the equivalence of ( 3 ) . Thus, it cannot happen that characteristic p might fail, yet there would be a positive theorem for characteristic 0 ND-rings. Both hold or both fail. Since we have been trying to balance isomorphism theorems with direct factor theorems, we now turn our attention to the splitting of G in V(RG). Let us say that "R has the direct factor property"

...

Unit groups and isomorphism theorems

177

if whenever G is an R-favorable group, then G is a direct factor of V(RG). Ullery has generalized Theorem 2.5 by showing that indecomposable rings of characteristic 0 have the direct factor property (the information on the complement F is different, however). We now show that direct factor results for ND-rings are remarkably similar to what we have seen earlier. Theorem 5 . 4 . If R has the direct factor property, then R is an ND-ring. The converse holds if the set of invertible primes in R is not the complement of a single prime. Proof. Suppose first that R is not an ND-ring. Then there exist rings R, ,...,Rn and primes pl,...,pn, such that R = R1 x . . x Rnl and pi is invertible in R. but not in R (1 5 i 5 n) Example 3 . 5 now provides an R-favorable group G such that G is not a direct factor of V(RG). Thus R does not have the direct factor property. Now suppose that R is an ND-ring in which the set of invertible primes is not the complement of a single prime, and let G be an R-favorable group. By Proposition 5.1, there is a ring homomorphism R + S for some indecomposable ring S having the same invertible primes as R. Thus G is S-favorable. Since S is indecomposable and cannot have prime power characteristic, S must have characteristic 0. The result of Ullery mentioned above implies that there is a projection V(SG) + G. By combining this with the induced map V(RG) + V(SG), we see that G is a direct factor of N

.

V(RG). / / We are thus led to state the Direct Factor Conjecture. A ring R has the direct factor property if and only if R is an ND-ring. Reviewing the proof of the theorem, the assumption on primes was used to guarantee that S had characteristic 0. In the excluded case, R could be mapped into a field of prime characteristic, and we could conclude that R had the direct factor property if the field had it. Hence we have Theorem 5.5. The Direct Factor Conjecture is true if and only if every field of prime characteristic has the direct factor property. We have been discussing properties that hold for groups that we are restricted by the ring R. It is natural to ask what rings possess a "universal isomorphism property" (for every G and H, RG RH implies G H) , or a "universal direct factor property" (for every G, G is a direct factor of V(RG)). This is easily answered.

.

178

W. May

Theorem 5.6. The following statements are equivalent. (1) R has the universal isomorphism property. (2) R has the universal direct factor property. (3) R is an ND-ring in which no prime number is invertible. Proof. Theorems 5.2 and 5.4 show that (3) implies both (1) and (2), and that (1) and (2) each imply that R is an ND-ring. Examples 3.2 and 3.4 show that no prime number can invert in R. / / Thus we have seen strong relationships between the isomorphism and direct factor properties, and ND-rings. REFERENCES [ l ] D. Beers, F. Richman and E. Walker, Group algebras of abelian groups, Rend. Sem. Mat. Univ. Padova 69 (19831, pp. 41-46. [ 2 ] S.D. Berman, Group algebras of countable abelian p-groups, Publ. Math. Debrecen 14 (1967), pp. 365-405. [3] R.K. Dennis, Units in group rings, J. Algebra 43 (19761, pp. 655-664. [ 4 ] P.F. Dubois and S.K. Sehgal, Another proof of the invariance of Ulm's functions in commutative modular group rings, Math. J. Okayama Univ. 15 (1972), pp. 137-139. [ 5 ] L. Fuchs, Infinite Abelian Groups, Vol. 11, Academic Press, New York, 1973. [6] G. Higman, The units of group rings, Proc. London Math. SOC. 46 (1940), pp. 231-248. [ 7 ] P. Hill, Isotype subgroups of direct sums of countable groups, Illinois J. Math. 13 (1969), pp. 281-290. [8] G. Karpilovsky, Commutative Group Algebras, Marcel Dekker, New York, 1983. [9] W. May, Commutative group algebras, Trans. Amer. Math. SOC. 136 (1969), pp. 139-149. 1101 W. May, Group algebras over finitely generated rings, J. Algebra 39 (1976), pp. 483-511. [Ill W. May, Isomorphism of group algebras, J. Algebra 4 0 (19761, pp. 10-18. [12] W. May, Modular group algebras of totally projective p-primary groups, Proc. Amer. Math. SOC. 76 (19791, pp. 31-34. [131 W. Ullery, Isomorphism of group algebras, Comm. Algebra, to appear. [I41 W. Ullery, A conjecture relating to the isomorphism problem for commutative group algebras, this volume.

Croup and Semigroup Rings

C. KarpiloMky (ed.) 0Elsevier Science Publishers B.V. (North-Holland),1986

179

TORSION UNITS I N GROUP RINGS AND A CONJECTURE OF H.J. ZASSENHAUS C. Polcino M i l i e s

I n s t i t u t o de Matemztica e E s t a t i s t i c a Universidade de SZo Paulo Caixa Postal 20.570 Ag. Iguatemi 01000 SZo Paulo S.P. Brasil

-

ABSTRACT:- H. J. Zassenhaus has conjectured t h a t every t o r s i o n u n i t i n an i n t e g r a l group r i n g o f a f i n i t e group i s r a t i o n a l l y conjugate t o a t r i v i a l one. I n t h i s paper, we survey the known r e s u l t s regarding t h i s conjecture; namely, t h a t i t holds when G i s a n i l p o t e n t class 2 group o r a s p l i t metacyclic group G = A M B w i t h some r e s t r i c t i o n s on the order o f A and 6.

1.

INTRODUCTION L e t G be a f i n i t e g r o u p .

We d e n o t e b y ZG t h e g r o u p r i n g o f G

o v e r t h e r i n g Z o f r a t i o n a l i n t e g e r s a n d b y U(ZG) t h e g r o u p o f u n i t s o f ZG. Let

E:

ZG

+ Z denote t h e augmentation function. V(ZG) = { a 6 U(ZG)

I

€(a) =

i s c a l l e d t h e g r o u p o f nohmaLized u n i t b o f ZG.

The s e t

11 I t i s e a s i l y seen

that

F o r a g i v e n g r o u p X,

eLementb o f X , G.

Higman,

i.e.

we s h a l l d e n o t e b y TX t h e s e t o f t o a n i o n

t h e s e t o f elements o f f i n i t e o r d e r i n X.

i n t h e f i r s t c l a s s i c a l p a p e r on u n i t s i n group rings

[6] showed, among o t h e r t h i n g s , t h a t i f G i s a b e l i a n , t h e n e v e r y t o r s i o n u n i t o f ZG i s t r i v i a l , o t h e r words,

i.e.

o f the form

+g, g 6 G or,

in

that TV(ZG) = G

When G i s n o t a b e l i a n ,

an o b v i o u s way t o e x i b i t new u n i t s o f

f i n i t e o r d e r i s t o c o n s i d e r t h e c o n j u g a t e s o f t h e t r i v i a l ones by e l e m e n t s u E U(ZG).

O f course,

t h e s e u n i t s w o u l d h a v e t h e same

o r d e r a s t h e e l e m e n t s i n G. D.S.

Berman C21 showed t h a t ,

o r d e r o f u i s a d i v i s o r o f ]GI.

i n general, Also,

S.K.

i f u € TV(BG) t h e n t h e S e h g a l has shown t h a t

C.P. Milies

180

If u E TV(ZG) i s a u n i t whose o r d e r i s a power o f a r a t i o n a l prime, t h e n t h e r e e x i s t s an element g 6 supp(u) such t h a t o ( u ) = o ( g ) (see [15,

theorem VI.2.13).

Hence, one m i g h t h a v e a hope t h a t evehy e l e m e n t u 6 TV(ZG) c o n j u g a t e t o an e l e m e n t i n G.

is

T h i s q u e s t i o n was f i r s t c o n s i d e r e d b y I. Hughes a n d K. R. P e a r s o n [7], who a t t r i b u t e i t t o P r o f . H. Zassenhaus, a n d showed t h a t t h e r e e x i s t t o r s i o n e l e m e n t s i n V(ZS,) w h i c h ahe n o t conjugate, i n ZS,, t o a t r i v i a l u n i t . S h o r t l y a f t e r w a r d s , C . P o l c i n o M i l i e s

[lo]

showed t h a t t h e same happens i n ZD4.

N e v e r t h e l e s s , i t i s easy

i n b o t h c a s e s , t h a t t h i s i s n o t s o i f we a l l o w conjugation t o t a k e p l a c e i n s i d e (PG, t h e r a t i o n a l g r o u p a l g e b r a o f t h e g i v e n t o show,

group. F i n a l l y , H.

Zassenhaus s t a t e d h i s c o n j e c t u r e p r e c i s e l y

n [181:

(1.1) L e t u E TV(ZG). Then, t h e r e e x i s t s a n i n v e r t i b l e e ement 1 6 Q G a n d a n e l e m e n t g E G s u c h t h a t ~ 1 - U C I = g.

CI

I n t h i s c a s e , we s h a l l s a y t h a t u i s r a t i o n a l l y c o n j u g a e t o g and w r i t e u%g. The f i r s t p o s i t i v e r e s u l t i n r e g a r d t o t h i s c o n j e c t u r e i s q u i t e recent. I t i s due t o A. K. B h a n d a r i and I . S . L u t h a r [ 3 ] who p r o v e d t h a t t h e c o n j e c t u r e h o l d s when G i s a m e t a c y c l i c g r o u p o f o r d e r pq, where p and q a r e b o t h p r i m e s and q d i v i d e s ( p - 1 ) .

2 . SOME GENERAL RESULTS B e f o r e p r o c e e d i n g t o s u r v e y t h e known r e s u l t s i n r e g a r d t o t h i s conjecture,

l e t us r e c a l l s e v e r a l well-known f a c t s .

A. Whitcomb [ 1 7 ] showed t h a t i f G i s a m e t a b e l i a n g r o u p , i . e . i f i t c o n t a i n s a normal subgroup A such t h a t b o t h A and G/A a r e abelian,

then G i s determined by i t s i n t e g r a l group r i n g .

o t h e r words,

i f H i s a n o t h e r group such t h a t Z G : Z H

L e t us denote by I ( A )

t h e n G = H.

t h e kernel o f t h e augmentation cA:ZA-+Z

and b y I ( G , A ) t h e k e r n e l o f t h e map ZG G/A. n a t u r a l homomorphism G

+

ZG/A i n d u c e d b y t h e

-+

I t i s e a s i l y seen t h a t :

(2.1)

I(G,A)

In

= ZG.I(A) =

I

E

Y a ( a - l ) l y a 6 ZG aA

I.

Tonion units in group rings

181

T h e essential part o f w h i t c o m b ' s a r g u m e n t c o n s i s t s in s h o w i n g t h a t , f o r e a c h u n i t u E TV(PG), t h e r e e x i s t s a u n i q u e e l e m e n t g € G such that u = g (mod I(G).I(A)).

(2.2)

This i s derived from the following remark which we record for l a t t e r use: G i v e n e l e m e n t s a,b E A

w e h a v e that:

(a-l)(b-1)

=

ab-a-b+l

(ab-1)-(a-1)-(b-1)

=

and hence: (a-l)+(b-l) :( a b - 1 )

( m o d I(G)I(A))

= a-1 - 1

(mod I (GI IA( 1 )

(2.3)

-(a-l)

A l s o , w e recall t h e f o l l o w i n g result. (2.4) P r o p o s i t i o n (G. C l i f f , S.K. Sehgal and A. W e i s s ['I]) - T h e g r o u p u(I+I(G)I(A)) i s torsion-free.

Hence , we have : (2.5) C o r o l l a r y Then

L e t u € T V ( P G ) a n d g E G be a s in f o r m u l a (2.2). o(u)

Proof

S e t o(u)

=

= o(g)

n, o(g) u m -= g m

=

m.

Then:

( m o d I(G)I(A))

S i n c e gm = 1 , w e h a v e t h a t um E 1 + I(G)I(A) a n d , b e c a u s e o f P r o p o s i t i o n (2.41, w e m u s t h a v e u=l. H e n c e nlm. In a s i m i l a r w a y w e a l s o g e t t h a t mln.

A

W e n o w t u r n o u r a t t e n t i o n t o s p l i t m e t a b e l i a n g r o u p s i.e. w e shall a s s u m e t h a t G is a s e m i d i r e c t p r o d u c t o f t h e f o r m A M B w h e r e A a n d B a r e abelian. Hence, w e h a v e a s p l i t e x a c t

s e q u e n c e o f groups:

l + A + G = B + l w h i c h , in turn, i n d u c e s a s e q u e n c e

o

-+

I(G,A)

-+

PG

ZB

-+

0

S i n c e t h e s e q u e n c e splits, t h e r e s t r i c t i o n t o u n i t s V(ZG)+V(ZB) is o n t o and t h u s w e h a v e a n o t h e r s p l i t s e q u e n c e : 1

-+

v(~+I(G,A))+V(ZG) 2 V(PB)

-+

I

182

C. P. Milies

Hence, w e see that V ( H G ) , groups: (2.6)

is again a semidirect product o f

V ( Z G ) = v(I+I(G,A))

Y

V(ZB).

We shall now show that the conjecture holds at least f o r some of t h e units in Z G . T h e technique involved goes back t o H i l b e r t ‘ s theorem 90. First we need the following lemma. (2.7) Lemma Let G be a finite group and let A be a normal p-subgroup o f G , where p is a rational prime. Then any element o f t h e form a = t+6, where t € Z i s not divisible by p a n d 6 6 I ( G , A ) , is invertible in Q G . Proof Since QG is semisimple, artinian, if cx was not invertible it should be a zero divisor. Hence, there would exist an element 6 E Z G such that: thus , Since (t,p) = 1 , w e can find integers r,s such that rt+ps so, if w e work modulo p (i.e. if we go to22 G I , w e obtain: P 6 E -rg6 (mod p). It follows immediately that: 3 E +r n 6 6 n Since I ( G , A ) hence

(mod p ) , f o r all n € N .

is nilpotent in Z G , w e see that 6 P 8 = pB1

with

= 1

B1

E

0

(mod p )

eZG.

Then, again w e must have that ~~a = 0 and the same argument ~ , B2€ZG. above shows that 1 3 ~ is o f t h e form B~ = p ~ with Inductively, w e would prove that all powers o f p divide the coefficients of 6.a contradiction. A Theorem ( C . Polcino Milies and S.K. Sehgal C 1 1 1 ) Let be such that (o(u), I A I ) = 1 . Then u is rationally conjugate to an element b E 8. (2.8)

G = A w B

Proof We shall proceed by induction on t h e number o f primes dividing I A I . So, let us first assume that A is a p-group, f o r some rational prime p. Let u € V ( Z G ) be a s in the statement o f t h e theorem and write u=vw, with v E V ( l + I ( G , A ) ) and w € V ( Z B ) and set t = o ( u ) . writing v =wvw”

Torsion units in group rings

183

we h a v e t h a t : u t = v.v

W

...

.v

W

t-1

.wt = 1

Hence:

wt = 1 v.vw.

... .vw t - 1

NOW, s e t :

z

...+v.vw. ... . v w

= l+v+v.vW+

Then

W

wzw-1 = l + v

= 1.

w

.

+. .+v .v

w2

and hence vwzw-1 = v+v

W

+...+

v.v

W

t-2

. ....v ,t-1

.....v ,t - 1

= z .

NOW, s i n c e v r l (mod I ( G , A ) ) , i t f o l l o w s t h a t z t (mod I ( G , A ) ) t h u s , i t e x i s t s 6 E I ( G , A ) s u c h t h a t z = t + 6 and, s i n c e ( t , p ) = 1 , t h e p r e v i o u s lemma shows t h a t z i s i n v e r t i b l e Consequently: z - 1 uz = z - ' ( v w ) z

= w

Since B i s abelian,

,

with

w 6 TV(ZB).

t h e r e s u l t b y G.

i n t r o d u c t i o n shows t h a t

Higman m e n t i o n e d i n t h e

w = b , f o r some b 6 B.

F o r t h e i n d u c t i o n s t e p , w r i t e A = A1

x

A2

a n d A 2 a p ' - g r o u p f o r some r a t i o n a l p r i m e p. we h a v e t h a t G1 = A 1 M G 2 . G 2 = I f uETV(ZG) i s such t h a t ( o ( u ) , l A l )

,

w h e r e A 1 i s a p-group Then w r i t i n g

= 1 then a l s o (o(u),IAll)=l

a n d we c a n u s e t h e a r g u m e n t a b o v e t o show t h a t t h e r e e x i s t s an From t h e i n v e r t i b l e e l e m e n t z E BG s u c h t h a t z - l u z = w E TV(ZG2). i n d u c t i o n h y p o t h e s i s , s i n c e G2 = A 2 * B y w i t s e l f i s c o n j u g a t e t o

A

an e l e m e n t g 6 G 2 a n d t h e p r o o f i s c o m p l e t e d .

We s t i l l h a v e a n o t h e r i n f o r m a t i o n a b o u t u n i t s i n V ( i 2 G ) w h i c h i s worth mentioning. (2.9)

Proposition

divisor of Proof

then the order o f u i s a

If u€TV(l+I(G,A))

IAI.

Assume u=1+6

w i t h 6 6 I(G,A).

u = 1+lya(a-l) a NOW, e a c h ya i s o f t h e f o r m :

Then u i s o f t h e f o r m :

,

y a €ZG

-l)a.+a.). J J

C.P. Milies

184

Hence:

and t h u s

and f o r m u l a s (2.3)

show t h a t

1y a ( a - l )

E a,-I

( m o d I ( G ) I ( A ) ) f o r some a, E A

a

and, h e n c e : u: a NOW, c o r o l l a r y ( 2 . 5 ) of

(mod

0

I(G).I(A)).

shows t h a t o ( u ) = o ( a o ) , w h i c h i s a d i v i s o r

IAl

A

No o t h e r g e n e r a l r e s u l t i s known s o f a r , s o we s h a l l h a v e t o c o n s i d e r some s p e c i a l c a s e s . However, we s h a l l f i r s t show how r e p r e s e n t a t i o n s can be used t o h e l p t o s o l v e t h i s problem. 3.

CONNECTION WITH REPRESENTATIONS

F i r s t , we s h a l l p r o v e a r e s u l t t h a t e n a b l e s u s t o c o n s i d e r t h e problem i n a b i g g e r f i e l d . (3.1)

Lemma

group.

L e t kcK b e f i e l d s o f c h a r a c t e r i s t i c 0 a n d G any f i n i t e

I f t w o g i v e n e l e m e n t s a , B € kG a r e c o n j u g a t e i n KG t h e n

t h e y a r e a l s o c o n j u g a t e i n kG. P r o o f L e t x,8 the equation

be g i v e n e l e m e n t s i n kG a n d l e t u s f i r s t c o n s i d e r ax = x B

with x = 1x.g i i i

w h e r e xi,

15i6n

a r e unknowns.

I f we w r i t e down e x p l i c i t l y b o t h s i d e s o f t h e e q u a t i o n , we s h a l l o b t a i n a l i n e a r system o f t h e form:

with X =

[

MX = 0 X.

and M € knxn

,

where n = I G I .

Since t h e assumptions o f t h e theorem i m p l y t h a t t h e r e e x i s t s a non t r i v i a l s o l u t i o n i n Knxl t h e n we m u s t h a v e d e t M = O and hence t h e r e e x i s t s a non t r i v i a l s o l u t i o n a l s o i n knxl. A c t u a l l y , we k c a n f i n d t i n d e p e n d e n t v e c t o r s vl,...,vtE such t h a t e v e r y tx1 s o l u t i o n i n knxl o f t h e s y s t e m i s o f t h e f o r m :

Torsion units in group rings

v = k v 1 1

, xi

+ ...+A~v~

185

E k

,

lsisn.

We w i s h t o show f i r s t t h a t a t l e a s t f o r one o f t h e s e s o l u t i o n s

v

=[.:.I]

t h e c o r r e s p o n d i n g e l e m e n t u = laigi

is

not

a zero d i v i s o r

i n kG? Hence assume t h a t a l l o f t h e m a r e . r e p r e s e n t a t i o n o f KG.

L e t T be t h e r e g u l a r

We s h a l l d e n o t e b y T ( v i ) ,

f o r briefness,

t h e r e p r e s e n t a t i o n o f t h e e l e m e n t c o r r e s p o n d i n g t o vi

i n kG.

Then :

S i n c e we a r e a s s u m i n g t h a t T ( v ) i s a z e r o d i v i s o r , f o r e v e r y c h o i c e o f A1,...,Xt6

k

we h a v e : t det( hiT(vi)) i=l

f

= 0

t

S i n c e k i s i n f i n i t e , t h i s means t h a t t h e p o l y n o m i a l d e t ( i n the

... , X t

i n d e t e r m i n a t e s X1,

1 XiT(vi))

i=l However, we know t h a t t h e r e

i s 0.

e x i s t s a s o l u t i o n o f t h e s y s t e m , i n K n X 1 , s a y ( rl,...yrt) t h a t t h e c o r r e s p o n d i n g element i s i n v e r t i b l e , and t h u s

such

t det(

1 riT(vi))

#

0

i=1 a contradiction.

Hence, we m u s t a l s o h a v e a s o l u t i o n Al,...,ht€ k such t h a t t h e c o r r e s p o n d i n g e l e m e n t u i s n o t a z e r o d i v i s o r and v e r i f i e s : au =

UB

.

S i n c e kG i s a r t i n i a n , s e m i s i m p l e , a n e l e m e n t u E kG w h i c h i s n o t a z e r o d i v i s o r i s i n v e r t i b l e . Thus u ' l a u = B a n d a,B a r e c o n j u g a t e i n kG, a s d e s i r e d . A A s an i m m e d i a t e c o n s e q u e n c e ,

i t follows:

Corollary Let u€TV(ZG). To show t h a t u i s r a t i o n a l l y c o n j u g a t e t o an e l e m e n t g E G i t s u f f i c e s t o show t h a t t h e r e e x i s t s (3.2)

an e l e m e n t a 6 U ( l f G ) s u c h t h a t a - l u a = g. Hence, we h a v e : (3.3)

Lemma

L e t u € TV(BG).

Then u i s r a t i o n a l l y c o n j u g a t e t o an

e l e m e n t g € G i f and o n l y i f f o r e v e r y i r r e d u c i b l e complex r e p r e s e n t a t i o n T o f G t h e m a t r i c e s T ( u ) and T ( g ) a r e c o n j u g a t e .

C.P. Milies

186

From W e d d e r b u r n ' s t h e o r e m we know t h a t OG i s o f t h e f o r m : t (IG = @ a n * x n i=l i i

Proof

L e t Ti

be t h e i r r e d u c i b l e r e p r e s e n t a t i o n corresponding t o t h e

i - t h s i m p l e component, a n d assume t h a t t h e r e e x i s t s Ai 6

anax,,, I

s u c h t h a t ATITi(u)Ai

= Ti(g).

I

t

We c a n assume t h a t t h e i s o m o r p h i s m $ i s g i v e n b y + =

@ Ti.

i=1 i f a € O G i s t h e e l e m e n t c o r r e s p o n d i n g t o (A l,...,At) i s o m o r p h i s m , we h a v e t h a t a - ' u a = g.

Then,

i n the

L e t u 6 V(ZG) and g 6 G b e e l e m e n t s s u c h t h a t k k o(u) = o(g). I f x(u = x(g f o r a l l p o s i t i v e i n t e g e r s k and a l l complex i r r e d u c i b l e c h a r a c t e r s x t h e n u i s c o n j u g a t e t o g i n (IG. (3.4)

Lemma

Proof I f T i s the i r r e d u c i b l e representation corresponding t o the character x , then: g

+

T(g)

and

g -+ u -+ T ( u )

a r e two r e p r e s e n t a t i o n s o f t h e c y c l i c group same c h a r a c t e r ;

which a f f o r d t h e

h e n c e , t h e y a r e s i m i l a r and t h u s T ( g ) s T ( u ) .

S i n c e t h i s happens f o r a l l i r r e d u c i b l e c o m p l e x r e p r e s e n t a t i o n s , t h e r e s u l t f o l l o w s f r o m lemma ( 3 . 3 ) . 4.

A

NILPOTENT CLASS 2 GROUPS

J. R i t t e r and S.K.

[ 1 3 1 h a v e shown t h a t t h e Zassenhaus

Sehgal

Conjecture holds f o r n i l p o t e n t c l a s s 2 groups. t h e i r r e s u l t i s as f o l l o w s : (4.1)

Theorem

- L e t X be

kernel o f

L e t G be a n i l p o t e n t c l a s s 2 g r o u p a n d s e t

L e t g b e t h e u n i q u e e l e m e n t o f G s u c h t h a t u z g (mod Then u i s r a t i o n a l l y c o n j u g a t e t o g.

u 6 TV(ZG). I(G)IG')). &f

x.

one such i r r e d u c i b l e c h a r a c t e r a n d l e t N a G be t h e F o r a n e l e m e n t uGZG we s h a l l d e n o t e b y

i n ZG/N. Since

Write

More p r e c i s e l y ,

I(G)I(G')

-

= I(G)I(G')

-u = g-

we h a v e t h a t

(mod I(g)I ( 5 ' ) 1.

u'

i t s image

Torsion units in group rings

I a7

I f a; # 0 f o r same i n t h e c e n t e r o f F , then a well-known t h e o r e m o f Berman [ 2 1 shows t h a t a c t u a l l y u' = i . B e c a u s e o f t h e u n i q u e n e s s o f t h e c o r r e s p o n d e n c e m o d u l o I(G)I(7;') we m u s t

have:

-u

h e n c e , o b v i o u s l y X(u:)

-

-

= x = g = X(g) i n t h i s case.

Then, we c a n assume t h a t a l l e l e m e n t s i n t h e e x p r e s s i o n o f

u'

are noncentral. Notice that also

?J

m u s t be n o n - c e n t r a l ,

s i n c e o t h e r w i s e we

w o u l d have

i-'. u' E 1 and

i'

mod(I(F)I(i*;')

would be o f f i n i t e o r d e r ,

Now, T.R.

Berger [11

c o n t r a d i c t i n g (2.4).

h a s shown t h a t i r r e d u c i b l e c o m p l e x

characters o f n i l p o t e n t c l a s s 2 groups vanish o u t s i d e t h e center. Hence we h a v e : X(G) = 0

Thus, x ( g ) = x ( u ) a l s o i n t h i s c a s e . Finally,

since

(mod I ( G ) I ( G ' ) ) ,

U

E

(mod ~ I(G)I(G'))

a l s o i m p l i e s t h a t uk,gk

A

o u r p r o o f i s c o m p l e t e b e c a u s e o f Lemma ( 3 . 4 ) .

The r e s u l t a b o v e m i g h t 3 u g g e s . t

that,

i n the metabelian case,if

u =g(mod I ( G ) . I ( A ) )

and t h e Zassenhaus c o n j e c t u r e h o l d s , t h e n u i s r a t i o n a l l y c o n j u g a t e p r e c i s e l y t o g. However, t h i s i s n o t t h e case. A c t u a l l y G.

Cliff,

S.K.

Sehgal and A.

Weiss gave i n [4]

a

w h o l e f a m i l y o f i d e a l s Iks u c h t h a t f o r e a c h u n i t u € T V ( Z G ) t h e r e e x i s t s an e l e m e n t g k E G s u c h t h a t u - g k (mod I k ) . I n t h e v e r y same p a p e r [ 1 3 ]

,

J . R i t t e r a n d S.K.

Sehgal gave

t h e f o l l o w i n g example: S e t D,, Then,

=

that:

< a > x cb>

a l l t h e i d e a l s Ikc o i n c i d e w i t h e i t h e r I ( G ) I ( A )

.

w h e r e A = 2 3 4 3 The e l e m e n t u = -a +a +a + ( - l + a ) b

I(A)I(G),

=

. or

i s a u n i t of o r d e r 5 such

C.P. Milies

188

uza3

mod I ( A ) I ( G )

u = a z mod I ( G ) I ( A ) . On t h e o t h e r hand, t h e y show t h a t u i s r a t i o n a l l y c o n j u g a t e t o b o t h a and a

4

.

5 . METACYCLIC G R O U P S As we m e n t i o n e d i n t h e i n t r o d u c t i o n , t h e f i r s t p o s i t i v e r e s u l t o n t h e Zassenhaus c o n j e c t u r e was o b t a i n e d b y A.K. B h a n d a r i and I. S.

L u t h e r [3]

who p r o v e d

G w h e r e p,q

it

f o r t h e m e t a c y c l i c group:

= = u < b >

1 (mod p ) .

T h i s r e s u l t has b e e n e x t e n d e d t o t h e f o l l o w i n g . Theorem (C. P o l c i n o M i l i e s , J . R i t t e r , S . K . S e h g a l C121) L e t G be t h e s p l i t m e t a c y c l i c g r o u p G = c a > M c b > w i t h ( o ( a ) , o ( b ) ) = 1. Then e v e r y u n i t u € TV(ZG) i s r a t i o n a l l y c o n j u g a t e t o an e l e m e n t g€G. A c t u a l l y , t h i s r e s u l t was a c h i e v e d i n s e v e r a l s t e p s .

With the

n o t a t i o n s o f t h e t h e o r e m , we h a v e : The Zassenhaus c o n j e c t u r e h o l d s i f i s a p - g r o u p , a p i - g r o u p and t h e a c t i o n o f b on i s f a i t h f u l C111. I n C131, t h e r e s t r i c t i o n a b o u t t h e f a i t h f u l l n e s s o f t h e a c t i o n was removed a n d i t was a l s o shown t h a t t h e c o n j e c t u r e h o l d s when o ( a ) = n an

odd i n t e g e r a n d o ( b ) = q, a p r i m e n o t d i v i d i n g n .

The f i n a l r e s u l t was o b t a i n e d i n C123. To g i v e an i d e a o f t h e m e t h o d s i n v o l v e d i n t h e p r o o f , l e t u s c o n s i d e r t h e f i r s t c a s e , w h e r e o ( a ) = pm a n d b a b - l = a’ and o ( j ) = s i n Z / p m Z .

,

o ( b ) = s , o(p,s)

G i v e n a n e l e m e n t g € G we c a n w r i t e i t i n a€

and

B€

.

= 1

t h e f o r m g =a3 where

.

h I f 3 # 1,itis o f the form 8 = b As b e f o r e , we see t h a t : , a j h ( ~ - l ) . B s = . ( l + j h + ...+jh(s-1) g s = a.a j h

. ...

And we have:

( 1 - 5 h ) ( l + j h +...+j h(s-l)) h

I f p l ( 1 - j ) we c a n w r i t e j h = l + k p , h e n c e

= l-jhs E

0 (mod p m ) ,

Torsion units in group rings

189

j h p= l + k l p 2

and, i n d u c t i v e l y

jh p m - l = I So we w o u l d h a v e s l h p m "

and s i n c e ( s , p ) = l , Then p t ( l - j h ) s o

a contradiction.

l + j h+

... +

T h i s a r g u m e n t shows t h a t ,

j h(s-l)

z 0

we h a v e t h a t s l h ,

(mod pm) we h a v e t h a t

f o r any element g € G

e i t h e r o ( g ) l s o r o(g)lpm. B e c a u s e o f C o r o l l a r y (2.51, t h e same i s t r u e f o r a u n i t u 6 TV(EG) a n d Theorem (2.8) shows t h a t we o n l y n e e d t o c o n s i d e r u n i t s u such t h a t o ( u ) l p m .

i f u=v.w w i t h v € V ( I + I ( G , < a > ) ) a n d w 6 V ( Z < b > ) , w i t h W * 1, t h e i n i t i a l p a r t o f t h e a r g u m e n t i n t h e o r e m (2.8) shows t h a t ( O ( u ) , s ) 1. Notice also that,

Hence, we o n l y h a v e t o c o n s i d e r u n i t s u € V ( I + I ( G , < a > )

and

P r o p o s i t i o n (2.9) shows t h a t t h e s e a r e c e r t a i n l y p - e l e m e n t s . T o t r e a t t h i s c a s e we s h a l l u s e r e p r e s e n t a t i o n s . i n Curtis-Reiner,

[5,p.3363

I t i s shown

that the irreducible representations

of G a r e among t h e ones g i v e n b y :

where

5 i s a pmth r o o t o f u n i t y . s-I

Write u i n the form u =

1 ak(a)bk

w h e r e .,(a)

€Z

lskis-I.

k=o If 4:

q(S)

+

q ( 5 ) denotes t h e automorphism such t h a t $ ( E l = S j ,

i t i s e a s y t o see t h a t :

Ti(u)

Now, we n o t i c e t h a t

=

C.P. Milies

190

T i ( b ) Ti(u) Ti(b)-’

=

T @i ( u )

m S i n c e T i ( u ) ~ 1 , t h e m a t r i x is diagonal izable a n d , b e c a u s e o f t h e o b s e r v a t i o n above. w e s e e t h a t if tri is an e i g e n v a l u e o f r.j is a g a i n an eingenvalue. Thus: Ti(u), t h e n 5 1

Ti(u)

T h e r e s t o f t h e proof, w h i c h i s still r a t h e r long, c o n s i s t s in showing t h a t a c t u a l l y t h e e l e m e n t a r i d o e s n o t depend o n t h e r e p r e s e n t a t i o n T i considered. In t h e general c a s e additional d i f f i c u l t i e s a r i s e , s i n c e t h e r e d u c t i o n t o t h e c a s e w h e r e u € TV(l+I(G,)) is m o r e d e l i c a t e and t h e r e p r e s e n t a t i o n s o f G up t o rational e q u i v a l e n c e , a r e given by: 0 (a) Td

=

1

. . * . *

IIp

‘dXtd

I 0

tdXtd

w h e r e d r u n s o v e r t h e d i v i s o r s o f n, 5 is a fixed n-th root o f unity, t d is t h e o r d e r o f j in Z/$, q is a p r i m i t i v e t/td-th root o f unity a n d LI = 0,1,2,...,t/td-l. 6.

R E C E N T RESULTS

Recently, s o m e r e s u l t s r e g a r d i n g t h e Z a s s e n h a u s C o n j e c t u r e have been o b t a i n e d f o r s o m e special c l a s s e s o f s p l i t m e t a b e l i a n g r o u p s , w h i c h w e w i s h t o mention. T h e o r e m ( S . K . Sehgal a n d A . W e i s s [16]) L e t G = A M B be a s p l i t m e t a b e l i a n group, w h e r e A is an e l e m e n t a r y a b e l i a n p-group and B is a n y a b e l i a n group. If B a c t s f a i t h f u l l y i r r e d u c i b l y o n A t h e n t h e Z a s s e n h a u s C o n j e c t u r e holds f o r ZG

.

T h e o r e m ( 2 . Marciniak, J. Ritter, S . K .

Sehgal and A . W e i s s [ 8 ] )

191

Torsion units in group rings

L e t G = A w B be a s p l i t m e t a b e l i a n g r o u p .

Then t h e Z a s s e n h a u s

c o n j e c t u r e h o l d s f o r HG i n t h e f o l l o w i n g t w o c a s e s :

(1)

When A i s a b e l i a n ,

B i s o f p r i m e o r d e r q and q < p

f o r every

prime p d i v i d i n g I A ( . ( i i ) When A =

,

B i s a b e l i a n o f o r d e r m and m < p

prime p d i v i d i n g

7.

f o r every

IAl.

FINAL R E M A R K T h e r e i s a n o t h e r c o n j e c t u r e due t o H.J.

Zassenhaus w h i c h i s

s t r o n g e r t h a n t h e one we h a v e c o n s i d e r e d , n a m e l y : E v e r y f i n i t e s u b g r o u p o f u n i t s i n V(ZG) i s

r a t i o n a l l y conjuga-

t e t o a s u b g r o u p o f G. I f we r e s t r i c t o u r s e l v e s t o c o n s i d e r o n l y m a x i m a l s u b g r o u p s o f V(ZG) i . e . s u b g r o u p s H c V(ZG) s u c h t h a t = [ G I t h e n some

IHI

r e s u l t s a r e known. F o r example,

i n 1 9 6 9 , S.K.

S e h g a l [ 1 4 1 showed t h a t t h i s i s

Peterson [ g 1 I n his showed t h a t i t i s a l s o t r u e f o r t h e s y m m e t r i c g r o u p s Sn.

t r u e f o r n i l p o t e n t c l a s s 2 g r o u p s a n d i n 1 9 7 6 G. c o n f e r e n c e i n t h i s same m e e t i n g , P r o f . K.W.

Roggenkamp h a s

announced t h a t t h e r e s u l t a l s o h o l d s f o r a r b i t r a r y n i l p o t e n t groups. No r e s u l t s a r e known i n r e g a r d t o t h e c o n j e c t u r e i n i t s f u l l g e n e r a l i t y as s t a t e d above. REFERENCES

[ l ] T.G. B e r g e r - C l a s s 2 p - g r o u p s as p o i n t f r e e a u t o m o r p h i s m group, 1 L L . J . Math. 14 (19701, 121-149.

-

Berman On t h e e u a t i o n X m = l i n an i n t e g r a l g r o u p r i n g , U k h a n i a n . Math. Zh. 7 7 1 9 5 5 ) , 253-261.

[ 2 3 S.D.

-

[ 3 ] A.K.

B h a n d a r i and I . S . L u t h a r Torsion u n i t s o f i n t e g r a l 1 7 (1983), g r o u p r i n g s o f m e t a c y c l i c g r o u p s , J. 06 Nurnbeh T h e o h y -

270-283.

-

S.K. S e h g a l a n d A . Weiss U n i t s o f i n t e g r a l group r i n g s o f m e t a b e l i a n g r o u p s , J . o h ALgebha 73 ( 1 9 8 1 1 , 1 6 7 - 1 8 5 .

[ 4 ] G. C l i f f ,

-

W . C u r t i s a n d I. R e i n e r Rephenentation T h e o k y 06 Finite Ghoupb and Abbociative Atgebhan, I n t e r s c i e n c e , New Y o r k , 1962.

[ 5 ] C.

-

The U n i t s o f g r o e p r i n g s , Phoc. London Math. S a c , (2) 46 ( 1 9 4 0 ) , 2 3 1 - 2 4 8 . [ 7 ] I. Hughes a n d K.R. P e a r s o n - The g r o u p o f u n i t s o f t h e i n t e g r a l g r o u p r i n g Z S j , Canad. Math. BULL. 1 5 (19721, 529-534.

[6] G. Higman

C.P. Milies

192

[8]

[9]

[lo] [113

Z. M a r c i n i a k , J . R i t t e r , S.K. S e h g a l a n d A. Weiss - T o r s i o n u n i t s i n i n t e g r a l g r o u p r i n g s o f some m e t a b e l i a n g r o u p s 11, (preprint). 6. P e t e r s o n A u t o m o r p h i s m s o f t h e i n t e g r a l g r o u p r i n g ZS,, Phoc. Ameh. Math. S o c . 59 ( 1 9 7 6 ) , 1 4 - 1 8 . C. P o l c i n o M i l i e s The u n i t s o f t h e i n t e g r a l g r o u p r i n g ZD4, 8 0 L . S O C . 8hahiLeiha de M a t . , 4 ( 1 9 7 2 ) , 8 5 - 9 2 . C . P o l c i n o M i l i e s a n d S.K. S e h g a l - T o r s i o n u n i t s i n i n t e g r a l r o u p r i n g s o f m e t a c y c l i c g r o u p s , J. 06 Numbeh Theohy, 19 1984) , 103-114. C . P o l c i n o M i l i e s , J . R i t t e r a n d S.K. S e h g a l On a c o n j e c t u r e o f Zassenhaus o n t o r s i o n u n i t s i n i n t e g r a l g r o u p r i n g s 11, Phoc. Ameh. Math. S a c . ( t o a p p e a r ) . J . R i t t e r and S.K. S e h g a l On a c o n j e c t u r e o f Zassenhaus on 264 (1983) t o r s i o n u n i t s i n i n t e g r a l g r o u p r i n g s , Math. A m , 257-270. S. K. S e h g a l On t h e i s o m o r p h i s m o f i n t e g r a l g r o u p r i n g s I , Canad. 1. 06 Math. 21 ( 1 9 6 9 ) , 4 1 0 - 4 1 3 . S.K. S e h g a l - T o p i c 4 i n G h o u p R i n g n , M a r c e l D e k k e r , New Y o r k 1968. S.K. S e h g a l a n d A . Weiss Torsion u n i t s i n i n t e g r a l group r i n g s o f some m e t a b e l i a n g r o u p s ( p r e p r i n t ) A. Whitcomb - The g r o u p r i n g p r o b l e m , Ph.D. T h e s i s , U n i v e r s i t y o f Chicago, 1968. H. J . Zassenhaus - On t h e t o r s i o n u n i t s o f f i n i t e g r o u p r i n g s , S t u d i e s i n Mathematics ( i n honor o f A. Almeida Costa), I n s t i t u t o d e A l t a C u l t u r a , L i s b o a , 1 9 7 4 , p.p. 1 1 9 - 1 2 6 .

-

-

4

[12]

[13]

[14] [I51 [16] [17] [l8]

-

-

-

-

Group and Semigroup Rings G. Karpilovsky (ed.) 0 Elsevier Science Publishers B.V.(North-Holland),1986

193

ON THE NILPOTENCY I N D E X OF THE RADICAL OF A GROUP ALGEBRA I X KAORU MOTOSE Department of Mathematics, Faculty of Science Hirosaki U n i v e r s i t y , Hirosaki, Japan

Dedicated t o Professor Katsumi Numakura on h i s 60th b i r t h d a y

1. INTRODUCTION

Let

be a f i x e d prime number, l e t

p

a p-Sylow subgroup

be a f i n i t e p-solvable group with

G

be a f i e l d of c h a r a c t e r i s t i c

K

P, l e t

be t h e nilpotency index of t h e r a d i c a l

J(KG)

p

t(G)

and l e t

of a group a l g e b r a

KG

of

G

K.

over

It follows from Morita's theorem [ 2 , Theorem 21 and Villamayor's theorem

G of p-length 1. Hence, i n o f Jennings' theorem [l, Theorems 3.7 and 5.51, t h e number t ( G ) i s

[ 5 , Theorem 11 t h a t virtue

t ( G ) = t(P)

f o r a group

G.

computable f o r such a group

However, groups o f p-length a t l e a s t 2 have

no longer t h i s property s i n c e t h e r e i s an example with group with

G of p-length 2 ( s e e [31). t ( G ) > t ( P ) . The i n e q u a l i t y

t ( G ) < t(P) for a

I n s p i t e of t h i s , we can f i n d no examples t(P)

2 t(G)

seems t o be i m p l i c i t l y

c o n j e c t u r e d b y some mathematicians who a r e i n t e r e s t e d i n t h e n i l p o t e n c y index of t h e radical.

W e have very few r e s u l t s about t h e numbers

t(G)

f o r groups

of p-length 2 ( s e e [ 3 ] and [ b ] ) . I n t h i s paper we s h a l l study t h e number

G of p-length 2.

f o r a s p e c i a l group t(P)

f o r t h e following group Let

q = pr, l e t

respectively, l e t element o f o r d e r

m m

F

E

and

be f i n i t e f i e l d s of (qp

-

( = , P = <W, U>

(walk= ua(k)wk

1

h E P>) = {ua

F over E.

5

-

qp

and

1) and l e t

F.

q

elements c

be an

We s h a l l consider t h e

F (see [ 3 1 ) .

V = {vb: x and

2. DETERMINATION OF Since

l)/(q

i n t h e m u l t i p l i c a t i v e group of

U = {u : x + x ia Cw: x

t(G)

G.

be a d i v i s o r of

next permutation groups on

W =

More e x p l i c i t l y , we s h a l l show

G

t(G)

I

CW,

xb

a(k) = a

I

b E },

U, V>.

t (P)

+

. .. + a'

+ T r ( F ) = E l where Tr

where a E

G =

*

a'

k-1

, we

have

i s t h e t r a c e function

K. Motose

194

-

PROPOSITION 1. t ( p ) = p r ( p 2 PROOF.

Let

+ p.

1)/2

b e an endomorphism of the a d d i t i v e group of

defined k sn(x) = l ~ = o ( - l ) k ( ~ ) x q

s

-

F

xq. It f o l l o w s from i n d u c t i o n t h a t by s ( x ) = x 1 (mod p ) . S i n c e and hence sP-l = Tr by t h e e q u a t i o n ( - l ) k ( p i l ) k k and sk ( F ) c o n t a i n s s ( F ) / ( K e r s n s (F)) i s isomorphic t o sk+’(F) = T r ( F ) = E = Ker s

q

k = 0 , 1,

for

k = 0 , 1,

for

...

- 1.

,p

k P ( k ) = [P(k-l),

PI.

Since

xq

,p -

1, we o b t a i n

-

x

sP-l(F)

Isk(F)l/lsk+’(F)I

We c o n s i d e r t h e lower c e n t r a l series

f o l l o w s from some computations t h a t

, p.

...

P(O)= P,

i s c o n t a i n e d i n s(F) f o r x E F , it P ( k ) = {ua la E sk ( F ) ] f o r k = 1, 2 ,

Hence w e can s e e t h a t Jennings’ M-series

(see

=

...

[l]) c o i n c i d e s w i t h t h e

lower c e n t r a l s e r i e s :

M1 = P,

= p(l),

$>

M2 = =

= 1 and

(mod J(KU)2) where

PROOF.

Let

C

C

is a V-conjugacy class i n V .

is the sum of a l l elements of

b e a V-conjugacy c l a s s of

u

and l e t

d =

m-1 lkzO

k

c

Then w e o b t a i n t h e f o l l o w i n g e q u a t i o n

5 m

from lemma 1 and polynomial. d i v i s o r of

d = (cm

- 1 + uad

- l ) / ( c - 1) =

Then it i s w e l l known t h a t QP(q) = (qp

- l ) / ( q - 1).

:m

Then

C.

(mod J(KU)2)

0 . Let ( x ) b e a cyclotomic P m 5 1 (mod p ) s i n c e m i s a

This completes o u r a s s e r t i o n .

.

On the nilpotency index n

h

and B = J(KW)VKG

We set A = J(KU)KG

195

where V

is the sum of a l l

elements of V.

LEMMA 3. Asswne m > 1. Then (1) J(KG) = A + B 2 (2) Bp z 0 (mod A ) (3) J(KG)P+l 0 (mod A2)

(4)

J(KG)’

0 (mod A)

(1) It is well known that A

PROOF.

is contained in J(KG).

Since

is isomorphic to KG/A and WV is a Frobenius group with the kernel V and a complement W, it is easy to see OUT assertion (1). ( 2 ) We have the following equation from lemma 2.

KWV

BP

=

(J(KW);KG)P

= (J(KW)~KU~)P-~J(KW)?KG =

(lc J(KW)GP-~J(KW)GKG

E J(KW)p?KG

(mod A2)

= o where C runs over all V-conjugacy classes in U. (3) Since A2 :0 and Bp E 0 (mod A 2) , we get our equation from (1).

(4) We have J(KG)’ THEOREM.

=

(A + BIP

A + Bp

t(G) 5 - t(P) = pr(p2 - 1)/2

+

A.

P.

PROOF. We may assume m > 1. It follows from lemma 3 ( 3 ) and (4) that 2 p r ( p -11/Z+P = ( KG )P ( J ( KG ) p+l)pr( P-1 12 J(KG)

c - A( A2)Pr(P-l) I2 = Apr(P-l)+l

= o

4. EXAMPLES In case m = 1, it is evident that t(G) = t(P). I d o n ‘ t know an example in case m > 1 such that t(G) = t(P). In general, it is well known that t(G),

rp(p

t(G) = r p ( p

- 1) + p.

- 1) + p

In case m = (qp (see [ 3 1 ) .

- l)/(q

-

11, we had already proved

In case p = 2, r = 3 and m = 3, we can

see t(G) = 10 < t(P) = 11 using a computer. I think that it is very difficult to compute explicitly the number t(G)

for a proper divisor m.

REFERENCES [l] S. A . JENNINGS, The structure of the group ring of a p-group over a

modular f i e l d , T r a n s . her. Math. SOC. 50 (19411, 175-185.

K. Morose

196

[2] K . MORITA, On group r i n g s over a modular f i e l d which possess r a d i c a l s e x p r e s s i b l e as p r i n c i p a l i d e a l s , S c i . Rep. Tokyo Bunrika Daigaku, 4 (1954)

, 177-194. [3] K. MOTOSE,

On t h e nilpotency index of t h e r a d i c a l of a group a l g e b r a 111,

J . London Math. SOC. ( 2 ) 25 (1982), 39-42.

[4]

K. MOTOSE,

On t h e nilpotency index of t h e r a d i c a l of a group algebra V ,

J . o f Algebra, 90 (1984), 251-258.

[5] 0 . E. VILIAMAYOR, On t h e semi-simplicity of group a l g e b r a s . 11, Proc. h e r . Math. SOC. 1 0 (1959), 27-31.

Group and Semigroup Rings G. Karpilovsky (ed.) 0 Elsevier Science Publishers B.V. (North-Holland), 1986

197

INVERSE SEMIGROUP ALGEBRAS

W.D. MUNN Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland, U.K.

A survey is given of work on inverse semigroup algebras concerning semiprimitivity, nonexistence of nonzero nil ideals, von Neumann regularity, and (for contracted algebras) simplicity. The subject of this paper is a class of semigroup algebras having much in common with group algebras, namely those arising from inverse semigroups. Such semigroups were introduced by Vagner C321 in 1952 (under the heading of 'generalised groups') and, independently, by Preston C281 in 1954. They have been studied intensively and the resulting theory is rich and elaborate (see C231). To keep matters as simple as possible, I have chosen to restrict the account to algebras over a field. The paper takes the form of a survey, the central theme of which is the response to the question:

'When is an inverse

semigroup algebra semiprimitive (that is, semisimple in the sense of Jacobson)?'.

Various related matters are also discussed, the choice of

material being largely a reflection of my own interests over the years: no attempt is made at comprehensive coverage of what is already a fairly wide research area. There are seven sections: 1. Inverse semigroups

2. Finite-dimensional inverse semigroup algebras 3. Matrix representations 4. Domanov ' s theorem 5 . Nonexistence of nonzero nil ideals 6.

Von Neumann regularity

7. Simple contracted inverse semigroup algebras.

For the reader's convenience, 11 consists of a summary of those properties of

inverse semigroups required for later sections, together with some examples to illustrate the scope of the concept. The earliest work on inverse semigroup algebras, namely the analogue of Maschke's theorem for the finite-dimensional case, is described in 12. It has implications for the theory of matrix representations of finite inverse semigroups and, in 13, some aspects of this are explored. The core of the paper is 54, which is devoted to a key theorem

W.D. Munn

198

of A . I .

Domanov ( 1 9 7 6 ) :

an i n v e r s e semigroup a l g e b r a i s s e m i p r i m i t i v e i f t h e

a l g e b r a o f each maximal subgroup o f t h e semigroup i s s e m i p r i m i t i v e .

A complete

proof of t h e theorem i s p r o v i d e d , t o g e t h e r with a n example c o n s t r u c t e d by Teply, Turman and Quesada which shows t h a t t h e converse i s f a l s e .

I n 95, some r e s u l t s

on t h e n o n e x i s t e n c e o f nonzero n i l i d e a l s are p r e s e n t e d , t h e t e c h n i q u e s h e r e b e i n g adapted from t h o s e used f o r group a l g e b r a s .

The problem of d e t e r m i n i n g

n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r a n i n v e r s e semigroup a l g e b r a t o be r e g u l a r i s touched on b r i e f l y i n 96;

and t h e f i n a l s e c t i o n , § 7 , p r o v i d e s a

s u f f i c i e n t c o n d i t i o n f o r a c o n t r a c t e d i n v e r s e semigroup a l g e b r a t o be s i m p l e . In

The n o t a t i o n i s based mainly on t h a t of C l i f f o r d and P r e s t o n [41.

p a r t i c u l a r , w e r e c a l l t h a t i f S is a semigroup t h e n So i s t h e semigroup d e f i n e d as f o l l o w s :

S

=

So i f S h a s

a z e r o and I S 1 > 1; o t h e r w i s e So i s t h e semigroup

o b t a i n e d from S by a d j o i n i n g a z e r o . L e t S be a semigroup and l e t F be a f i e l d .

The semigroup aZgebra o f S o v e r

F ( d e f i n e d i n t h e u s u a l way) w i l l be denoted by F[S].

and t h a t z is t h e z e r o of S.

Now suppose t h a t S

So

Then F [ z l . i s a one-dimensional i d e a l o f FLSI.

By t h e contracted semigroup aZgebra o f S over F w e mean t h e f a c t o r a l g e b r a F[S]/F[z]

and we denote t h i s by FoCS].

with b a s i s S \ z

We can t h i n k o f Fo[S] as t h e a l g e b r a

i n which m u l t i p l i c a t i o n is induced by t h e m u l t i p l i c a t i o n i n S

and z i s i d e n t i f i e d w i t h t h e z e r o of t h e u n d e r l y i n g v e c t o r s p a c e . Again l e t S be an a r b i t r a r y semigroup. FoCSol = FCSl.

If S = So w e s h a l l h e n c e f o r t h d e n o t e i t s z e r o by 0.

a nonzero element o f FCSl ( r e s p . FoCS]). IiZlaixi

Note t h a t i f S # So t h e n

f o r some n €IN, some d i s t i n c t elements x l , ,

and some a,,

..., an €

FXO.

w i l l be denoted by s u p p a .

Let a be

Then w e may write a u n i q u e l y as

.., xn

of S ( r e s p . S\O)

The support o f a is t h e s e t {xl,

..., xn} and

If a = 0 w e t a k e s u p p a t o be 0.

F i n a l l y , i f A is an a l g e b r a o v e r F and n €IN, w e d e n o t e t h e a l g e b r a o f a l l n x n matrices over A by A n .

1.

INVERSE SEMIGROUPS An inverse semigroup i s a semigroup S i n which t o each element a E S t h e r e

corresponds a unique element x E S such t h a t axa = a and xax = x . This s e c t i o n is mainly devoted t o a summary o f t h o s e p r o p e r t i e s of i n v e r s e semigroups t h a t w i l l be r e q u i r e d l a t e r i n t h e paper:

for t h e d e t a i l s ( p r o o f s ,

s o u r c e s , e t c . ) , r e f e r e n c e can be made t o t h e t e x t b o o k s C41, 1.1.

Alternative description:

[ E l , C231.

p r o p e r t i e s of i nver ses

The f o l l o w i n g r e s u l t p r o v i d e s a n a l t e r n a t i v e d e s c r i p t i o n o f a n i n v e r s e semigroup.

is an inverse semigroup i f and onZy i f ( i ) for a22 a E S there e x i s t s x E S such that axa = a and ( i i ) any two idempotents of S commute.

A semigroup S

199

Inverse sem@wp algebras

Let S be an i n v e r s e semigroup and l e t a E S. axa = a and xax Evidently aa

-1

, a -1 a

a r e idempotents;

For a l l a , b , e E S with e 2

= a, 1.2.

The unique x E S such t h a t

x i s c a l l e d t h e inverse of x and i s denoted by a-’. but, i n general, these a r e d i s t i n c t .

e we have t h a t

(ab)-’ = b-la-l,

e

-1

= e,

= a-lea.

(a-lea)’

Semilattices

By a semizattice w e mean a commutative semigroup of idempotents.

Clearly,

s e m i l a t t i c e s a r e i n v e r s e semigroups ( s e e 1.1). Let E be a s e m i l a t t i c e .

-

Then t h e r e l a t i o n 5 defined on E by x 5 y

xy-x

i s a p a r t i a l o r d e r on E under which every p a i r of elements has a g r e a t e s t lower bound (namely t h e i r p r o d u c t ) . Conversely, any such p a r t i a l l y ordered set E i s a s e m i l a t t i c e with r e s p e c t t o m u l t i p l i c a t i o n defined by xy = g . l . b . { x ,

y}.

The s e t of a l l idempotents of an inverse semigroup S w i l l be denoted by ES. Since t h e idempotents of S commute, ES i s a subsemigroup of S .

the semilattice of S.

It i s c a l l e d

W e assume throughout t h a t ES i s p a r t i a l l y ordered i n

t h e manner described above.

For any subset A 1.3.

c

S w e w r i t e EA f o r

ES I7

A.

Maximal subgroups

L e t S be an i n v e r s e semigroup and l e t e E ES.

He={xES:x

-1

Then H e ,

defined by

-1

x = e = x x 1,

i s t h e g r e a t e s t subgroup of S with i d e n t i t y e .

The groups He(e F ES) a r e

c a l l e d the mximaz subgroups of S . We say t h a t S i s combinatorial i f and only i f each of i t s maximal subgroups is t r i v i a l . 1.4.

The symmetric i n v e r s e semigroup on X

Let X be a nonempty s e t and l e t

9,

denote t h e s e t of a l l one-to-one p a r t i a l

transformations of X ( i n c l u d i n g t h e empty transformation w: 0

9,

-+

0).

The s e t

i s closed under composition of r e l a t i o n s and, with r e s p e c t t o t h i s operation,

ix is an i n v e r s e semigroup i n which, the relation a .

f o r each element a , a

-1

i s t h e converse of

We c a l l j X the s y m e t r i c inverse semigroup on X .

I t is t h e

n a t u r a l analogue for i n v e r s e semigroups of t h e symmetric group on X . A r e s u l t similar t o Cayley‘s theorem on groups holds:

Every inverse semigroup S can be embedded i n

W.D. Munn

200

1.5. Homomorphisms; fundamental inverse semigroups Let S be an inverse semigroup, let T be a semigroup and let 0 : S

T be a homomorphism. Then im 0 is an inverse subsemigroup of T and, for all x E S, is We say that 0 is idempotent-separating if and only if 01 x-10 = (x0)-'. ES injective. An inverse semigroup S is termed fundamental if and only if, for all semigroups T, every idempotent-separating homomorphism from S to T is injective. Fundamental inverse semigroups are also called antigroups (e.g. in C231). Evidently a fundamental inverse semigroup is a group only if it is trivial. It can readily be shown that every combinatorial inverse semigroup is fundamental. We also note the following characterisation: An inverse semigroup S is fundamental if and only if, for all x, y E S, [(Vf E ES) x-lfx = y-lfyl

+

* x = y.

1.6. Clifford semigroups

An inverse semigroup

S

is called a Clifford semigroup if and only if every

idempotent of S is central. Such semigroups can be classified in terms of groups, semilattices and group homomorphisms.

Let E be a semilattice, let (Ga)aEE be a family of pairwise-disjoint groups indexed by the elements of E, for each pair ( a , 8 ) E E x E with a > B let be a homomorphism, and for a l l a E E let $a a denote the identity $ a , B : Ga + G 6 automorphism of Ga. Suppose further that if a t 6 2 y then $ a36 $B,v = %,Y U Ga and define a multiplication on S by the rule ( a , 6, y E E ) . Let S that if xa E Ga and y

B

aEE E G

B

then

XaYB = (xa +a,y) (YB06,J

3

where y at? and the product on the right is the usuai! product in G Then S Y' is a Clifford semigroup with Es E E and maximal subgroups the given groups Ga ( a E E). Conversely, to trithin isomorphism, every Clifford semigroup is of this type. For obvious reasons, a Clifford semigroup is often called a semilattice of groups. Clearly an inverse semigroup is commutative if and only if it is a Clifford semigroup with abelian maximal subgroups. 1.1. 0-simple inverse semigroups; Brandt semigroups

An inverse semigroup S is called 0-simple if and only if ideals are

S

so and its only and {Ol; moreover, it is said to be completely 0-simple if and S

only if it is 0-simple and contains an idempotent that is minimal (under 5) in the set ES L O (in which case every element of E S X O is minimal). every finite 0-simple inverse semigroup is completely 0-simple.

In particular,

201

Inverse sem@oup algebras

Now l e t G be a group and l e t I be a nonempty s e t .

x G x

I)

I;

{O)

f o r a l l i , j , k , 9. E I and a l l

and d e f i n e a m u l t i p l i c a t i o n on S t h u s : X,

Let S = ( I

Y E G, [i,

,;

j = k,

if

9.)

( i , x , j ) ( k , y , 9.) =

j # k,

if

o ( i , x, j ) = (i, x, j)o

=

0'

0.

W e denote it by M(G, I ) .

Then S is a completely 0-simple i n v e r s e semigroup.

Any semigroup of t h i s t y p e is c a l l e d a Brandt semigroup.

Note t h a t , i n

M(G, I ) , ( i ,x , j )

-1

= ( j , x -1 , i ) , 0-l = o .

The f o l l o w i n g r e s u l t i s a s p e c i a l case o f a well-known theorem due t o D. Rees.

Every completely 0-simple inverse semigroup i s isomorphic t o a Brandt semigroup. Note t h a t a l l t h e maximal subgroups o f M(G, I ) , a p a r t from { O ) , are isomorphic t o G .

For G

c a l l the semigroup of I

1, t h e t r i v i a l g r o u p , w e o b t a i n M ( 1 , x

I ) , which w e

I matrix units.

..., n )

I f I i s f i n i t e t h e n w e u s u a l l y t a k e it t o be {l,

f o r some n :IN

and w r i t e M(G, n ) i n p l a c e o f M(G, I). 1.8.

#,-classes;

principal factors

Let U be a n i d e a l o f a semigroup T . (TXU) V { O ) w i t h m u l t i p l i c a t i o n xy i f xy

x o y =

I

0

4

U,

( 0 )

Then T/U d e n o t e s t h e semigroup

d e f i n e d by

x00 = 0 o x

Now l e t S b e a n i n v e r s e semigroup.

(x, y E T Y U ) .

W e d e f i n e an equivalence r e l a t i o n

S by t h e r u l e

(x, y ) E

(VX, y E S )

For any #,-class

0 o 0 = 0

i f xy E U,

j#

-

f

On

SXS = s y s .

J of S w e write

P ( J ) = SxS f o r some ( a n y ) x E J , N(J) = {x E P ( J ) : SxS # P ( J ) ) , A(J) = {x E S: SxS

1 P(J)}.

E v i d e n t l y P ( J ) is a n i d e a l o f S ( t h e principaz i d e a l g e n e r a t e d by J).

It is

e a s i l y s e e n t h a t if N(J) and A(J) are nonempty t h e n t h e s e are a l s o i d e a l s o f S.

I l'PJ)/N(J)

W e d e f i n e t h e principal f a c t o r Q i J ) of S corresponding t o J b y

Q(J) =

i f N(J) # 0, i f N(J) = 0.

This d e p a r t s from t h e s t a n d a r d d e f i n i t i o n g i v e n i n

C41 i n s o f a r as w e t a k e

Jo

r a t h e r t h a n J i f N(J) = 0, t h a t is, i f J i s t h e l e a s t i d e a l of S.

Each principal f a c t o r of an inverse semigroup i s a 0-simple inverse semigroup. F i n a l l y , suppose t h a t S i s a n i n v e r s e semigroup w i t h a ' p r i n c i p a l series'

W. 0.Munn

202 S = S,

=)

S,

=)

...

3

Sn ( e a c h Si i s a n i d e a l of S and it i s n o t p o s s i b l e t o

i n c r e a s e n by i n s e r t i n g e x t r a i d e a l s i n t o t h e s e r i e s ) . Si\Si+l

(i

1,

..., n

-

...,in -

1) are j u s t t h e

i n g semigroups S x and Si/Si+l

I,

(i

-classes of S and t h e correspond1) are, t o w i t h i n isomorphism,

Note t h a t Sn is t h e least

t h e p r i n c i p a l f a c t o r s o f S as p r e v i o u s l y d e f i n e d .

Such a series e x i s t s i f S i s a f i n i t e i n v e r s e semigroup, i n

i d e a l of S.

which case Sn i s a maximal subgroup o f S and Si/Si,l ( 1 . 7 ) f o r i = 1, 1.9.

Then t h e sets Sn and

..., n -

9 -classes;

t h e b i c y c l i c semigroup

W e define a relation (Vx, y E

s)

i s a Brandt semigroup

1.

8 on

(x, y)

-

a n i n v e r s e semigroup S by

€9

(32 E

It i s n o t d i f f i c u l t t o v e r i f y t h a t i f ( x , y) E B then ( x , y )

E

9.

s)

xx

-1

zz

-1

and z

-1

-1

z =y

y.

i s a n e q u i v a l e n c e r e l a t i o n on S and t h a t

F u r t h e r , it can be shown t h a t any t w o maximal

subgroups o f S c o n t a i n e d i n t h e same

-class o f S a r e isomorphic. -class i s termed bisimpze.

An i n v e r s e semigroup c o n s i s t i n g o f a s i n g l e C l e a r l y e v e r y group i s a b i s i m p l e i n v e r s e monoid.

Consider t h e monoid B g e n e r a t e d by two symbols p and q s u b j e c t t o t h e Then B is a

s i n g l e d e f i n i n g r e l a t i o n pq = 1, where 1 i s t h e i d e n t i t y .

c o m b i n a t o r i a l b i s i m p l e i n v e r s e monoid. Each element can be w r i t t e n u n i q u e l y r s 0 0 i n t h e form q p (r i: 0 , s 2 O ) , with t h e convention t h a t q = p = 1; s r n n (qrps)-’ q p f o r a l l nonnegative i n t e g e r s r , s; and EB = {q p : n E 0 ) . W e c a l l B t h e bicyclic semigroup.

Many semigroups c o n t a i n isomorphic c o p i e s o f B , as t h e f o l l o w i n g r e s u l t shows.

Let S be a semigroup containing an idempotent e , l e t eSe denote the subsemigroup {exe : x E Sl and l e t a , b E eSe be such that ab = e , ba # e . Then a and b generate a bicyclic subsemigroup of S with identity e. An i n v e r s e semigroup S is c a l l e d 0-bisirple i f and only i f S = Soand i t h a s e x a c t l y two

9 -classes,

namely S \ 0 and { O } ,

i n v e r s e semigroup i s 0-simple.

E v i d e n t l y e v e r y 0-bisimple

By c o n s i d e r i n g a Brandt semigroup, we can

also e a s i l y prove t h e f o l l o w i n g .

Every comp Ze t e Zy 0-simp l e inverse semigroup is 0-bisimp l e 1.10.

.

Completely semisimple i n v e r s e semigroups

W e have a l r e a d y remarked i n 1.8 t h a t each p r i n c i p a l f a c t o r o f a n i n v e r s e

semigroup i s a 0-simple i n v e r s e semigroup.

An i n v e r s e semigroup S is termed

completely semisimple i f and o n l y i f each p r i n c i p a l f a c t o r o f S i s completely 0-simple. Examples of completely semisimple i n v e r s e semigroups i n c l u d e C l i f f o r d semigroups ( i n p a r t i c u l a r , groups and s e m i l a t t i c e s ) and Brandt semigroups.

203

Inverse semigroup algebras

The b i c y c l i c semigroup is not completely semisimple:

i n f a c t we have t h e

following c h a r a c t e r i s a t i o n .

An inverse semigroup is completely semisimple if and only if it has no

bicyclic subsemigroup. Free inverse semigroups

1.11.

A

free inverse semigroup on a nonempty s e t X is an i n v e r s e semigroup T such

t h a t for a given i n j e c t i v e mapping

T and any mapping a : X

-f

S, where S

i s an i n v e r s e semigroup, t h e r e e x i s t s a unique homomorphism 4 : T

+

S such t h a t

14 =

a.

X

I:

-+

It is easy t o check t h a t i f such an i n v e r s e semigroup T e x i s t s then

hence w e may speak of the f r e e i n v e r s e

it must be unique up t o isomorphism: semigroup on X .

For any nonempty set X, the free inverse semigroup on X exists. It is a

combinatorial completely semisimple inverse semigroup, whose principal factors (discounting multiplicities) are the finite semigroups of matrix units M ( 1 , n) ( n 2 , 3, ...) . An e x p l i c i t c o n s t r u c t i o n for t h e free i n v e r s e semigroup on X can be given i n terms of t h e f r e e group on X .

2.

FINITE-DIMENSIONAL INVERSE SEMIGROUP ALGEBRAS The study of t h e a l g e b r a s of f i n i t e i n v e r s e semigroups was l a r g e l y i n s p i r e d

by t h e c l a s s i c a l r e s u l t of Maschke on group a l g e b r a s , which can be s t a t e d as follows :

Let G be a finite group and let F be a field. Then FCGl is semisimple if and only if F has characteristic 0 o r a prime not dividing the order of G. In view of t h e f a c t t h a t i n v e r s e semigroups have many group-like f e a t u r e s ,

it i s n a t u r a l t o ask whether t h e r e i s an analogue of Maschke's theorem f o r i n v e r s e semigroup a l g e b r a s .

Such an analogue does indeed e x i s t and was found

independently by Oganesyan C l E ] , mid 1950s.

Ponizovskii [ 2 5 ] and t h e author

[lo]

in the

The r e s u l t , s t a t e d below, i s an easy consequence of a more g e n e r a l

one obtained by Ponizovskir [25] and t h e author [9] on t h e semisimplicity of finite-dimensional semigroup a l g e b r a s . not include t h e 'only i f

'

(Note t h a t t h e formulation i n

[lo]

did

part. )

Let S be a finite inverse semigroup and let F be a field. Then F[S] is semisimple if and only if F has characteristic 0 o r a prime not dividing the order of any m i m l subgroup of G. 2.1.

THEOREM.

The ' i f ' p a r t of t h i s theorem i s a consequence of l a t e r , more general r e s u l t s ( i n §§4,5). d i r e c t proof.

However it may be of i n t e r e s t t o have a n o u t l i n e o f a

Let S = S,

=- S,

3

...

2

Sn be a p r i n c i p a l s e r i e s f o r S:

thus

W.D. Munn

204

Sn is a maximal subgroup o f S a n d , f o r i = 1,

..., n

-

1, S i / S i t l is a

0-simple i n v e r s e semigroup and s o i s isomorphic t o a Brandt semigroup M(Gi)

n i l , s a y , where G i is a maximal subgroup o f S ( s e e 1.7 and 1 . 8 ) .

FCS,]

2

FCS,]

3

...

3

simple i n d u c t i o n shows t h a t FCS] is semisimple i f and o n l y i f FCS,] f a c t o r a l g e b r a F[SiI/F[Si+,1 FCSi]/FCSitl]

But, f o r i = 1,

is semisimple.

FoCM(Gi, n i ) ]

(F[Gi])n

. i

simple i f and only i f FCGil is semisimple ( i

and each

..., n

is isomorphic t o t h e c o n t r a c t e d a l g e b r a Fo[Si/Sitl]

t u r n , FoCSi/Sitl]

Now

i s a descending c h a i n o f i d e a l s o f FCS] and a

FCS,]

..., n

-

1).

1,

and, i n

Thus FCSi]/FCSitl] 1,

-

is semiHence FCS] i s

semisimple i f and o n l y i f a l l o f t h e group a l g e b r a s FCSnl and FCGi] (i

1,

..., n

-

1) are semisimple.

The r e s u l t now f o l l o w s from Maschke's

theorem. The number of simple i d e a l s i n a f i n i t e - d i m e n s i o n a l semisimple a l g e b r a A o v e r F is c a l l e d t h e c l a s s number o f A and w i l l b e denoted by cL(A).

I t is

a l s o t h e number o f e q u i v a l e n c e classes of i r r e d u c i b l e m a t r i x r e p r e s e n t a t i o n s

of A o v e r F. Again l e t S and F be as i n 2 . 1 and suppose t h a t F[S] i s semisimple.

It is

e a s i l y v e r i f i e d t h a t , w i t h t h e n o t a t i o n above, n-1

and C ! L ( F ~ C S ~ / S ~ =+ ~c!L(FIGil) ])

( i = 1,

..., n -

1).

These e q u a t i o n s are r e l e v a n t t o t h e r e s u l t s on m a t r i x r e p r e s e n t a t i o n s o f f i n i t e i n v e r s e semigroups s t a t e d i n t h e n e x t s e c t i o n . The f o l l o w i n g is a n immediate consequence of 2 . 1 : 2.2.

COROLLARY.

F be a f i e l d .

Let S be a f i n i t e combinatorial inverse semigroup and l e t

Then FCSI is semisimple.

I n p a r t i c u l a r , we see from 2 . 2 t h a t i f E i s a f i n i t e semilattice and F a f i e l d t h e n F[E] h a s a u n i t y .

(Of c o u r s e , E i t s e l f need n o t have a n i d e n t i t y . )

It f o l l o w s r e a d i l y t h a t i f S is a f i n i t e i n v e r s e semigroup and F a n a r b i t r a r y f i e l d t h e n FCS] h a s a u n i t y , namely t h e u n i t y o f FCESI.

Moreover, a formula

f o r t h i s element (quoted i n [ l o ] ) was o b t a i n e d by Penrose.

We have

2.3. THEOREM. Let S be a f i n i t e inverse semigroup and l e t F be a f i e l d . Then FCSl has a u n i t y , namely

s1

-

s2 t

... t

(-1) n-1

'n

where n is t h e number of maximal elements of ES and sp i s the sum o f a l l products of r d i s t i n c t m x i m l elements of Es ( r Proof.

L e t u = s1 - s 2 +

set of maximal elements o f ES.

' . . a

t

(-l)n-lsn€FIEsl

Then -

1,

..., n ) .

and l e t M d e n o t e t h e

Inverse sem&oup algebras u - 1 i n an o v e r r i n g of FIESl with u n i t y 1. such t h a t f 2 g and s o (1

i s t h e u n i t y of FEES]. and, s i m i l a r l y , ux = x.

-

g ) f = 0.

205

n ( 1 - e ) eEM Now, f o r a l l f E ES, t h e r e e x i s t s g E M Hence, f o r a l l f E Es,

F i n a l l y , f o r a l l x E S, xu

x(x

-1

uf = f and s o u

= x

x)u = x(x-'x)

Thus u i s a l s o a u n i t y f o r FCS].

In t h e case where S = M(G, n ) , a f i n i t e Brandt semigroup, t h e i d e n t i t y of FCS] (and o f Fo[S])

i s t h e sum of t h e nonzero idempotents of S, namely

(1, e , 1) + ( 2 , e , 2 ) +

+ (n, e, n),

where e is t h e i d e n t i t y o f G . We remark, i n p a s s i n g , t h a t Wenger C351 has shown t h a t , f o r an i n f i n i t e i n v e r s e semigroup S , F[S] has a u n i t y i f and only i f ES has f i n i t e l y many maximal elements and every element of ES i s under a maximal element.

I f F[S]

has a u n i t y , it i s given by t h e same formula as i n 2 . 3 .

3.

MATRIX REPRESENTATIONS Let S be a semigroup and F a f i e l d .

By a ( r n a t r k ) r e p r e s e n t a t i o n of S of

degree n over F we mean a homomorphism from S i n t o t h e m u l t i p l i c a t i v e semigroup of Fn.

Evidently every r e p r e s e n t a t i o n of S extends by l i n e a r i t y t o a

r e p r e s e n t a t i o n of FCSl and, conversely, every r e p r e s e n t a t i o n of F[S] induces

a r e p r e s e n t a t i o n o f S. is zero.

We c a l l a r e p r e s e n t a t i o n n u l l i f and only i f i t s image

The term ' i r r e d u c i b l e ' , a s a p p l i e d t o a r e p r e s e n t a t i o n of S, w i l l

have i t s standard meaning:

by convention, however, we s h a l l not regard t h e

n u l l r e p r e s e n t a t i o n of degree 1 a s i r r e d u c i b l e . r e p r e s e n t a t i o n s a r e defined i n t h e usual way;

Equivalence and d i r e c t sums of and a r e p r e s e n t a t i o n i s c a l l e d

completely r e d u c i b l e i f and only i f it i s equivalent t o a d i r e c t sum of r e p r e s e n t a t i o n s each of which i s i r r e d u c i b l e o r n u l l . 0 ) w i l l be denoted by

For a r e p r e s e n t a t i o n p o f S t h e set {x E S: p ( x ) V(p).

Clearly i f V ( p ) # 0 then it i s an i d e a l of S .

r e p r e s e n t a t i o n i f and only i f V ( p ) = A ( J ) f o r some

We c a l l p a p r i n c i p a l

4

- c l a s s J of S ( s e e 1 . 8 ) .

F i n a l l y , i f S = So t h e n p i s s a i d t o be 0 - r e s t r i c t e d t i f and only i f p ( 0 )

0.

Now suppose t h a t S i s a f i n i t e i n v e r s e semigroup and t h a t F i s a f i e l d of c h a r a c t e r i s t i c 0 o r a prime not d i v i d i n g t h e o r d e r of any maximal subgroup of S.

By 2 . 1 , F[S] i s semisimple and s o every r e p r e s e n t a t i o n of FCS], and hence

o f S, i s completely r e d u c i b l e .

Thus a l l r e p r e s e n t a t i o n s of S a r e e s s e n t i a l l y

known when a complete s e t of inequivalent i r r e d u c i b l e r e p r e s e n t a t i o n s i s known-as

i n t h e c l a s s i c a l case of group r e p r e s e n t a t i o n s .

W e now d e s c r i b e a process for o b t a i n i n g such a s e t of i r r e d u c i b l e

1- This term has a d i f f e r e n t meaning i n [ 4 ] .

W. D. Munn

206

r e p r e s e n t a t i o n s of S i n terms of i r r e d u c i b l e r e p r e s e n t a t i o n s of t h e maximal This i s i n two s t e p s :

subgroups o f S over F.

t h e f i r s t , due t o C l i f f o r d C31,

d e a l s with t h e case of a f i n i t e Brandt semigroup, while t h e second, due t o Munn

[lo]

( s e e a l s o [25] and [26]),provides a complete s e t of inequivalent

i r r e d u c i b l e r e p r e s e n t a t i o n s of S i n terms of 0 - r e s t r i c t e d i r r e d u c i b l e r e p r e s e n t a t i o n s of t h e various p r i n c i p a l f a c t o r s of S (0-simple i n v e r s e semigroups )

.

The r e s u l t s a r e s t a t e d here without proof. 3.1. THEOREM. Let S M ( G , m ) be a f i n i t e Brandt semigroup and l e t F be ,kl a f i e l d of characteristic 0 or a prime not dividing I G ) . Let I p r : r 1, be a complete s e t of inequivalent irreducible representations of G over F and l e t pr have degree dr ( r = 1, . . . , k ) . For each r define a mapping

. ..

Ply :

s

+

Fmdr by

; r ( ( ig ,, j)) [ p ( g ) l ; j ( 9 E GI, P r ( 0 ) = 0 , where Cp(g)lij denotes the m x m matrix of dr x dr blocks with p ( g ) as the ( i ,j ) t h block and a l l other blocks zero. Then { f i r : r = 1, . . , k l i s a complete s e t of inequivalent 0-restricted irreducible representations of S over F .

.

Note t h a t

V(6,)

0 ( r e f l e c t i n g t h e f a c t t h a t S is 0-simple) and t h a t every

nonzero matrix i n t h e image of

6,

h a s ’ r a n k dr, a d i v i s o r of t h e degree of

6r.

Let S be a f i n i t e inverse semigroup, with 1-classes where Jn i s the l e a s t ideal of S, and l e t the idempotents i n J i be e i , , ei-, e d i ( i 1, ..., n ) . Let F be a f i e l d of characteristic 0 or a prime not dividing the order of any maximal subgroup of S and l e t Ibir : r 1, kil be a complete s e t of inequivalent 0-restricted irreducible representations o f the principal f a c t o r Q ( J i ) ( i 1, ..., n ) . For each p a i r (i, r ) Let d i r denote the degree of ijir and define a mapping 3.2.

J,,

J,,

THEOREM.

..., J n ,

...,

...,

is the naturcl homomorphism from P ( J i ) t o Q ( J i ) if 1 C i 6 n where y t-+ and the i d e n t i t y on Jn i f i = n. Then r’r

Ipir:

r

1,

..., ki;

i = 1,

-

..., nl

i s a complete s e t of inequivalent irreducible representations of S over F . $0 B Moreover, f o r each ! i , r ) , pirlJi = E;irlJiand V ( p i r ) = A ( J i ) . To combine t h e r e s u l t s of 3 . 1 and 3.2 we proceed as follows.

$-class of S, l e t el,

...,

L e t J be a

em be t h e elements of EJ and l e t G = He

Q(J) is 0-bisimple we can choose ri E J such t h a t rirfl

e l and

1

.

Since

rylri = ei

1

207

Inverse semigmup algebras

(i = 1, ..., m) and it is readily checked that the mapping 8: Q(J) defined by

I

ye = (i, r.yrT1, j ) 1

oe = o

1

+

M(G, m)

if yy-l = e and y- 1y = e i j’

is an isomorphism. Now let p be an irreducible representation of G, let p^ be the corresponding 0-restricted irreducible representation of M(G, m) given by i((i, g, j)) = [p(g)lij (with the notation of 3.1) and let tion of Q(J) defined by G(y)

6

(g E G),

P(o)

o

be the 0-restricted irreducible representa-

;(ye) for all y E Q(J).

Let x E S\A(J).

Then

...,

xei E P(J) (i = 1, m). Suppose that j is such that xe E J. Then j -1 xe.(xe.)-l = xe.x-1 = ek, say, and (xe.)-lxe x ej, since ej is 3 1 7 1 - j ejx minimal in EJ. Further rk(xe.)r-l = rkxrjl. Hence 3 j p(xe.1 = ;((xe.)B) = 6((k, r xrT1 j)) = b ( rkxr-l j ”kj‘ 1 1 k I’ By means of this the irreducible representations 02, in 3.2 can be constructed explicitly from irreducible group representations of maximal subgroups of S. With the notation of 3.2 it can be shown further that if

tip = min{rank pi,(x)

:x E S x

V(P:~)I then dip = miti, and, for all x E S,

(2) is a multiple of tip. ir In the special case where S is a finite Clifford semigroup, which we can

rank

p“

take to be in the form described in 1.6, a complete set of inequivalent irreducible representations of S can be constructed in a particularly simple fashion from the irreducible representations of the groups Ga and the homomorphisms Q

a,B

C9I.

The final statement in 3.2 asserts that every irreducible representation of a finite inverse semigroup S over F is a principal representation. This holds although it is false in general.

under much wider hypotheses on S (see Ill]),

A theory of matrix representations of arbitrary inverse semigroups has been developed by the author (c121, C131) and Preston C291; but it is beyond the scope of the present brief survey. We remark here, however, that the only inverse semigroups to admit faithful (that is, injective) matrix representations are those with finitely many idempotents, as the following result shows. 3.3.

THEOREM. Let S be an inverse subsemigroup of the multiplicative semi-

group of Fn for some field F and some n € IN. Then lEsl S 2n. Proof. Suppose that (ESI > 2”. Let A,, ..., A be distinct elements of ES, m where m > 2n. Since each Ai is diagonable over F (being idempotent) and since

A.A.

1 3

T-lAiT

A.A. for all i, j there exists a nonsingular matrix T in Fn such that 1 1

Di, where Di is a diagonal matrix (i = 1,

.. . , m)

C7].

Now each Di is

208

W.D. Munn hence i t s d i a g o n a l e n t r i e s are e i t h e r 0 or 1.

idempotent;

But t h e r e are

e x a c t l y 2n d i a g o n a l m a t r i c e s w i t h t h i s p r o p e r t y and s o t h e D i cannot a l l be distinct.

Thus t h e Ai cannot a l l be d i s t i n c t , c o n t r a r y t o h y p o t h e s i s .

Note t h a t an i n v e r s e semigroup w i t h a f i n i t e semilattice i s n e c e s s a r i l y completely semisimple:

f o r it cannot c o n t a i n a b i c y c l i c subsemigroup ( s e e 1.10).

For t h e remainder of t h i s s e c t i o n w e c o n s i d e r t h e case F r e p r e s e n t a t i o n ' we s h a l l mean a m a t r i x r e p r e s e n t a t i o n o v e r C.

a.

By a 'complex

The h e r m i t i a n

c o n j u g a t e ( c o n j u g a t e t r a n s p o s e ) o f a complex m a t r i x A w i l l be denoted by A". A complex r e p r e s e n t a t i o n p of an i n v e r s e semigroup S i s termed semiunitary ( p ( x ) ) < ' f o r a l l x E S.

i f and o n l y i f p ( x - l )

a u t h o r c141.a

This terminology i s due t o t h e

Semiunitary r e p r e s e n t a t i o n s have a l s o been c a l l e d ' s t a r

r e p r e s e n t a t i o n s ' by Barnes [2]. Let p be a s e m i u n i t a r y r e p r e s e n t a t i o n o f S.

Then each m a t r i x p ( x ) ( x E S )

i s a p a r t i a l isometry:

f o r p (x ) = p(xx-lx) = p (x) p( x- l ) p( x)

Thus, f o r a l l x E S , p ( x

-1

i n v e r s e o f a complex m a t r i x A C241. p ( x ) is a u n i t a r y m a t r i x ( x

p(x)(p(x))+(p(x).

( p ( x ) ) t , where A t d e n o t e s t h e Moore-Penrose

!(

Evidently i f p(x) is nonsingular then

s).

The f o l l o w i n g r e s u l t ( c 1 4 1 , Theorem 1) g e n e r a l i s e s t h e familiar Peter-Weyl theorem on u n i t a r y r e p r e s e n t a t i o n s o f groups.

Every semiunitary representation of an inverse semigroup i s completely reducible 3.4.

THEOREM.

.

For a complex r e p r e s e n t a t i o n p o f a semigroup S l e t p i j ( x ) d e n o t e t h e

( i , j ) t h e n t r y o f p ( x ) ( x E S).

W e s a y t h a t p i s bounded i f and o n l y i f t h e r e

e x i s t s a p o s i t i v e r e a l number k such t h a t , f o r a l l i , j and a l l x F S, Ipij(x)l 4 k.

I t i s e a s y t o v e r i f y t h a t i f p and u are e q u i v a l e n t complex

r e p r e s e n t a t i o n s t h e n p is bounded i f and o n l y i f a i s bounded. A simple argument a l s o shows t h a t i f A

that A

AA':'A

then laij

I

5 1 f o r all i, j .

= La..] i s a complex m a t r i x such 11

Thus e v e r y s e m i u n i t a r y r e p r e s e n t a -

t i o n o f a n i n v e r s e semigroup i s bounded. These remarks p r o v i d e t h e e a s y h a l f o f t h e f o l l o w i n g theorem (C141, Theorem 2 ) , which g e n e r a l i s e s a n o t h e r classical r e s u l t (Auerbach-Weyl) on complex group r e p r e s e n t a t i o n s . 3.5. THEOREM. A complex representation of an inverse semigroup i s equivalent t o a semiunitary representation i f and only i f it i s bounded. From 3.4 and 3.5 we immediately o b t a i n

Every bounded complex representation of an inverse semigroup is completely reducible. 3.6.

THEOREM.

Inverse semigroup algebras

209

DOMANOV'S THEOREM

4.

The Jacobson r a d i c a l o f a n a l g e b r a A w i l l be denoted by J ( A ) . A i s semiprimitive i f and o n l y i f J(A) = 0.

W e say that

Over a p e r i o d o f some twenty

y e a r s , much e f f o r t was devoted t o d e t e r m i n i n g s u f f i c i e n t c o n d i t i o n s f o r t h e s e m i p r i m i t i v i t y o f t h e a l g e b r a FCG] o f a n a r b i t r a r y group G o v e r a f i e l d F. This culminated i n t h e f o l l o w i n g r e s u l t , due t o Amitsur [l] f o r f i e l d s o f c h a r a c t e r i s t i c 0 and t o Passman [20] f o r f i e l d s o f prime c h a r a c t e r i s t i c .

Let G be a group a d l e t F be a f i e l d t h a t i s not algebraic over i t s prime

subfield ( i n p a r t i c u l a r , an uncountable f i e l d ) . Suppose, f u r t h e r , t h a t if F has prime characteristic p then G has no element of order p . Then J ( F [ G ] ) = 0 . I n p a r t i c u l a r , f o r any group G , I R [ G ]

and CCG] are s e m i p r i m i t i v e .

(The

r e s u l t f o r C had, i n f a c t , been e s t a b l i s h e d by R i c k a r t C301 i n 1950.)

The

q u e s t i o n of whether QCG] i s s e m i p r i m i t i v e remains open. Since Maschke's theorem on f i n i t e - d i m e n s i o n a l group a l g e b r a s h a s a n analogue f o r f i n i t e - d i m e n s i o n a l i n v e r s e semigroup a l g e b r a s ( s e e 2.1) i t i s r e a s o n a b l e t o c o n j e c t u r e t h a t t h e Amitsur-Passman theorem s t a t e d above a l s o h a s an analogue f o r i n v e r s e semigroup a l g e b r a s . A.I.

This was shown t o b e t r u e by

Domanov i n 1 9 7 6 , as a consequence o f a n i m p o r t a n t theorem ( [ 6 ] ,

Theorem 1 ) s t a t e d below as 4.1.

I n view o f t h e f a c t t h a t Domanov's paper C6l

i s somewhat i n a c c e s s i b l e , h i s proof of 4 . 1 - i n

paraphrased form

- is

reproduced h e r e . THEOREM. Let S be an inverse semigruup and l e t F be a f i e l d .

4.1. J(FCG1)

= 0 f o r each maximal subgroup G of S then

If

0.

J(F[S])

The proof depends on s e v e r a l lemmas. 4.2.

LEMMA.

For a l l f F- Es and a l l x, y E S: -1

f 6 m

Proof.

and x-lfx 5 yy-'

++ f 6 xy(xy)-'.

Suppose t h a t f 6 xx-l and x - l f x 5 yy

-1

.

x y ( x y ) - l f x y ( x y ) - l = x y y -1 (x-lfX)yy-lx-l and so f 5 xy(xy)

-1

.

Then

= xx -1 f x x - l = f

Conversely, suppose t h a t f 6 x y ( x y ) -1 -1

fxx-l = f(x y y

x

)xX-l = f ( X y y - l x - l )

-1

.

Then

= f

and (x-lfx)yy-l = x-lfx(x-lx)(yy-l) Thus f 6 xx 4.3.

-1

and x - l f x 6 yy

-1

= x-lf(xyy

-1 -1

x

)x = x

-1

fx.

.

LEMMA.

Let e Es and l e t D denote the$)-class of S containing e . -1 For each f E: ED l e t r E S be chosen so t h a t r r-l e and r f r f = f. Let

f

x , y F: S and l e t f E ED.

f f

Write g = x-lfx and h = y-lgy.

210

W.D. Munn -1

-1

then g E ED and rFrg E He.

(a) Iff 4 a

-1

( b ) If f 4 a

Proof. Also

( a ) Let f 6 xx -1

-1 -1

r f x r g (r x r ) f g

-1

.

Fs

(rfxrg )

fx(fx)-l, g = (fx)

Then f

-1

= rgx

rfxrg

-1

f x and s o g E ED.

f = r f f x x -1rf-1 = rf fr-'

rfxgx-'riL

-1 -1

syril) = r f ( " y ) r i l .

then g , h E ED and ( r r-l)(r

and g 6 yy-'

-1

fxr

-1

= r gr

g

g

-1

g

rfrfl = e, -1

= rr

e.

g g

-1

Hence rf x r g

He*

( b ) Let f 5 xx

-1

and g 2 yy -1

-1

rfxFg r g Y r h Again l e t e E E

S

E ED and s o h E ED, by ( a ) .

Then g

-1

= rffxyrh

rfxgyrh

Further,

rf xYrh l .

and l e t D denote t h e Q-class o f S c o n t a i n i n g e .

a r i g h t FIHel-module W.

Consider

For each f E ED l e t Vf be an isomorphic copy o f W and

Vf be a ( f i x e d ) isomorphism.

L e t V = @ V f , a vector space d i r e c t fCE, Without l o s s o f g e n e r a l i t y we assume t h a t e a c h if is a subspace of V .

l e t Of :W sum.

.

-1

-1

-P

For each f E ED l e t rf be chosen as

W e now t u r n V i n t o a r i g h t F[S]-module.

i n 4.3.

Then, by 4 . 3 ( a ) , f o r each f E ED and each ( v , x ) E Vf x S we can

d e f i n e vx E V by t h e r u l e t h a t

yf

-1

(rfxrg )Bg

(4.4)

vx

if

f 2 xx

if

f

L e t f E ED, v E Vf and x , y

E

suppose t h a t f 2 xy(xy)-'.

Then v(xy)

-1

fx,

xx-l.

Write g = x

=

and g = x

-1

f x and h = y

vtl;l(rf(xy)rh1)€Ih,

Also, from 4 . 2 , f 5 xx-l and g 5 y y - l .

(xy)-lf(xy).

h

S.

+

-1

-1

gy.

First

since Thus vx E V

g

and s o ,

by 4 . 3 ( b ) ,

Hence v ( x y ) = ( v x ) ~ . Now suppose t h a t € by 4 . 2 , e i t h e r f and s o ( v x ) y 4.4,

(vx)y

= 0.

0.

4

xx-' or g

$

yy-l.

4

xy(xy)-'.

Then v(xy)

If f 2 xx-l and g

On t h e o t h e r hand, i f f

4

1 yy-I

xx-l t h e n vx = 0

0.

Also,

t h e n Vx F :1

Vf

lb

and s o , by

Thus w e have shown t h a t , f o r a l l f 5 ED, a l l v E Vf and a l l

x , y E S, ( v x ) y is d e f i n e d and i s e q u a l t o v ( x y ) .

It f o l l o w s t h a t t h e

m u l t i p l i c a t i o n i n 4.4 e x t e n d s by l i n e a r i t y t o an a c t i o n of F[Sl on V and s o V becomes a r i g h t F[S]-module.

W e d e n o t e t h i s module by V(e, W).

Note t h a t , for a l l f , g E ED and a l l x E H e ,

inverse sem&roup algebras

21 1

With t h e above n o t a t i o n , w e have 4.6. LEMMA. Let the r i g h t FCHel-module W be irreducible. Then the r i g h t FCSI-module V ( e , W ) i s irreducible. Proof. Note t h a t d i s t i n c t elements f l , ..., f n E ED ( n €IN) can always

be numbered i n such a way t h a t f i Let f l , f ,

, ... , f n ,

$

fi+k for a l l i , k with 1 2 i < i + k 5 n .

g E ED be such t h a t f i

be elements of V with vi E Vf

= 1,

‘ 0 ( i

$

f i + k and l e t v l , v 2

..., n )

and w E Vg.

,. . .,vn, w

It is enough t o

i show t h a t t h e r e e x i s t s t E FCS] such t h a t v l t = w and v i t = 0 i f 2 5 i 5 n . We show f i r s t t h a t t h e r e e x i s t s t l such t h a t v l t l

Since W is

w.

such t h a t v O - l s we-’. Write ff g Since a t y p i c a l element of supp t l 1s of t h e form

i r r e d u c i b l e t h e r e e x i s t s s E FCH,] -1

r sr

tl

fl g‘ by 4 . 5 , it follows t h a t r f-l1x r g ( x E He) and v l ( r i 1 x r 1 = v 8-lx8 1 g 1 f, g’ -1 v t = v l ( e f l s e g ) = W.

1 1

Now assume t h a t n t 2 .

Suppose i n d u c t i v e l y t h a t f o r 2 5 m 5 n t h e r e e x i s t s

tm-lE FCS] such t h a t vltm-l

w and v it m-1

0 i f 2 5 i 5 m - 1.

...,

w where w t Vh ( j 1, k) f o r some k ‘j=1 j ’ j j hk E ED. Since W i s i r r e d u c i b l e , f o r each j t h e r e e x i s t s

We have t h a t vmtm-l and some h,, s

j

E FCH,]

...,

-1

such t h a t v O-ls = w j e h j . m fm j

Define t, E FCSl by

1j =k 1rf-1m s1. rh j ’

tm= tm-l S i n c e , f o r each x C He,

it f o l l o w s from 4.4 t h a t

k -1 vi(Cj=lrfmsjrhj)

=

o

( i = I,

..., m

-

1)

and s o vitm = vitm-, F u r t h e r , by 4 . 5

(i

1,

..., m

- 1).

,

Thus vltm = w and vitm = 0 i f 2 5 i 5 m.

The r e s u l t follows by i n d u c t i o n on

m. A f a m i l y of modules o v e r an a l g e b r a i s termed f a i t h f u l i f and o n l y i f t h e

i n t e r s e c t i o n o f t h e a n n i h i l a t o r s of a l l t h e modules o f t h e f a m i l y is z e r o . 4.7.

LEMMA.

Let l’be a subset of Es with exactly one element from each

212

W.D. Munn

(a-class of S . For each e E T l e t & be a f a i t h f u l family of r i g h t FCHe]modules and l e t = (NeY q q q , e E T .

J4

Then& is a f a i t h f u l f a m i l y of r i g h t F[Sl-moduZes. Proof. L e t y E F[S]\O and l e t f b e maximal i n t h e s e t {xx-’: x E supp y). . Let D be t h e 0-class o f S c o n t a i n i n g f , l e t e F: T ll D and l e t t h e elements -1 r (g E ED) be as i n 4.3. Suppose t h a t x E s u p p y i s such t h a t xx f . Let g -1 -1 -1 -1 g = x x. Then x rf r f x r g rg and r x r E He. Thus w e c a n w r i t e f

g

...,

..., gik

( i = 1, k) are d i s t i n c t elements of ED, agi E F[He] -1 and i f z E supp y ‘ t h e n f $ zz , Without loss of g e n e r a l i t y , assume t h a t

where gil, a

# 0.

g1

Since

,%

i s f a i t h f u l and a

g1

# 0 t h e r e exists W E

L e t Vg, Bg ( g E ED) b e as b e f o r e . v E Vf such t h a t vB-la

g1

# 0.

Then V f B-la f g,

& such t h a t

W a

# 0.

g1 # 0 and s o t h e r e e x i s t s

Now vy’ = 0 , by t h e c h o i c e of f .

Hence

.. .

e E Vgi ( i = 1, , k ) and v e - l a 0 # 0. Thus vy # 0 gi gi g1 g1 and s o y d o e s not l i e i n t h e a n n i h i l a t o r of V(e, W). Consequently,u i s a But vo-la

by 4.5.

f

f a i t h f u l f a m i l y o f r i g h t F[S]-modules. pro0 f of 4.1. be as i n 4 . 7 .

Suppose t h a t FCH,]

For each W E M e form t h e F[S]-module

V(e, W).

each V(e, W) is i r r e d u c i b l e and, by 4 . 7 , t h e f a m i l y (V(e, W))

faithful.

Let T

Then f o r each e E T t h e r e e x i s t s a f a i t h f u l f a m i l y A e o f

i r r e d u c i b l e FIHel-modules. 4.6,

is s e m i p r i m i t i v e f o r a l l e E ES.

WE% ,eET

By is

Hence F[S] i s s e m i p r i m i t i v e .

Combining 4 . 1 w i t h t h e Amitsur-Passman theorem, w e immediately o b t a i n 4.8. COROLLARY. Let S be an inverse semigroup and l e t F be a f i e l d that is not algebraic over i t s prime subfieZd. Suppose, f u r t h e r , that if F has prime characteristic p then no subgroup of S has an element of order p . Then J(FCS1)

=

0.

The r e s u l t f o r t h e case F = C i s a l s o e consequence o f work of Barnes [2]. 4.9.

a field.

COROLLARY.

Let S be a combinatorial inverse semigroup and l e t F be

Then J(FCS1) =

0.

I n p a r t i c u l a r , since a free i n v e r s e semigroup is c o m b i n a t o r i a l ( l . l l ) , t h e a l g e b r a of such a semigroup o v e r a n a r b i t r a r y f i e l d i s s e m i p r i m i t i v e .

This

Inverse sem&roup algebras

213

was e s t a b l i s h e d by a d i f f e r e n t method i n C 1 6 l . It t u r n s o u t t h a t t h e converse o f Domanov's theorem i s false: each maximal subgroup G o f S.

f o r an

= 0 d o e s not imply J ( F [ G ] ) = 0 f o r

i n v e r s e semigroup S and a f i e l d F , J ( F [ S ] )

The e a r l i e s t examples i l l u s t r a t i n g t h i s although a t t h e

phenomena were s u p p l i e d by Teply, Turman and Quesada [ 3 1 ] -

time t h e s e a u t h o r s were a p p a r e n t l y unaware o f Domanov's work.

W e conclude

t h i s s e c t i o n w i t h a b r i e f d e s c r i p t i o n o f one s u c h . EXAMPLE.

4.10.

Let F b e a f i e l d o f prime c h a r a c t e r i s t i c p and l e t G be a

# 0.

group with a subgroup H such t h a t J ( F [ G ] ) = 0 w h i l e J(F[H])

1 (t,+ )

we may t a k e G t o be t h e wreath product ( Z p , + )

F o r example

(see ( [ 2 2 ] ,

Ch.7,

Let E d e n o t e

4 . 1 2 ) and H t o b e t h e d i r e c t product of 1231 c o p i e s of ( Z p , + ) .

t h e semilattice c o n s i s t i n g o f t h e i n i t i a l segment [ O , w] o f t h e o r d i n a l s under For a l l a € E l e t Ga be d e f i n e d by

the usual order.

Ga =

{

{a) x H

if

a

{a}

if

a < w.

G

x

w,

Then each Ga i s a group under componentwise m u l t i p l i c a t i o n and Gw G

s G

For a l l (a,

if a < w.

6) E E

C l e a r l y , f o r a l l a , B , y E E w i t h a 2 B 2 y w e have t h a t Q S

=

a l s o J(F[Gw]) f 0 and J(F[G,])

E v i d e n t l y S \ G w is an i d e a l o f S. o f F[S] a n d , by 4 . 1 , J(K)

0.

Then a

( i = 1,

K and so ( s u p p a )

..., xn

. . ., n ) .

and l e t al,

x.e

1 B

B

E N n

= +a,v. Let

Thus K is an i d e a l

n

Since N

n

K is a quasi-

Now suppose t h a t N # 0.

Gw # 0.

..., an E

Take

Let t h e d i s t i n c t e l e m e n t s o f

E be such t h a t x i E G

Choose B E E such t h a t w > B

L e t eg denote t h e i d e n t i t y of G B .

=

i > r,

if

i s i n j e c t i v e , aeB # 0.

w,B K, contradicting t h e fact t h a t N

Then

1 S i 5 r,

if

ae

B,Y

ai Without l o s s o f g e n e r a l i t y , assume t h a t , f o r some r,

and ( i f r # n ) B > a i f o r i > r .

and s o , s i n c e 4

by K .

Let N = J ( F [ S ] ) .

ar = w , w h i l e ai < w i f i > r.

a 1 = a2 =

4

0 if a < w.

Denote F[S\G,]

r e g u l a r i d e a l o f K w e have t h a t N f l K = 0.

s u p p a be x l ,

a,B

( I n [31], 4 . 1 i s proved f o r t h e s p e c i a l case

of a l g e b r a s of C l i f f o r d semigroups.)

a E NXO.

H , while

Then S is a C l i f f o r d

U G, and d e f i n e a m u l t i p l i c a t i o n on S as i n 1 . 6 .

a€ E semigroup:

g

E such t h a t a Z B d e f i n e a homomorphism

x

But s u p p ( a e

n

K = 0.

Hence B SS\Gw. Consequently N 0 , as

required. Another example of a C l i f f o r d semigroup a l g e b r a , w i t h some remarkable p r o p e r t i e s , h a s been provided by Ponizovski:

[ 271.

W.D. Munn

214 5.

NONEXISTENCE OF NONZERO N I L IDEALS It f o l l o w s from 4.8 t h a t i f S i s a n i n v e r s e semigroup and F = Q: orIR, s a y ,

t h e n FCS] h a s no nonzero n i l i d e a l s .

However w e cannot u s e 4.1 t o prove t h a t

f o r w e do n o t know whether group a l g e b r a s o v e r

QCS] h a s no nonzero n i l i d e a l s ;

Q are s e m i p r i m i t i v e and so cannot deduce t h a t QCS] is s e m i p r i m i t i v e .

It is

known, however, t h a t , f o r any group G , QCG] h a s no nonzero n i l i d e a l s E l ] : indeed t h i s r e s u l t i s a key s t e p i n A m i t s u r ' s proof o f t h e s e m i p r i m i t i v i t y o f FCG] f o r F a f i e l d o f c h a r a c t e r i s t i c 0 t h a t i s n o t a l g e b r a i c o v e r Q .

Unaware of Domanov's work, t h e a u t h o r ClS] i n v e s t i g a t e d t h e n o n e x i s t e n c e o f nonzero n i l i d e a l s i n i n v e r s e semigroup a l g e b r a s , w i t h a view t o t a c k l i n g t h e This s e c t i o n comprises a b r i e f account o f some o f

s e m i p r i m i t i v i t y problem. t h e main r e s u l t s .

Let S be an inverse semigroiq, l e t T be a nonempty f i n i t e subset of S and l e t e be maximal i n {xx-': x E TI. Then 5.1.

LEMMA.

xy-l e rq x y. ( ~ xy , E T) Suppose t h a t x , y E T are s u c h t h a t xy-l = e. Then

Proof. xx

-1

e

=

xx

xy

-1

xy

-1

= e;

t h a t i s , e 5 xx

-1 -1

ex = x . fore y

-1

Also yx-l = ( x y

-1

e = y

-1 -1

x = ( y

)

-1

.

Hence y

= e

)

-1

-1

= e.

-1

.

Hence e = xx

Thus, as above, e

( x y - l ) = y - l and ( x y

-1

)x = x.

-1

yy

and s o -1

and t h e r e -

Consequently,

=y.

The next lemma i s modelled on (C211, Theorem 3 . 2 ) .

Let S be an inverse semigroup and l e t F be a subfield of C d o s e d under complex conjugation ( i n particuZar, Q , I R or C). Then FCS] has no nonzero nit ideaZs. Proof. F o r a l l a E F l e t u denote t h e complex c o n j u g a t e of a . We first LEMMA.

5.2.

observe t h a t t h e mapping

': : FCS]

(1aixi)" i

is an i n v o l u t i o n .

=

-1

: i

1,

( a i E F , xi E S )

i

Let a E FCSl\O.

.. . , n l .

FCS] d e f i n e d by

1 gixf'

are d i s t i n c t elements of S and a l , {xixi

-f

Then a = liqlaixi

..., an E

FXO.

s a y , where x l ,

Then t h e c o e f f i c i e n t o f e i n aa?': is -1

t h e sum is t a k e n o v e r a l l p a i r s ( i , j ) such t h a t xixj

= e then i

i f x.x-'

l j

j.

( i = 1,

..., n ) ,

= e.

1 a.E. 1 I'

= e.

where

But, by 5 . 1 ,

Hence t h e c o e f f i c i e n t o f e i n aa" i s

t h e sum b e i n g t a k e n o v e r a l l i f o r which xixi'

... , xn

Choose e maximal i n

1l"iI2,

Since ai # 0

t h i s c o e f f i c i e n t is nonzero and s o aa" # 0.

Thus we have

shown t h a t

(Va E FCSl)

aa" = O

c4

Now l e t A be a n i l i d e a l of FCSl and l e t a E A . (aa"'Ik

=

0 f o r some k E N .

a=O. Then aa': E A and s o

Let m denote t h e l e a s t s u c h i n t e g e r k.

Suppose

Inverse sem&oup algebras

m-1 t h a t m > 1. Write b = ( a a ” )

.

Then bg2 = b and s o bb*

Thus a a ” = 0 and s o , again by (1), a

0.

Therefore A

Since CCS] has no nonzero n i l i d e a l s it i s semiprime. argument ( s e e , f o r example, “211, 5.3.

(aaa)2m-2 - 0 ,

Hence, by (l), b = 0 and t h i s c o n t r a d i c t s t h e d e f i n i t i o n of

s i n c e 2m - 2 b m.

m.

215

0.

By a standard

Theorem 3 . 3 ) we deduce

Let S be an inverse semigroup and l e t F be a f i e l d of

THEOREM.

characteristic 0.

Then F [ S I i s semiprime.

From 5.2 (with F = Q) and 5.3 we can proceed t o deduce 4.8 f o r t h e zero

For t h e

c h a r a c t e r i s t i c c a s e , by e x a c t l y t h e method used f o r group a l g e b r a s . d e t a i l s , s e e C161. We now t u r n t o t h e prime c h a r a c t e r i s t i c case. incomplete:

The r e s u l t s h e r e a r e

a t t h i s s t a g e we a r e unable t o d e a l with a r b i t r a r y i n v e r s e semi-

groups and we confine our a t t e n t i o n t o completely semisimple i n v e r s e semigroups

(1.10). The following r e s u l t ( [ 2 1 ] , Lemma 3 . 4 ) i s due t o Passman. 5.4. LEMMA. Let S be a semigroup, l e t F be a f i e l d of prime c h a r a c t e r i s t i c p and l e t q = pr f o r some r E I N . Then, f o r any x l , xn E S and

...,

where c i s a linear combination of elements of FCS] of the fomn uv u, v

in the subsemigroup of

S generated by

{x,,

-

vu, with

..., x n l .

In o r d e r t o apply 5.4 t o t h e completely semisimple i n v e r s e semigroup c a s e , we r e q u i r e 5.5.

Let S be an inverse semigroup, l e t F be a f i e l d and l e t A be

LEMMA.

Then there e x i s t s e E Es and a E A\O

a nonzero i d e a l of F C S ] . e E supp a

Proof.

= eSe.

Choose b E A \ O and l e t t h e elements of s u p p b be x l , x 2 ,

L e t e be maximal i n 1x.x

Let yi

-1

- 1i i

suppose t h a t e e E supp a

xlxl

.

: i

..., n}

= 1, 2 ,

Take a

-1

ebx,

.

ex.x

-1

( i = 1, 2 ,

11

Clearly a E A.

We show t h a t

..., n ) .

= y . e ( i = 1,

..., n )

Then supp a E { y , ,

. .. , yn}.

Suppose t h a t yk = e .

Since yk

t h e g r e a t e s t element of {exicexi) T = {exi : i = 1,

..., n))

exk(exl) -1

: i

= 1,

-1

t h a t exk = exl = x l . -1

t h a t i s , e S $‘c,

.

and e (

..., n} -1

Thus e = xkxk

=

e t h e n k = 1.

exl(exl)-’) i s

it follows from 5 . 1 (with -1

Hence exkxk

, by

Also

Now y1 = e .

and s o suppa 5 eSe.

Hence t o show t h a t e E s u p p a it s u f f i c e s t o prove t h a t if yk

= e;

..., x n *

and, without l o s s of g e n e r a l i t y ,

c eSe.

eyi = yi = y 1 . x 1x 1

x1x;’

such that

=

exk(exk)-l =

t h e choice of e .

216

W. D. Munn exk = x1 and s o k

Therefore xk

1, a s r e q u i r e d .

The n e x t lemma i s a d a p t e d from ([21],

Lemma 3 . 5 ) .

Let S be a completely semisimple inverse semigroup and l e t F be a f i e l d of prime characteristic p . Let e E Es and l e t a be a nilpotent 5.6.

LEMMA.

element of element of Proof. x1 = e and and t h a t a'

FCS] such that e E supp a Z e S e . '

order p . Let a al,

=

1

where x1 ,

aixi,

i=l ..., an E FXO.

. .., xn

are d i s t i n c t elements o f e S e ,

Since a i s n i l p o t e n t it i s clear t h a t n L 2 Hence, by 5 . 4 ,

0 f o r some power q o f p .

o

Then the group He contains an

q = ale +

n

1

a4.5 + c , i-2 where c i s a l i n e a r combination o f elements o f t h e form uv

-

vu, w i t h

u , v E eSe ( s i n c e supp a E e S e ) . Suppose t h a t e E supp c.

Then t h e r e e x i s t u , v € eSe such t h a t e = uv # vu

and s o u and v g e n e r a t e a b i c y c l i c subsemigroup o f S w i t h i d e n t i t y e (1.9). 9

But t h i s i s i m p o s s i b l e s i n c e S i s c o m p l e t e l y semisimple (1.10). Hence xi = e fo;

some i 2 2.

Also e x i = x i

=

x i e and s o x i E He;

moreover, xi # x1

e.

Hence some power o f xi is a n element of He o f o r d e r p . From 5.5 and 5.6 we now d e r i v e

Let S be a completely semisimple inverse semigroup and l e t F be a f i e l d of prime characteristic p , where p i s not the order o f an element i n a subgroup of S. Then FCS] has no nonzero n i l i d e a l s . Proof. Suppose t h a t t h e r e s u l t i s f a l s e and t h a t F[S] h a s a nonzero n i l i d e a l A. By 5 . 5 , t h e r e e x i s t s e E ES and a C: A X 0 such t h a t e E s u p p a E eSe. 5.1.

THEOREM.

Hence, by 5.6,He c o n t a i n s a n element o f o r d e r p - c o n t r a r y

t o hypothesis.

An a p p l i c a t i o n of t h e t h e o r y of p l a c e s , s i m i l a r t o t h a t used i n t h e case of group a l g e b r a s , y i e l d s t h e f o l l o w i n g r e s u l t (C161, Theorem 3 . 4 ) ,

t h e proof o f

which is o m i t t e d h e r e .

Let S be a completely semisimple inverse semigroup and l e t F be a f i e l d of characteristic 0 . Then FCSl has no nonzero nil i d e a l s . 5.8.

THEOREM.

It would be of i n t e r e s t t o know whether 5 . 1 and 5.8 h o l d w i t h o u t t h e h y p o t h e s i s o f complete s e m i s i m p l i c i t y . r e c e n t l y shown t h a t t h i s is s o . ] 6.

[Added i n p r o o f .

The a u t h o r h a s

VON NEUMANN REGULARITY

An a l g e b r a A i s s a i d t o be regular, i n t h e s e n s e of von Neumann, i f and o n l y i f f o r a l l a E A t h e r e e x i s t s x E A such t h a t a x a = a .

It i s n o t hard

t o show t h a t e v e r y r e g u l a r a l g e b r a i s s e m i p r i m i t i v e . The problem o f f i n d i n g n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r a group

inverse semigroup algebras

217

a l g e b r a t o be r e g u l a r was s o l v e d , independently, by Villamayor C33] and Connell C51.

Their r e s u l t may be s t a t e d as follows.

L e t G be a group and l e t F be a f i e l d .

Then FCGI i s regular i f and only i f G i s locally f i n i t e and F has characteristic 0 or a prime t h a t is not the order of an element o f G. Regularity of semigroup a l g e b r a s was f i r s t i n v e s t i g a t e d by Weissglass C341. In p a r t i c u l a r , he e s t a b l i s h e d r e s u l t s 6 . 1 , 6 . 2 and 6 . 3 below. 6.1.

LEMMA.

Let S be a semigroup and l e t F be a f i e l d .

I f FCSl i s

regular then ( i ) every subgroup of S i s locally f i n i t e , ( i i ) F has characteristic 0 or a prime t h a t i s not the order of an element i n a subgroup of S. By analogy with t h e group c a s e , we say t h a t an i n v e r s e semigroup S is

locally f i n i t e i f and only i f every f i n i t e subset of S i s contained i n a f i n i t e i n v e r s e subsemigroup. 6.2. THEOREM. Let S be an inverse semigroup and l e t F be a f i e l d . Suppose, further, that ( i ) S i s locally f i n i t e , ( i i ) F has characteristic 0 or a prime t h a t i s not the order of an element i n a subgroup of S. Then F C S l i s regular.

This r e s u l t follows e a s i l y from 2 . 1 and t h e f a c t t h a t a semisimple finite-dimensional algebra is necessarily regular. From 6 . 1 and 6 . 2 Weisglass deduces 6.3.

THEOREM.

Let S be a C l i f f o r d semigroup and l e t F be a f i e l d .

Then

F C S ] i s regular i f and only i f

( i ) every subgroup of S i s locally f i n i t e , ( i i ) F has Characteristic 0 or a prime t h a t i s not the order of an element

of

s.

Condition ( i ) i n a C l i f f o r d semigroup S is i n f a c t equivalent t o t h e l o c a l f i n i t e n e s s of S.

It i s conjectured t h a t , f o r t h e algebra of an a r b i t r a r y i n v e r s e semigroup S , a necessary condition f o r r e g u l a r i t y i s t h a t S i s l o c a l l y f i n i t e . (See 6 . 2 . ) S i g n i f i c a n t progress towards e s t a b l i s h i n g t h i s has been made by O k n i k k i C191. Recall t h a t a semigroup S i s periodic i f and only i f f o r each x E S t h e r e m m+n x Evidently every l o c a l l y f i n i t e i n v e r s e

e x i s t m , n €IN such t h a t x semigroup is p e r i o d i c :

.

but t h e converse is f a l s e .

F u r t h e r , it can be shown

t h a t every p e r i o d i c i n v e r s e semigroup i s completely semisimple. The following r e s u l t s are s p e c i a l c a s e s o f Theorems 1 and 2 of C 1 9 1 .

218

W.D. Munn

6.4. THEOREM. Let S be a semigroup and l e t F be a f i e l d . reguZar then S i s periodic. 6.5.

THEOREM.

I f FCSI i s

Let S be a semigroup and l e t F be a f i e l d o f characteristic

0. If FCS] is regular then S i s ZocaZly f i n i t e . Combining 6.2 and 6.5, we o b t a i n 6.6.

THEOREM.

characteristic 0.

Let S be an inverse semigroup and l e t F be a f i e l d of Then FCSl i s regular i f and only i f S i s locally f i n i t e .

The prime c h a r a c t e r i s t i c case remains unresolved.

7.

SIMPLE CONTRACTED INVERSE SEMIGROUP ALGEBRAS A s i s w e l l known, t h e algebra FCS] o f a semigroup S over a f i e l d F i s

simple only i n t h e t r i v i a l case where I S 1 = 1: f o r i f I S 1 > 1 then { c a i x i : ai E F, xi E S , l a . = 0) i s an i d e a l of FCS] o t h e r than 0 and FCS]. However i f S = So then t h e contracted a l g e b r a Fo[S] can c e r t a i n l y be simple. A f a m i l i a r example i s obtained by t a k i n g S = M(1, n ) , t h e semigroup of n matrix u n i t s , s o t h a t Fo[S] i s t h e f u l l matrix a l g e b r a Fn.

x

n

More g e n e r a l l y , i f

we t a k e S = M(1, I ) f o r any nonempty s e t I then FoCS] i s t h e simple algebra c o n s i s t i n g o f a l l I x I matrices over F with a t most f i n i t e l y many nonzero entries. In t h i s s e c t i o n we e s t a b l i s h a r e s u l t giving a s u f f i c i e n t condition on an inverse semigroup S be simple.

So f o r i t s contracted algebra over an a r b i t r a r y f i e l d t o

This was announced i n C151.

Before proceeding, we n o t e t h a t a simple contracted inverse semigroup algebra i s n e c e s s a r i l y p r i m i t i v e .

The following theorem on p r i m i t i v e

contracted i n v e r s e semigroup algebras can be deduced by adapting t h e proof of Domanov's theorem (4.1).

7.1.

I am g r a t e f u l t o I . S . Ponizovskir f o r t h i s remark.

Let S be a O-bisimple inverse semigroup and l e t F be a f i e l d . If FCGl is primitive f o r any nonzero maximal subgroup of G then FOES] i s primitive. In particutar, Fo[Sl i s primitive i f S i s combinatorial. THEOREM.

The formulation of t h e main r e s u l t ( 7 . 2 ) r e q u i r e s two f u r t h e r concepts. (1) An i n v e r s e semigroup S = So i s termed quasicancelkrtive i f and only i f (Va, b , c E S )

ab = a c # 0 and bb-l = c c

-1

*b =

c.

By considering i n v e r s e s , we r e a d i l y see t h a t t h i s condition is equivalent t o (Va, b , c E S )

ba = c a # 0 and b-lb

c-'c

*b =

c.

Thus q u a s i c a n c e l l a t i v i t y i s a property with l e f t - r i g h t symmetry. checked t h a t a Brandt semigroup i s n e c e s s a r i l y q u a s i c a n c e l l a t i v e .

It can be So a l s o i s

a 0-bisimple inverse monoid S i n which t h e ' r i g h t u n i t subsemigroup'

219

Inverse semigroup algebras {x E S: xx-l

1) s a t i s f i e s t h e l e f t c a n c e l l a t i o n law.

(The r i g h t c a n c e l l a -

t i o n l a w holds automatically.) ( 2 ) A semilattice E

Eo i s s a i d t o be strongly disjunctive i f and o n l y i f

f o r every f i n i t e c o l l e c t i o n e l ,

...,

en o f d i s t i n c t nonzero e l e m e n t s of E

t h e r e e x i s t s f E E such t h a t e x a c t l y one o f t h e elements e i f ( i = 1, is nonzero.

..., n )

E v i d e n t l y t h e s e m i l a t t i c e o f a Brandt semigroup h a s t h i s p r o p e r t y ,

Another example i s given l a t e r . Clearly a necessary condition f o r t h e c o n t r a c t e d a l g e b r a o f an i n v e r s e semigroup S = So t o be s i m p l e i s t h a t S i s 0-simple.

A sufficient condition

i s provided by THEOREM.

7.2.

Let S be a 0-simple inverse semigroup and let F be a field.

Suppose also that ( i ) S is fundamental (in particular, combinatorial), ( i i ) S is quasicancellative, ( i i i ) Es is strongly disjunctive.

Then F o [ S ] is simple. Proof.

I supp a [

Let A be a nonzero i d e a l o f FoCSl and l e t a E A'O

is a s small as p o s s i b l e .

I supp a 1 .

Write n

be such t h a t

W e s h a l l show t h a t

n = 1. Suppose t h a t n > 1 and l e t supp a

..., x,}.

{x,, x 2 ,

By ( i i i ) , t h e r e

e x i s t s e E ES such t h a t e a n n i h i l a t e s a l l b u t one o f t h e e l e m e n t s o f t h e s e t -1

{xixi

: i

= 1,

r 2 1,

..., n ) . x 1x -1 1

and t h a t exixfl

Without l o s s o f g e n e r a l i t y , we assume t h a t , f o r some

... =

-1

x x

-1

xixi

z

xrxilif i > r

-1

# 0 i f 1 2 i 2 r, w h i l e exixi

j By ( i i ) , i f i , j E 11, thus exl, ex2,

=

0 i f i > r.

exi#O

if

1 5 i 5 r ,

o

if

i > r.

exi

i = j;

,

..., r} are such t h a t ..., ex, are d i s t i n c t

exi

e x j t h e n xi

elements o f S ' O .

lsupp eal = r and s o , s i n c e ea E A , w e must have r

Thus

x and s o j Hence

n , by t h e m i n i m a l i t y o f

n. S i n c e e x l # ex2 it f o l l o w s from ( i ) (see 1 . 5 ) t h a t t h e r e e x i s t s f E ES such that t h a t is, -1

x1 ef x1

t x;lefx2.

-1

Consider t h e idempotents xi e f ? ( i = 1, e q u a l , a 1 least one is nonzero. nonempty p r o p e r s u b s e t K o f 11,

..., n ) .

S i n c e t h e s e are n o t a l l

Hence, by ( X i ) , t h e r e e x i s t s g E ES and a

..., n )

such t h a t

W.D. Munn

220

-

-1

xiefxig#O and

( V i , j E K)

x

-1

i

i E K

efxi = x

-1

j

efx

j'

From t h e first o f t h e s e , e f xig # 0 i f and o n l y i f i E K .

F u r t h e r , by ( i i ) , i f

i , j E K are such t h a t ef xig t h a t i s , i = j.

ef x . g t h e n e f xi ef x . and s o , a g a i n by ( i i ) , 1 3 Thus lsupp ( e f a g ) I = I K I < n . But ef ag E A and

xi 'j; t h i s c o n t r a d i c t s minimality o f n. Hence n = 1 and so a

and A E F . 0

Ax f o r some x E S.0

Since S is 0-simple, t h e r e e x i s t u , v E S such t h a t uxv

E A.

y = (A-'u)av

This i m p l i e s t h a t Fo[S] = A .

L e t y E S.

y.

Thus

Consequently, Fo[S] is simple.

I am i n d e b t e d t o Dr P . R . Jones f o r p o i n t i n g o u t t h a t my o r i g i n a l h y p o t h e s i s

i n 1 . 2 t h a t S i s c o m b i n a t o r i a l can be r e p l a c e d by t h e weaker h y p o t h e s i s t h a t S i s fundamental. To c o n c l u d e , we a p p l y 7.2 t o an i n t e r e s t i n g t y p e o f semigroup first s t u d i e d by Nivat and P e r r o t C171. monoid on I .

Let I be a nonempty set and l e t I" denote t h e free

(The elements o f I" are t h e words i n t h e a l p h a b e t I , t o g e t h e r

with t h e empty word 1, and m u l t i p l i c a t i o n i s c o n c a t e n a t i o n . ) S = (I" x

I>$) U

Let

{ O ) and d e f i n e a m u l t i p l i c a t i o n on S by t h e r u l e :

a , b , c , d E I", ( a , b)(c, d) =

I

(aw, d )

if

c = bw f o r some w E I",

( a , dw)

if

b

0

for all

c w f o r some w E I",

otherwise,

( a , b)O = O(a, b ) = 0'

0.

Then S i s a c o m b i n a t o r i a l 0-bisimple i n v e r s e monoid ( c a l l e d t h e poZycycZic

semigroup on 1). Note t h a t i f /I[= 1 t h e n S

2

Bo, where B i s t h e b i c y c l i c

semigroup ( 1 . 9 1. E v i d e n t l y ES = { ( a , a ) : a E I") U { O ) and

( V a , b E I")

(a, a) 2 (b, b)

*

From t h i s it f o l l o w s r e a d i l y t h a t

ES i s s t r o n g l y d i s j u n c t i v e

-

a

bw f o r some w E I*.

I is i n f i n i t e .

I t i s a l s o e a s y t o see t h a t , f o r a l l a , b E I", ( a , b ) - l = ( b , a ) .

Now

suppose t h a t a , b , c , d , e , f E Ift are such t h a t ( a , b ) ( c , d ) = ( a , b ) ( e , f ) # O and (c, d ) ( c , d ) - l and so c where c = e

e.

(e, f ) ( e , f)-l.

From t h e second e q u a t i o n , ( c , c )

(e,e)

Hence from t h e f i r s t e q u a t i o n e i t h e r ( i ) ( a u , d )

(au, f ) ,

bu ( u E I") o r ( i i ) ( a , d v ) = ( a , f v ) , where b = c v

ev ( v

I n e i t h e r case w e have d = f . quasicancellative.

Thus ( c , d )

(e, f).

T h i s shows t h a t S i s

E

22 1

Inverse semigroup algebras 7 . 3 . THEOREM. Let S be the polycyclic semigroup on a nonempty s e t I and l e t F be a f i e l d . Then ( i ) FoCSl is p r i m i t i v e ,

( i i ) FoCSl is simple if and only if I i s i n f i n i t e .

Proof.

( i )This i s an immediate consequence of 7.1.

( i i ) Since S is a combinatorial 0-bisimple i n v e r s e semigroup it i s a

Also, as shown above, it is quasi-

fundamental 0-simple i n v e r s e semigroup. cancellative. disjunctive.

Now suppose t h a t I i s i n f i n i t e . Hence, by 7 . 2 , Fo[S]

Then ES is s t r o n g l y

i s simple.

I t remains t o show t h a t i f I i s f i n i t e then Fo[S]

have f i n i t e l y many elements, x l ,

. . . , xn,

define s E Fo[Sl by a,b

say.

i s not simple.

Let I

For a l l ( a , b ) E I"' x Is"

n Sa,b

1

( a , b) -

(axi. bxi). i=l

We compute t h e product s a Y b ( c , d ) ( a , b , c , d E I"). Case ( i ) : b = cw f o r some w E I".

There a r e t h r e e c a s e s .

Here n

bw f o r some w E I", w # 1.

Case ( i i ) : c

and some v E I";

We have t h a t w = xkv for some k

hence

sa,b(c, d)

n (axkv, d )

1

-

(axi, bxi)(bxkv, d ) i=l

= (axkv, d ) - (axkv, d ) Case ( i i i ) :

f o r a l l w E I", c # bw and b # cw.

a l l v E I", c # bxiv and bxi # cv ( i

1,

0.

Since we a l s o have t h a t , for

..., n ) ,

it follows t h a t

s ~ , ~ ( dc) , = 0.

Now l e t A denote t h e subspace of Fo[S]

spanned by { s a Y b: ( a , b ) E I" x I"}.

The c a l c u l a t i o n s above show t h a t A i s a r i g h t i d e a l of Fo[S]. argument shows t h a t it i s a l e f t i d e a l . (1, 1)

l(a,b)ET X a , b ~ a , b ,

A similar

Suppose t h a t

where T i s a nonempty f i n i t e subset of I" x 1"'

and X

E F \ O f o r a l l ( a , b ) E T. Choose c E I" of l e n g t h exceeding t h a t a ,b of each b with ( a , b ) E T. Then s ( c , 1) 0 f o r each ( a , b ) E T and s o a ,b 0 = (1, l ) ( c , 1) ( c , l), which i s f a l s e . Hence ( 1 , 1) A. Thus we have shown t h a t A is a nonzero i d e a l of Fo[S] Fo[S]

such t h a t A # FoCSl.

Consequently,

i s not simple.

ACKNOWLEDGEMENT

This paper is an extended version of t h e t e x t of a l e c t u r e d e l i v e r e d a t t h e Conference on Group and Semigroup Rings held i n t h e University of t h e

222

W.D. Munn

Witwatersrand, Johannesburg i n J u l y 1985.

I am g r a t e f u l t o P r o f e s s o r G .

Karpilovsky f o r i n v i t i n g me t o t a k e p a r t i n t h i s c o n f e r e n c e and t o t h e Department o f Mathematics of t h e U n i v e r s i t y o f t h e Witwatersrand f o r p r o v i d i n g generous h o s p i t a l i t y d u r i n g my v i s i t .

REFERENCES [l] Amitsur, S.A. On t h e s e m i - s i m p l i c i t y of group a l g e b r a s . Michigan Math. J. 6 (1959) 251-253. [ 2 ] Barnes, B.A. R e p r e s e n t a t i o n s o f t h e R1-algebra o f a n i n v e r s e semigroup.

Trans. A m e r . Math. SOC. 218 (1976) 361-396. Clifford, A.H. Matrix r e p r e s e n t a t i o n s o f completely simple semigroups. A m e r . J . Math. 64 (19q2) 327-342. The a l g e b r a i c t h e o r y o f semigroups. [4] C l i f f o r d , A . H . and P r e s t o n , G . B . Math. Surveys 7 (Amer. Math. SOC., Providence R . I . , 1961 ( ~ 0 1 . 1 )and 1967 (vol.11)). [5] Connell, I . G . On t h e group r i n g . Canad. J. Math. 15 (1963) 650-685. [6] Domanov. A . I . On s e m i s i m p l i c i t y and i d e n t i t i e s of i n v e r s e semigroup algebras. 'Rings and Modules', Mat. I s s l e d . Vyp. 38 (1976) 123-137 [Russian]. Some theorems on commuta[7] Drazin, M.P., Dungey, J.W. and Gruenberg, K.W. t i v e matrices. J. London Math. SOC. 26 (1951) 221-228. [8] Howie, J.M. An i n t r o d u c t i o n t o semigroup t h e o r y (Acad. P r e s s , London, 1976 1. [9] Munn, W.D. On semigroup a l g e b r a s . Proc. Cambridge P h i l . SOC. 5 1 (1955) 1-15. [lo] Munn, W.D. Matrix r e p r e s e n t a t i o n s of semigroups. Proc. Cambridge P h i l . SOC. 53 (1957) 5-12. [ll] Kunn, W.D. I r r e d u c i b l e m a t r i x r e p r e s e n t a t i o n s of semigroups. Q u a r t e r l y J . Math. Oxford ( 2 ) 11 (1960) 295-309. A class of i r r e d u c i b l e m a t r i x r e p r e s e n t a t i o n s of a n a r b i t r a r y [ 1 2 ] Munn, W.D. i n v e r s e semigroup. Proc. Glasgow Math. Assoc. 5 (1961) 41-48. [131 Munn, W.D. Matrix r e p r e s e n t a t i o n s of i n v e r s e semigroups. Proc. London Math. SOC. ( 3 ) 14 (1964) 165-181. [14] Munn, W.D. Semiunitary r e p r e s e n t a t i o n s o f i n v e r s e semigroups. J . London Math. SOC. ( 2 ) 18 (1978) 75-80. C151 Munn, W . D . Simple c o n t r a c t e d semigroup a l g e b r a s . Proceedings of a c o n f e r e n c e i n honor of A l f r e d H . C l i f f o r d , Tulane U n i v e r s i t y , 1978 (Tulane U n i v e r s i t y , N e w O r l e a n s , 1978) pp. 35-43. C161 Munn, W.D. S e m i p r i m i t i v i t y of i n v e r s e semigroup a l g e b r a s . Proc. Roy. SOC. Edinburgh A 93 (1982) 83-98. C171 Nivat, M. and P e r r o t , J . - F . Une g & r a l i s a t i o n du monoyde b i c y c l i q u e . C.R. Acad. S c i . Paris (S6r.A) 271 (1970) 824-827. Cl81 Oganesyan, V.A. On t h e s e m i s i m p l i c i t y of a system a l g e b r a . Akad. Nauk Armyan. SSR Dokl. 2 1 (1955) 145-147. C191 Okniiiski, J. On r e g u l a r semigroup r i n g s . Proc. Roy. SOC. Edinburgh A 99 (1984) 145-151. [ 2 0 ] Passman, D.S. On t h e s e m i s i m p l i c i t y o f t w i s t e d group a l g e b r a s . Proc. A m e r . Math. SOC. 25 (1970) 161-166. [ 2 1 ] Passman, D.S. I n f i n i t e group r i n g s (New York, Dekker, 1971). c221 Passman, D.S. The a l g e b r a i c s t r u c t u r e o f group r i n g s (Wiley, New York, 1977 1. C231 P e t r i c h , M. I n v e r s e semigroups (Wiley, New York, 1984). C241 Penrose, R. A g e n e r a l i z e d i n v e r s e f o r matrices. Proc. Cambridge P h i l . SOC. 51 (1955) 406-413. C251 Ponizovski!, I.S. On m a t r i x r e p r e s e n t a t i o n s o f a s s o c i a t i v e systems. Mat. Sbornik 38 (1956) 241-260 [Russian].

[a]

223

Inverse sem&oup algebras

Ponizovski?, I.S. On irreducible matrix representations of finite semigroups. Uspehi Mat. Nauk 13 ( 1 9 5 8 ) 139-144 [Russian]. C271 Ponizovski:, I.S. An example of a semiprimitive semigroup algebra. Semigroup Forum 2 6 ( 1 9 8 3 ) 225-228. C281 Preston, G.B. Inverse semi-groups. J. London Math. SOC. 29 ( 1 9 5 4 ) C261

396-403.

[33]

Preston, G.B. Matrix representations of inverse semigroups. J . Austral. Math. SOC. 9 ( 1 9 6 9 ) 29-61. Rickart, C. The uniqueness of norm problem in Banach algebras. Ann. Math. 5 1 ( 1 9 5 0 ) 615-628. Teply, M.L., Turman, E.G. and Quesada, A. On semisimple semigroup rings. F’roc. Amer. Math. SOC. 79 ( 1 9 8 0 ) 157-163. Vagner, V.V. Generalised groups. C.R. Acad. Sci. URSS 84 ( 1 9 5 2 ) 1119-1122 [Russian]. Villamayor, O.E. On weak dimension of algebras. Pacific J. Math.

[34]

Weissglass, J.

[29] [30]

[31] C321

9 ( 1 9 5 9 ) 941-951.

Regularity of semigroup rings. Proc. Amer. Math.

SOC.

25 ( 1 9 7 0 ) 499-503.

Self injective semigroup rings for finite inverse semigroups. Proc. Amer. Math. SOC. 2 0 ( 1 9 6 9 ) 213-216.

[35] Wenger, R.H.

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Group and Semigroup Ring8 G.Karpilovsky (ed.) 0Elsevier SciencePublishen B.V.(North-Holland),1986

225

SECTIONS AND IRREDUCIBLE MODULES M. F. O'Reilly Rhodes University Grahamstown, 6140 South Africa In this talk we summarize some results about p'-sections in modular group algebras and indicate a new development. 1.

NOTATION AND PRELIMINARY RESULTS Let G be a finite group, k a splitting field of characteristic p where

p divides the order of G and A = kG, the group algebra of For H 5 G, let

41 =

{y

E

A: h-lyh = y,h

E

HI

i.e

G

41

over k.

is the space of

H-invariant elements of A when H acts by conjugation. G = Cgi -1ygi where the sum is over the set of coset For y e 41 let yH G G representatives gi of H in G. Then AH = {a E A: a = yH, y E 41) is an ideal of the centre AG

of A.

For subgroups H, K of G, y

41, B

E

E

+, we

use the Mackey formula to

obtain products:

where the sum is over double KgH coset representatives g, in G. For a finite subset

x

let

=

-the conjugacy class sums. Each Di

ci

Z X. Then A has basis {~i,i=l,...ul xex g is a non zero multiple of xgi where

is a Sylow p-subgroup of the centralizer C(xi) of xi, xi

E

Ci. n

such that a A vertex is always a p-subgroup and is unique up to conjugation in G. For a

E

AG, a vertex of a

Each x E G element and

s

is a minimal subgroup H

E i$.

can be written uniquely x = rs = sr where r is a p-regular a p-element. A p'-section Sr containing r is the set of

all elements which have p-regular part conjugate to r. Let rl...,r t be a set of representatives of the p-regular conjugacy classes and S1, St

...,

the corresponding p'-sections with ri 2.

si'

PROPERTIES OF p'-SECTIONS In C5l. Reynolds showed using characters that

Isi,

...

The set i=l, t) spans an ideal of AG. Prop 1: A character-free proof is given in 3 which gives a formulation of a p'section sum which facilitates algebraic manipulation in A.

Thus

226

M. F. O'Reilly

-Sr

Prop 2:

Br

[C(r):D]

=

-G

= Br(rDID

.

k

E

where D

is a Sylow p-subgroup of

C(r)

and

Using Prop 2 and Mackey's formula it is then shown that any product - G G S (b ) decomposes into p'-section sums (r'E')D, with D' 5 D. r Q (A generalization of this can be found in [4] where it is shown that: Prop 3:

the space Im

For any integer m ,

where P 5 D 5 C (r),r

(Fy);

p-subgroup of

,

N(Py)

D:P

spanned by all elements

the p-regular part of y dividing pm,

E

N(P)

, D a Sylow

is an ideal of AG .)

In [61, Tsushima using characters theory proved

n Ann

Prop 4 :

xr = radA

We give here a character-free proof of this result.

(It has recently been

pointed out to me that a proof using the same idea has been given by Kiilshammer [2] in the more general situation of symmetric algebras).

The proof will

depend on the two subsequent lemmas. Let

T

be the subspace of A

S

{e: ep

=

i

E

spanned by all ab

S , i > 0 1 and J = radA.

- bar

a, b

i = 1. ..t}.

p ( 1 ) = 1, p(x)

Define =

and

p

0 for x

G

E

as linear functions A

pi

-

A,

Since Jn = (0) for some n , J c T.

It i s known (cf 111 p 100) that T is spanned by the set {x x ESi,

E

-

4

ri' k by

{l)

and q(x) = 1 Then ker pi

for x

E

Si

,

p

i

(x) = 0 for x

is spanned by the set

T =

(G

-

Si)

fl ker p

i

E G U {x

- Si ' - ri, x

E

and

Sil

(1)

i

Then Lemma 1.

J

Proof: Let

i s maximum as a (two-sidedlideal of

A

lying in T.

I be an ideal of A which is not contained in J .

Then

(I + J)modJ, being semisimple has a simple component A which is isomorphic 1 to a full matrix algebra over the splitting field k. Let $: I + J +. A1 be the natural surjective homomorphism. Choose an idempotent element f E A1 E I + J such that $(e) = f. Then

which has non-zero trace and choose e for all i ,

i

$(ep ) =

{$(e)}P

i

=

f and so has non-zero trace.

But elements

y E S have trace {$(y)1 = 0 , since$(y) i s the sum of elements of the form i $(a)$(b) - $(b) $(a), each of which has trace 0. Thus ep S for any i

T and I & T , completing the proof of Lemma 1. -1 Let Si be the p'-section containing r i' so

e

E

is maximum as an ideal of A lying in kerpi. Lemma 2. Ann We have that i~ and p i are related by pi(x) = p(xzi). Thus

Proof: ann

s-i

i s contained in

kerpi then

-

IS;

kerpi.

ideals; for if p(Cax x) = 0, ax

I i s an ideal of

Also i f

is an ideal lying in E

k, x

kerpi. E

G and

But

r

contained in

kerp contains no proper

if y

E

G

is such that

227

Sections and irreducible modules

# 0 then ~(y-~,Xa~x) = u # 0 and so no non-trivial subset of ker l~ Y Y closed under multiplication by elements of r . So I Ei = (0) whence a

is

I c a n n 5 i- ' Proof of Prop 4 .

ui for each i by ( 1 ) . So by Lemma ann 5 i c ker pi = T and by lemma 1,

J c T c ker

J c

i.

So

3.

prSECTIONS AND IRREDUCIBLE G-MODULES Let W

be the ideal of AG

Let R1,...,Rt

2,

J c ann

J =

generated by the p'section

5-

arms-i = 7 ann 5 i'

n

...,-St-

sums sl,

be the distinct irreducible G-modules (The number

equals the number of p'-conjugacy classes).

for each

i

t

of these

Any two-sided ideal I of A

can

be made a G x G module by setting Thus A, J(A)=radA,

A J (A)

v.(x,y)

A

=

-1

x vy,

x,y

G, v

E

E

I

J ~ A )and SocA (the socle of A) are all G x G-modules. As

iil

R,, where R i is the sum of all minimum right SOC A = @ ideals of A isomorphic as G-modules to Ri. Further there is an algebra isomorphism A @ IR; where the R; are the simple components of the semisimple algebra A/J(A) with K; Ri as indecomposable G x G-modules. such,.

xA)

Let AG = {(x,x) {m

E

M: m.(x,x)

E

G x GI.

For a G x G-module M

Then InvAG R;

= m).

=

centre of

let

InvGxG M

=

R; which is of dimension 1.

Let c be a basis of Inv R Then {c l,.... ct} is a basis of InvAG i AG i' (SOC A) which must therefore have dimension = t. But W c InvAG(Soc A) hence W = InvAG(Soc A). Theorem 1:

This gives

The correspondence Ri++ Ci

gives a natural 1-lcorrespondence

between the irreducible G-modules and certain 1-dimensional subspaces of W. We will call each ci (unique up to scalar multiplication) the core of A Sylow p-subgroup of the vertex of

ci will be the core group of

Ri.

Ri or Ri.

It would be nice to obtain local correspondence properties of core elements as well as local correspondences or counting results of core elements with a given core group along the lines of the Brauer correspondence for blocks. Towards this end we give some results about central eigenvectors, that is elements v

of AG

eigenfunction AG 4.

+

such that va = X(a)v,

a

E

AG. X

is the corresponding

k.

CENTRAL EIGENVECTORS.

....

+ + E be a decomposition of the identity element 1 E A 1 into the sum of primitive central idempotents E (the block idempotents). j Each central eigenvector v lies in some block AE For vE. = X(E )v and j' I j since X(E.) # 0 for some j (otherwise v = 1 v E = I X(E )v = O ) , for such 1 = E

Let

j, v

3

E

AE

Prop 5:

j'

(i)

j

Every central eigenvector lies in W

j

j

5

228

M. F. O'Reilly

(ii)

-

W

j

WE

j

is a space, each element of which is a

central eigenvector with common eigenfunction given by the central character associated with E j Proof: (i) Let v be a central eigenvector with eigenfunction A , For a

E

radAG, since am = 0 for some integer m ,

A(a)

0. Then v

anni-

hilates radAG and thus lies in the socle of AG = AG fl SOC A = W. (ii) Let a E AG For v f W v = vE and va = (vE)a = v{w(a)E + ql 1 1 j' where q E radA 4. But since v E W c ann J , vq = 0 and so va = da)v. Since each core element ci lies in Ri and for some block idempotent RiE = Ri we have

.

Corollary 1. Each core element c is a central eigenvector with associated eigenfunction given by the central character of the block containing C. Corollary 2 .

The core group a of a core element c is contained in the

defect group of the block containing c. S

From Proposition 5, W = Ul

C

W

j = 1 j'

, W'(G)

Let Wj(G)

j

be the subspaces

of W consisting of central eigenvectors with vertex 5 Q , < Q respectively j and W(G), W' (G) the corresponding subspaces of W Each W' (G) can be

-

complemented in Wj(G)

.

.

j

Wj(G) ij(Q) ls W;(G) Similarly W(G) W(Q) 8 W'(G). C i (GI Also since W(G) has as basis thepl-sections 3

Then i(G) =

@

-j

of vertex D, dim W(G)

.

equals the number of p-regular classes with vertex D.

Hence we have Prop

6:

C dimIW (G)modWj(G)l j j

equals the number of p-regular classes of

vertex D. Corollary

: The number of central eigenvalues of vertex

111 equals the

number of p-regular classes of defect 0. Local properties: The most useful map to obtain properties is a variation of the Brauer map. Let Q be a p-subgroup of G and H = Q C (9). N = N(Q). Define the linear map u: kG 4 kN by u(x) x if x E H , u(x) 0 if x f G - H . Then u is an H x H - module homomorphism. Also, however,

-

u has two important properties.

s

If is a p'-section sum of kG with vertex Q, then is a p'-section sum of kN.

Theorem 2 : u(l)

Proof:

s

Let = B(r5); , B f k , where r is a p-regular element in C(D). Then r f o ( S ) and hence rD and all its conjugates by N are preserved lies in H ; for if so by 0 . Also if g d N then no element of (rD)' by taking suitable powers rg E H. But the only conjugates of r in H are those which are conjugate to r in N since D is a Sylow p-subgroup

Sections and irreducible modules

of C(r). Theorem 3:

229

Thus o{(rE)g} = 0. Hence ~ ( $ 1= (rD)D. - N If K E AG is a class sum and S a p'-section then u(s) o ( t ) =

= o(Sic)

Proof:

Let

s = 6(6r)D G

and

z

=

(a),, G

B,y

E

k. Then Zi? = (By) (&)D G aL G =

where the sum is over double D x L cosets. But (Erax)DnL~ G

By Z.

is the sum of p-section sums of vertex < D unless rax E C(D)

whence D

n Lx

=

D

-

(DraX)DnL~ G

ax E C(D) and this term is a single p'-section sum whence o(6rax)z i.e.

=

(Erax):. Thus

o ( S t ) = 66

&(&aX):

= 6y

classes of ax

E

= o(S)

5.

$ax a EC(D)

C(D) u(f)

IRREDUCIBLE MODULES

...

t generates Ri Each core element ci, i = 1, following result is easily established

Prop 7 :

Let w =

Ccr

c

i i'

G x G-module.

is the G x G module generated by

Then if <w> <w> =

as a

@

ai#O

The w,

Ri

REFERENCES 1. 2. 3. 4. 5. 6.

Feit, W, The representation theory of finite groups. North-Holland, Amsterdam, 1982. Khihammer, B, Bemerkungen fiber die Gruppenalgebra als symmetrische Algebra. J. Algebra 72, 1-7, 1981. O'Reilly, M.F., Ideals in the entre of a group ring, Lecture Notes in Math 372, Springer, Berlin, 536-540, 1974. O'Reilly, M.F., A class of ideals of the centre of a group ring. J. Austral. Math. SOC. (Series A) 3 1 (1982) 146-151. Reynolds, W., Sections and ideals of centres of group algebras. J. Algebra, 20, 176 -181. Tsushima, Y. On the p'-section sum in a finite group ring. Math J. Okayama Univ., 20, 83-86, 1978.

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Group and Semigroup Rings G. Karpilovsky (ed.) 0 Elsevier Science Publishers B.V.(North-Holland), 1986

231

NOETHERIAN SEMIGROUP RINGS WITH SEVERAL OBJECTS bY Gunther RICHTER, UniversitHt Bielefeld, Bielefeld, West Germany.

Whether the following generalized version of Hilbert's Basis Theorem GHBT: C - W locally noetherian 9 C[X]-w locally noetherian, holds o r not depends on the small category 2(_ and possibly likewise on of modules over a ring C w i t h several objects i n the category the sense of B. Mitchell [ill; C[XI :=C@Z[Xl generalizes the concept of semigroup rings. There i s only l i t t l e hope t o characterize a l l small categories X f o r which GHBT holds because of the special case C = k being a commutative f i e l d and X = ( G , . ) being a group. Then i s just the category of l e f t modules over the group r i n g k[G]. I t i s a rather old unsolved problem t o distinguish a l l groups f o r which k[G] is l e f t noetherian [12,13]. We consider a class of certain order enriched small categories including a l l p a t h categories of quivers, f r e e commutative monoids, posets, and others; GHBT holds f o r such a category and a l l C i f f i t f u l f i l s the ascending chain condition on subfunctors of i t s covariant morphisn functors. The classical Hilbert Basis Theorem i s a special case of t h i s result. AMS: 05C20 , 16A33 , 16A90 , 18A25 , 18B35 , 18D20

c-m

c[X]-m

INTRODUCTION B. Mitchell considers small additive categories C as rings with several obj e c t s [ill. From t h i s point of view, an arbitrary small category X i s j u s t a semigroup with several objects. So we get a natural generalization of semigroup rings as follows C[XI := c@z[Xl 0.

where Z[x] denotes the f r e e additive category generated by

X.

Recall

R[Sl = R@Z[Sl f o r any semigroup r i n g R[Sl. Our aim i s t o characterize small categories alized version of Hilbert's Basis Theorem h o l d s GHBT:

C-@

locally noetherian

r,

X

f o r which the following gener-

C[X]-w locally

noetherian,

c-m

i s the category of a l l covariant additive functors from C.$l 0 to where the category & of abelian groups, and ' l o c a l l y noetherian' f o r C-W means j u s t ACC on additive subfunctors of a l l covariant morphism functors HomC(C,-) : c ->

-

&, C

E

C [14,15].

If X = ( N , t ) is the f r e e monoid w i t h one generator and C=R a ring with 1 , then GHBT specializes t o the original Hilbert Basis Theorem:

R l e f t noetherian

I ,

R[Xl

l e f t noetherian,

G. Richter

232 because R

l e f t noetherian

c)

R-Mod l o c a l l y noetherian.

Unfortunately, there i s o n l y l i t t l e hope f o r a general s o l u t i o n o f the problem above because o f the hard case o f group r i n g s where x = G i s a group and

&=

k

a commutative f i e l d 112,131. Fortunately, there are some important classes o f small categories 11, f o r instance a l l path categories o f o r i e n t e d multigraphs (quivers) [ 5 1 o r a l l posets [61, i n which complete i n t e r n a l characterizations o f those X f u l f i l l i n g GHBT are a v a i l a b l e . I n the f o l l o w i n g we consider a common g e n e r a l i z a t i o n o f these cases and most o f the well-known case o f commutative monoids 141. 1.

GENERAL RESULTS For convenience we use t h e natural isomorphism X C [ X I - M = (C@Z[XI)-W EL C-Mod-,

where the l a t t e r denotes the category o f a l l covariant functors from

c[x]

X

to

C-W.

Instead o f covariant morphism functors from to we look a t the 'canonical C - M -X , X E X , C E C , defined by

generators TX(C) i n

TX(C)(Y) := f o r objects X E

X,

@ HomC(C,-) fEh(X,Y) and i n the obvious way f o r X-morphisms h : Y ->

1.1 Lemna L e t C , X , X l be such t h a t GHBT holds. Then GHBT holds f o r

XOX',

resp.

Proof.

X'

c-WXxL' 9 (c-MX)and

C-MoJXuX'

X ag - w

Z E K(Y,Z).

C and X x X l x

&-Mod-X '

or

.

The l a t t e r isomorphism c a r r i e s canonical generators i n G-W-X W X ' over t o p a i r s X o r C-MX' , resp. , and subfunctors t o p a i r s o f canonical generators i n

&-w

o f subfunctors. The f o l l o w i n g generalizes the w e l l known f a c t t h a t i s a q u o t i e n t o f the monoid S.

R[Sl i f

f

R[fl

i n h e r i t s ACC from

1.2 Lemna L e t C ,X be such t h a t GHBT holds. If H : X -> and f u l l then GHBT holds f o r C and

8.

i s s u r j e c t i v e on objects

Proof. By assumption, H i s a s u r j e c t i o n on morphisms, too. Therefore,

Iz

C-ModH : C - M - ->

--

A

C-W-X A

M +>M*H

'

233

Noetherian semigroup rings

i s an embedding which preserves monomorphisms (subfunctors) and r e f l e c t s i s o -

.

Thus i t i s enough t o show t h a t TA(C) 0 H i s morphisms, hence o b j e c t s w i t h ACC X H(X). However, t h i s i s t r i v i a l because H induces always a q u o t i e n t o f TX(C) ,

t=

a pointwise s u r j e c t i v e n a t u r a l transformation u = (uy)yEX such t h a t the diagrams

commute. Every proper subclass o f objects i n a (small) category X defines a f u l l subcategory X', thus a new semigroup w i t h several objects. This c o n s t r u c t i o n t r i v i a l i z e s f o r monoids considered as categories w i t h one o b j e c t . Therefore, the following has no analogue on the monoid l e v e l 1.3 Lemma Let

C,X be

such t h a t GHBT holds. I f E

then GHBT holds f o r

C

and

X'.

Proof. Consider the r e s t r i c t i o n Kan extension L 4

c-mE[ l o ] .

: X ' L > 11 i s

C - M E : C-MX- ->

a f u l l embedding

X' C-w -

and the l e f t

By assumption on E we have, w i t h o u t l o s s o f

generality,

(c-ME)O L

( *)

Moreover, L preserves 'canonical

'

L(TX'(C)) for a l l X' E

X' , C E C . uo

=

id

c-m-X ' .

generators, i.e. TqXI)(C)

=

Every ascending chain

> -

u

> -

U 1f\

T~ I

...

un

->

..yun ... cci

o f subfunctors o f TXl(C) y i e l d s an ascending chain i n I m L(uo) ->

Im L(ul)L>

L(TX'(C))

...

=

...

C - e -X

Im L(un)L>

TE(X')(C)

where I m L(un) i s given by t h e image f a c t o r i z a t i o n o f L(un) :

Y

...

G. Richter

234

The l a t t e r chain t e n i n a t e s by assumption. But i t s r e s t r i c t i o n t o

.

X'

i s just

the o r i g i n a l chain, according t o (*) 1.4 Remark I f X i s d i s c r e t e , i . e . every X-morphism i s an i d e n t i t y , then GHBT holds for

X and

a 1oc.noeth.

C-M+0

( t r i v i a l ) examples f o r i n f i n i t e

iff

X is

f i n i t e . On the other hand there are

X fulfilling

GHBT, because GHBT i s s t a b l e under

categorical equivalence. So, i f we add t o an X-object X i n f i n i t e l y many i s o morphic copies, we get an i n f i n i t e category equivalent t o X. ( L a t e r on, we consider l e s s t r i v i a l i n f i n i t e examples.) This shows t h a t GHBT i s n o t i n h e r i t e d by a r b i t r a r y subcategories!

1.5 Proposition L e t X be an a r b i t r a r y small category, then ( i )

(i)

GHBT holds f o r

& and

(ii)

GHBT holds f o r

X

a l l rings

C with

and some n o n t r i v i a l

*

(ii) * (iii)

( i v ) , where

several objects.

C=# 0 such t h a t

&-wi s l o c a l l y

noetherian. (iii)

X

i s Zeft congruence noetherian, i.e. every covariant morphism f u n c t o r

: -> Set f u l f i l s ACC on i t s congruences considered i n the f u n c t o r category Set-X

X(X,-) (iv)

.

21 i s Zeft idea2 noetherim,

i . e . every X(X,-)

f u l f i l s ACC on i t s sub-

functors. Proof. C l e a r l y ( i ) congruences on X(X ,-)

*

(ii). For ( i i )

Ro ->

*

( i i i ) consider an ascending chain o f

...

R1 ->

...

Rn ->

and t h e corresponding chain o f q u o t i e n t s X(X,-) -

->> /RO

->>

X ( X y - 1

-

/R1

This y i e l d s a chain o f quotients o f TX(C) f o r some C $(C)

where

-->>

Qx(C) 1 ->>

...

...

->>

X(X,-)

-

/Rn

40 Q;(C)

->>

,

...

Noetherian sem&roup rings

235

By construction, t h i s chain terminates i f f t h e o r i g i n a l one does, because

i s n o n t r i v i a l . Now, C D 1 - M i s an a b e l i a n category, and we have a onezto-one correspondence between subobjects and quotients. Therefore, every chain o f q u o t i e n t s o f TX(C) as above terminates. To prove (iii) * ( i v ) observe t h a t every subfunctor HomC(C,-)

s c > X(X ,-) defines a Rees congruence RS on K(X,-)

as f o l l o w s

f RS(Y)g :w f = g

or

f,gES(Y)

because f

f o r every h : Y

3

RS(Y) 9

Z in

*

(h.f)

Rs(Z)(hog)

X.

1.6 Remark Condition ( i v ) above i s n o t s u f f i c i e n t f o r ( i ) i n general, because a group X=G has o n l y t r i v i a l l e f t ideals. But ( i i i ) i s j u s t ACC on subgroups which i s supposed t o be s u f f i c i e n t f o r ( i i ) ifC= k i s a commutative f i e l d 1131. I f we look a t monoids X=M, subfunctors o f c o v a r i a n t morphism f u n c t o r s correspond t o l e f t ideals, and congruences i n SetM t o l e f t congruences on M, resp. Chain conditions on i d e a l s and congruences have been considered by several authors, f o r instance L. Budach [21 and E. Hotzel [81. There i s an order t h e o r e t i c expression f o r using t h e r i g h t d i v i s o r r e l a t i o n on t h e s e t <X c m o n domain X f ( g :wS h : h o f = g .

X t o be l e f t i d e a l noetherian +X> o f a l l X-morphisms w i t h

Obviously, t h i s defines a r e f l e x i v e and t r a n s i t i v e r e l a t i o n , hence a preorder on <X

+x>.The equivalence r e l a t i o n f Og

:w

f < g and g

y i e l d s t h e u n i v e r s a l associated poset <X


,*

which has a smallest element,

namely t h e class o f t h e i d e n t i t y i d X . 1.7 P r o p o s i t i o n A small category X i s l e f t i d e a l noetherian i f f every poset

<X+X>,* i s stmngZy artinian, i . e . i t s a t i s f i e s DCC and contains no i n f i n i t e t o t a l l y unordered s e t [71. To prove t h i s , we need a very useful property o f s t r o n g l y a r t i n i a n posets, o r i g i n a l l y due t o I . Kaplansky 11, p. 1811 and independently recovered by M. Hoppner [61. 1.8 L m a A poset (1,s) i s s t r o n g l y a r t i n i a n i f f every i n f i n i t e subset contains an i n f i n i t e ascending chain.

G. Richter

236 Proof o f 1.7. L e t

X be

l e f t i d e a l noetherian. We have t o show t h a t there i s

n e i t h e r a s t r i c t l y decreasing sequence nor a sequence o f pairwise incomparable elements i n <X f&-/+, X E Now l e t ([fi : X -> be s t r i c t l y decreas-

X.

ing, i . e . fi


0. Then 0

I

fi

g : xi -> Y1 SX(X,Y)

defines an ascending chain o f subfunctors which does n o t terminate because

'Si,l(Xi)

fiE S p i )

E N i s pairwise incomparable we look a t t h e ascending chain

f o r i 2 1. I f ( [ f i l ) i

i Ti := u Sk

(Sk as above)

k=O

and get the corresponding c o n t r a d i c t i o n . Now l e t SOL>

...

SIL>

SnL>

...

&(X,-)

be an ascending chain o f subfunctors ( w i t h natural i n c l u s i o n s between them) and suppose t h a t there i s a morphism fi : x ->

xi

E S.(X.) 1 1

si,l(xi)

f o r each i 2 1. By assumption and 1.8 we g e t an increasing subsequence o f ([fil*), e s p e c i a l l y we g e t natural numbers i $ j such t h a t fi < f j , i . e . there i s a morphism h : Xi -> X . w i t h h o f . f J 1 j. Now the f o l l o w i n g diagram commutes

X(X,h)

X(X,Xi)

T,

h0

-

&(X,Xj)

>

Si(Xj)

S i (h)

si(Xi 1

I

>

I

e s p e c i a l l y we have

E S.(X.) 1 J

Si(h)(fi)=hofi=fj which c o n t r a d i c t s f .

J

2.

4

Sjml(Xj)

Sj-l(Xj).

ORDER ENRICHED CATEGORIES I n t h i s section we introduce a type o f (small) categories

& which

general-

i z e s path categories, e s p e c i a l l y f r e e monoids, free commutative monoids, as well as posets. Therefore, l e t @ be the category o f well-ordered s e t s and i s c a l l e d welt-order enriched i f i t s s t r i c t l y increasing maps. A category covariant morphism functors f a c t o r i z e (not necessarily uniquely) over

X

x X ( X s-1

------

\-

e:

> woSet

l i e r l y i n g set functor

Set

.

Noetherian sem&roup rings

2.1 Lemna well-order enriched iff every morphism s e t X(X,Y) such t h a t

X is

5xY

237

f$xy 9

-

c a r r i e s a well-order

h * f # X Z h.9

f o r every h : Y -> Z. From t h i s we g e t imnediately 2.2 C o r o l l a r y Let

(i) (ii)

X

be well-order enriched, then

every morphism i n jf i s mono, every automorphism i n

X

i s trivial.

The l a t t e r c o n d i t i o n t e l l s us t h a t t h e r e i s no n o n t r i v i a l group contained i n X as a subcategory. Nevertheless t h e r e a r e a l o t categories: 2 . 3 Lemna

o f well-order enriched

Subcategories, a r b i t r a r y coproducts and f i n i t e products o f we1 1-order enriched categories a r e again w e l l -order enriched. Proof. The assertions on subcategories and coproducts are immediate consequences o f 2.1. I f Xl,j2 are well-order enriched then

defines a well-order on

(Xl ~ 2 , )((XlyX2),(Yl,Y2))

which f u l f i l s the compati-

b i l i t y c o n d i t i o n mentioned i n 2.1. 2.4 Examples

&= (1,s) a r e well-order enriched because IX(X,Y) [ s 1.

(a)

Posets

(b)

Path categories &=p(Q)o f quivers Q are well-order enriched because every path p E X(X,Y) from X t o Y has a unique representation p = xm * as a composition o f edges xi. edges and d e f i n e f o r q = y n P $ q :-

.{

... *xl

Now take a well-order

... oyl

E X(X,Y),

s on

the set o f

yi edges,

m # n or xi $ yi

for

i := m i n ( j

I x J. 9 y .JI

if m=n.

Obviously, t h i s i s a well-order which f u l f i l s the c o m p a t i b i l i t y cond i t i o n i n 2.1. I t i s j u s t t h e lexicographical order f o r paths w i t h the same l e n g t h i f we read from t h e r i g h t t o t h e l e f t . Especially, f r e e monoids M(1) w i t h a s e t I o f generators a r e well-order enriched. (c)

The f r e e comnutative monoid

X= Mc(I)

w i t h a s e t I o f generators c a r r i e s

G. Richrer

238

a s i m i l a r well-order because every m E M c ( I ) has a unique representation

a€ I w i t h ma E N and m a = O f o r a l l b u t f i n i t e l y many a . Now l e t n =

Let

2!

'a a

Mc(I), then the desired well-order i s defined by

be another element i n

(d)

n

a€ I

be f r e e l y generated by t h e quiver

with relations

,

Y i + l o x i = xitloyi

Then every s : X k ->

Xl

iE

has a unique representation

s =

0

...

Ykmlt

(e)

Let

Y

. ..

OXkm

w i t h 0 5 m 5 l - k . Now take s , s ' : xk -> s s s '

Z.

Xl

OXk

and d e f i n e

m 3 m ' .

be f r e e l y generated by the quiver

with relations

i E Z.

yio xi =

I n t h i s case, every s : X k ->

s = y-l

0

Xl

has a unique representation

... o y k o xkn

and

s $ s'

:Y

n

n'

defines the desired we1 1-order on

X(Xk ,Xl)

.

There a r e a l o t o f s i m i l a r examples. 3.

SPECIAL RESULTS

3.1 Theorem A well-order enriched small category l e f t i d e a l noetherian.

f u l f i l s GHBT f o r a l l

C iff

it i s

239

Noetherian semigroup rings

Proof. We have t o show t h a t every TX(C) i s noetherian, i . e . t h a t every ascending chain So->

S1

-7

... Sn

... TX(C)

->

o f subfunctors o f TX(C) t e r n i n a t e s . Without any l o s s o f g e n e r a l i t y we suppose t h a t the monomorphisms above are natural i n c l u s i o n s . Now assume t h a t t h e chain (Sn)nEN

i s s t r i c t l y ascending, i.e. f o r every n E N there i s an o b j e c t Yn E

y

and an o b j e c t Cn E C such t h a t

5 Sn+l(Yn) (Cn)

Sn(Yn) (Cn)

Hence we have an ( x j n I ) E TX(C)(Yn)(Cn) with

By assumption on

11 i t

=

*

d

f€X(X,Yn)

HomC(C,Cn) -

i s possible t o choose (x(")

such t h a t f fn :=max{f E x(x,Y,) I x f( n ) 01 becomes minimal. Using 1.7 and 1.8 we g e t an increasing subsequence ( [ f i j l + )

*

-'

o f ([fnl+). Hence, t h e r e a r e X-morphisms h j : 'ij 'ij t i w i t h h j 0 fi = f i Now consider t h e ascending chain o f subfunctors o f HomC(C,-) generated bv

C-W t h i s sequence w L : Ci

By assumption on

and a f i n i t e

chain terminates, i.e. -> C i ,L=O,.,.,k-1, k

e

with

hence

I

0

otherwise.

j+l.

there i s a n a t u r a l number k o f C-morphisms such t h a t

G. Richter

240

X is well-order enriched. Therefore, by 2.1

we have

But (xg

for L

=

) E SiLt1(YiL)(Cil), hence

0,

...,k-1. Therefore,

(

which contradicts the choice of x iik))

. o

3.2 Corollary (M. Hoppner [ 6 1 ) . GHBT holds for a poset (Iys)and all C iff (1,s) is ZocaZly strongly artinian, i.e. every non-empty subset J c_ I which is bounded from below is strongly artinian. Proof. (1,s) is locally strongly artinian iff the subsets of all successors of fixed elements CL I , = Ii E I I a s i )

x=

are strongly artinian. But I,

,

n EN,

where q : X -> Y i s some path. This c o n t r a d i c t s 1.8. Moreover, t h e r e a r e o n l y f i n i t e l y many n o n t r i v i a l path components i n Qx because paths w i t h f i n a l vertex i n d i f f e r e n t components a r e incomparable. Next we show t h a t t h e r e are o n l y f i n i t e l y many v e r t i c e s i n Q, w i t h more than one edge ending a t them. I f not, we may assume t h a t there i s an i n f i n i t e ascending chain o f paths i n < X c X > ending a t such v e r t i c e s Yn, n E N , o u t s i d e t h e f i n i t e l y many and f i n i t e path components. Hence we have an " i n f i n i t e " elementary path j o i n i n g a l l Yn, path:

0 0

-

\>

Po

and t h e r e are edges en ending a t Yn o u t s i d e t h i s 0

el 0

+ 7 0

yo

p1

Now take any path qn : X ->

... - 0

>

y1 p2

Yn v i a en , n

.-

rn* - Pn+l

-> 'n

a

9,

..... .

Pn+1

E N . Obviously, the paths n E

Y

F4

are pairwise incomparable, which i s a c o n t r a d i c t i o n . By a s i m i l a r argument we g e t t h a t t h e r e a r e o n l y f i n i t e l y many v e r t i c e s i n

Q, w i t h more than one edge beginning a t them. Observe t h a t t h e r e a r e o n l y f i n i t e l y many elementary paths between X and a f i x e d vertex Y i n Q,, because such paths a r e pairwise incomparable. Now consider t h e f i n i t e f u l l subquiver spanned by a l l v e r t i c e s on elementary paths ending a t v e r t i c e s i n path components o r v e r t i c e s w i t h more than one edge ending o r beginning a t them, and the f u l l subquiver spanned by a l l the others. By construction, the connected corn

r,

ponents o f the underlying unoriented graph o f But they are chains i n

$, too,

0->0 Y. Thus, t h e chains are f i n i t e o r isomorphic t o (N,

Y E <X

where q : Y ' -->

ZX,, By assumption on <X z @ ,we g e t the f o l l o w i n g s i t u a t i o n

Y denotes a path, p = p'

o

q. Now q must be contained i n

Tx.

$,

otherwise Y ' would be outside But q cannot be contained i n <x because Y and Y ' belong t o d i f f e r e n t connected components o f The desired f i n i t e f u l l subquiver may now be spanned by t h e f i n i t e l y

QXfin

N

many f i n i t e chains i n Q,,

tX.

t h e f i n i t e l y many smallest v e r t i c e s i n the other chains,

and the remaining f i n i t e l y many v e r t i c e s outside

.

Tx .

3.5 Corol 1a r y L e t X be an a r b i t r a r y small category such t h a t every X-morphism i s epi,

(i.e. X i s r i g h t cancetlative). Then X i s l e f t i d e a l noetherian i f f GHBT holds f o r every coma category ( X ZK) 1101 and a l l C. Proof. By assumption on

X, the

comma categories ( X

( c a t e g o r i c a l ) equivalent t o the posets <X i a l automorphisms i n ( X

+X>,*.

z X) are

preordered and

Note t h a t there a r e o n l y t r i v -

cx).

3.6 Corollary. A r i g h t c a n c e l l a t i v e monoid (M,.) (regarded as category w i t h one o b j e c t *) i s l e f t ideal noetherian i f f GHBT holds f o r i t s comma category ( * c M ) and a l l C.

3.7 Corollary. L e t (M,.) be a f r e e monoid and R a l e f t noetherian r i n g w i t h 1. The semigroup r i n g R [ M I i s l e f t noetherian i f f (M,.)

GI

(",+)

.

Proof. (, 1, set I, = per(@"), and let +,:Mi

-+ F(G/G[p"]) be the

F-algebra homomorphism induced by the quotient map G + G/G[p"]. Then, as ideals of Mi, both I, and Ker(9,) are generated by S, = {g - llg E G[p"]}. Thus, I, = Ker(9,) for all n 3 1. m f . If Q' is any subgroup of G, it is a standard fact that the F-algebra homomorphism induced by G + G/G' has kernel generated by {h - llh E G'}. Thus Clearly Ker(+,) is generated by S, and we need only show that I, = Ker(9,). S, c I,. So, Ker(+,) c I,. To show the reverse inclusion, suppose a E I,. Let {gJlj E J} be a set of coset representatives for G[pn] in G and write

g F for all j E J and g E G[p"]. Then, 0 = ap" implies that zg rj , where r J v E 0 for all j E J. Thus, for each j E J, zg r J V g g= Cg+. rj,g(g - 1) E <S,>. Thus, a E <S,> = Ker(4,) and I, C Ker(9,).

We now establish a connection between generating sets for I, = Ker(B") and GIp"].

As a corollory, we see that minimal generating sets for I, exist.

L e m a 1.3. Suppose n > 1 and S, is a subset of G[p"]. G[p"] if and only if {g - llg E S,} generates I,.

Proof.

If S, generates G[p"] , then {g - llg E S,)

Then, S, generates

generates the kernel of

the natural map M 4 F(G/G[p"]), which is I, by Lenma 1.2. Conversely, if {g - llg E S,) generates I,, let H be the subgroup of G[p"] generated by S,. Then, {g - llg E S,) generates the kernel of the natural map 9 : FG -+ F(G/H) = 0 implying x - 1 E Suppose x E G[p 1. Then, (x - 1)'" Thus Ker(+) = I,. Thus XII = ell, and so x E H. Therefore, H = G[p"] and the proof I, = per(+). 0 is camplete. Corollary.

Proof.

.

For all n > 1, I, has a minimal generating set.

Since G[p"] is a bounded group, it is a direct sum of cyclic groups. Thus, G[p"] has a minimal generating set S, obtained by selecting a single generator from each of the cyclic surrmands. From L e a a 1.3 it follows that {g - llg E S,} must be a minimal generating set for I,. 0

250

W. Ullery

We now prove the rain result of this section. We state it in a more general form than is needed later, but this added generality causes no difficulty in the proof. Theorem 1.4. Let R be a commutative ring with identity and suppose the prime number p does not invert in R. If G and H are abelian groups with

RG

I

RH as R-algebras, then lGpl

IHpI.

Proof. Since p $? inv(R), there is a maximal ideal M of R such that F = R/M is a field of characteristic p. Moreover, Iw RH implies that F OR RG I F OR RH. Hence, there is an F-algebra isomorphism f : FG + FH. Let B (respectively e') be the Frobenius endomorphism of FG (respectively By the Corollary to FH). For each n > 1 set I, = Ker(e") and 1, Ker((e')"). L e m a 1.3, we may select minimal generating sets T, and TA for I, and 1; respectively. Let S, and SA be minimal generating sets for G[p"] and H[p"] respectively. Suppose Sk is infinite for some k > 1. Then, for every n k, S, is infinite and every minimal generating set for G[p"] has cardinality IS,I by Lemma 1.1.

Thus, Lemaa 1.1 and 1.3 imply that IT,I

= IS,I

for each n > k.

Since

f(In) = 1; for each n, f carries T, onto a minimal generating set for 1;. Thus L e m a 1.1 implies that ITAI = IT,I for each n k. Consequently, for each n > k, IG[p"]I = IS,I IT,I = ITAI = ISAl = IH[pn]l. Therefore lGpl G[pn]l = I U b k H[pn]l = IHpl. If S, is finite for all n 1, it follows, from Lemma 1.3 and the fact that f(1,) = I,, that lGpl and lHpl are at most countably infinite. By the Theorem I %>k

in May [4], the divisible parts of 0, and H, are isomorphic, and the UlmKaplansky pinvariants of the reduced parts of 0, and H, are the same. Thus Ulm's Theorem (see Kaplansky [3] for a reference) implies that Gp H,. In particular, l Q p l = IHpl. 2.

0

PROOF OF THE MAIN THEOREM

Before getting to the proof of the Main Theoren itself, we need some additional terminology and two technical lemas. Let A(RG) be the subalgebra of RG consisting of those elements integral over

R.

Let V(RG) be the group of units of A(RG) of a w n t a t i o n 1. In general, if an abelian group, we write Xo for the torsion subgroup of X, and Xp for the p-component of Xo. p is a prime number and if X is

W 2.1. Let F be a field of characteristic p > 0. Set VF = @ {V(FG),,l q E inv(F)). Then, VF is contained in the subalgebra F(GF) of FG.

Proof. F(Gp).

If

Suppose q is a prime different f r a p. It suffices to show V(W), c a E V(W),, aqr = 1 for sr > 0. By Proposition 1 in May [ 6 ] , a

A conjecture relating to the isomorphism problem

25 1

and a-l are both contained in F(G0) + N, where N is the nilradical of FG. Say P + Y and a’l = B’ + Y’ where b,b’ E F(Go) and Y,Y’ E N. Choose integers s, a, b so that yPn = ( Y * ) ~ * 0 and aq’ + bpn = 1. If b < 0 , a = (p’)-bp* E

a =

0, = Bbpa E F(Go). In either case, a E F(Go) and we F(Go), and if b t conclude that V(FG), 5 F(Go). Hence, there is an integer t so that both ap and are in F(GF). Select integers c and d so that cpt + dq‘ = 1. If c > 0 , then a = acPt E F(GF), and if c < 0, a = E F(GF). In either

case a E F(GF), showing V(FG),

5 F(GF) as required.

0

We now use kma 2.1 to prove

lena 2.2. Let F be a field of characteristic p as F-algebras. Then, F(G/GF) a F(H/HF).

Proof.

> 0, and

suppose FG a FH

Let f : FG + FH be an augmentation preserving F-algebra isomorph-

ism, and let I (respectively 1’) be the kernel of the natural map FG + F(G/GF) (respectively FH 4 F(H/HF)). It suffices to show that f(1) = 1‘. Set VF = 8 ) {V(FG)qlq E inv(F)l and V; = @ {V(FH)qlq E inv(F)). By L-a 2.1, GF 5 V F 5 F(GF) and HF 5 v; 5 F(HF). Since f(VF) = V;, we conclude that F(Hf). Since I (respectively 1’) is generated by the elements of f(F(GF)) F(GF) (recrpectively F(HF)) of augmentation 0, f carries a set of generators of I M an ideal of Mi onto a set of generators of I’ as an ideal of FQ‘. There fore, f(1) = 1’. In the proof of the k i n Theorem, which appears next, we use the following mpecial case of the Theor- in May (51: Theorem (May [5]). Let K be an algebraically closed field of characteristic 0 . Then, a camplete system of invariants for KG as a K-algebra is the cardinal number lGol and the isomorphism clase of G/Go. Main Theorem. The following statements are equivalent: (1) Every ND-ring satisfies the Isomorphiem Theorem. (2) Every ND-ring of characteristic 0 satisfies the Isomorphism Theorem. (3) Every field of prime characteristic satisfies the Isomorphism Theorem. (4) Every algebraically closed field of prime characteristic satisfies the IsoMrphism Theorem.

-

Proof. Clearly (1) implies (2), and (3) implies (4). To show that (2) implies ( 3 ) , let F be a field of characteristic p > 0, and suppose FG a FH as F-algebras. We need to show that (2) implies that G/GF H/HF. By Lema 2.2, we m y (USthat Go = G,, Ho = H,, and show that G a H. By Theorem 1.4, lGol = l Q p l = lHpl IHol. Moreover, by the Theorem in May [4], G/Go a H/Ho. Consequently, CG CE by the Theorem in May [S], where C is the field of

carplex numbers.

Set R = C

x

F.

Then, CG a CH and FG a FH imply that RG

RH

W. Ullery

252

am Ralgebras. Moreover, OR and HR are trivial, and R is an ND-ring of characteristic 0. Therefore, if (2) holds, G H. Finally, to show that (4) implies (l), let R be an ND-ring. If inv(R) is not the complement of a single prime, then R satisfies the Isomorphism Theorem by the Main Theorem in Ullery [8] (regardless of what (4) says). Thus we may assume that there is a unique prime number p with p f€ inv(R). Select a maximal

ideal M of R which contains p, and let K be an algebraic closure of R/M. Evidently K has characteristic p and inv(K) = inv(R). Thus, if RG a RH, then K OR RG K OR RH, and so KG a KH as K-algebras. Therefore, if (4) holds, 0 G/G, = G/GK H/HK = B/HR and (1) holds also. The question of whether fields of prime characteristic satisfy the Isomorphism Theorem seems difficult, and very little is known in this direction. FH as F-algebras. In Suppose F is a field of characteristic p > 0 and FG this case, we would like to conclude that G/GF m H/HF. In view of Lemma 2.2, it suffices to consider the case where G and H have only ptorsion and show G a H. In this situation, it is shown in May [4] that G/G, a H/H, and that the divisible parts of G, and H, are isomorphic. Also, in [4], Berman and Mollov [l], and Dubois and Sehgal [2], it is shown that Gp and H, have the same UlmKaplansky invariants. Moreover, in Theorem 1.4 we show that lGpl = IH,I. However, it is unknown in general if even Gp a H,, let alone whether or not G @ H. In conclusion, we reiterate our conjecture: The rings which satisfy the Isomorphism Theorem are precisely the ND-rings. It is now apparent, in view of the Main Theorem, that our conjecture cannot be verified or rejected without considerably more information concerning isomorphism questions over fields of prime characteristic.

REFERENCES S. D. Berman and T. Zh. Mollov, The group rings of abelian p-groups of arbitrary power (Russian), Mat. Zametki 6 (1969), 381-392; Math. Notes 6 (19691, 686696. P. R. Dubois and S. K. Sehgal, Another proof of the invariance of Ulm’s functions in colutative modular group rings, Math. J. Okayama Univ. 15 (1972), 137-139. I. Kaplausky, Infinite Abelian Groups, Univ. of Michigan Press, Ann Arbor, Mich. (1969). W. May, Comutative group algebras, Trans. her. Math. SOC. 136 (1969), 139-149. -, Invariants for corutative group algebras, Ill. J. Math. 15 (1971), 525-631. -, Group algebras over finitely generated rings, J. Algebra 39 (1976), 483-511. -, Isomorphism of group algebras, J. Algebra 40 (1976), 10-18. W. Ullery, Isamorphiom of group algebram, C-ications in Alg., to eppear.

Group and Semigroup Rings G. Karpilovsky (ed.) 0 Elsevier Science Publishers B.V. (North-Holland),1986

RINGS GRADED BY A SEMILATTICE

253

-

APPLICATIONS TO SEMIGROUP RINGS

P. WAUTERS(*) Katholieke U n i v e r s i t e i t Leuven, Departement Wiskunde, Celestijnenlaan 200B, 8-3030 Leuven, Belgium INTRODUCTION L e t S be a comnutative semigroup.

Then S i s uniquely expressible as a

s e m i l a t t i c e union o f Archimedean semigroups, i . e . S = Archimedean,

r

u

a €

r

i s a s e m i l a t t i c e and S S c Sap f o r a l l ~ r , bE a B

S a y each Sa i s

r

( c f . 111).

Hence the semigroup r i n g R[S] equals Now the r i n g s R[S,]

CB RISa! and becomes a r-graded r i n g . a ~ r are o f t e n b e t t e r understood than R [ S I . E.g. i f S i s sepa-

r a t i v e , then the Archimedean components Sa are c a n c e l l a t i v e .

We show i n

Section 1 t h a t some r i n g t h e o r e t i c a l p r o p e r t i e s h o l d f o r R [ S l i f and o n l y i f t h e corresponding p r o p e r t i e s h o l d f o r each R[Sa] on the s e m i l a t t i c e 1a t ti ce

r).

(and e v e n t u a l l y a c o n d i t i o n

This explains our i n t e r e s t i n r i n g s graded by a semi-

.

Rings graded by a s e m i l a t t i c e were introduced by J. Weissglass i n 1131 who c a l l s these r i n g s "supplementary s e m i l a t t i c e sum o f r i n g s " .

We r e f e r t o 1111

and [131 f o r some r e s u l t s on semigroup r i n g s using the Archimedean decomposit i o n o f S.

I n 1983, W.D.

Munn characterised t h e Jacobson r a d i c a l o f a commu-

t a t i v e semigroup algebra K [ S I over a f i e l d K, by using the Archimedean decomp o s i t i o n o f S ( c f . [7]).

This meant a new s t a r t i n g p o i n t i n the search f o r

a c h a r a c t e r i s a t i o n o f the Jacobson r a d i c a l o f a semigroup r i n g R [ S I , where R i s an a r b i t r a r y r i n g and S i s a commutative semigroup.

several steps ( c f . [4],[5],[9]).

This was done i n

We r e f e r t o [6] f o r a survey o f i t .

I n the

same vein, E. Jespers and the author s t u d i e d when a semigroup r i n g R [ S l o f a comnutative semigroup i s l o c a l o r semilocal ( c f . [121).

This paper i s a con-

(*) The author i s a s e n i o r research a s s i s t a n t a t the Belgian Fund f o r S c i e n t i f i c Research (N. F. W .O. )

254

P. Waufers

t i n u a t i o n o f [121 : i n p a r t i c u l a r we study when R [ S l i s (semi-) p e r f e c t o r (semi-) primary.

For t h i s purpose, we i n v e s t i g a t e these p r o p e r t i e s f i r s t on Note t h a t i n 181 necessary and s u f f i c i e n t con-

r i n g s graded by a s e m i l a t t i c e .

d i t i o n s are given such t h a t R [ S l i s a p e r f e c t r i n g , f o r a r b i t r a r y semigroups Nevertheless , using the Archimedean decompo-

S, so even non-commutative ones.

s i t i o n o f S, we o b t a i n a completely d i f f e r e n t p r o o f which

-

t a t i v e semigroups

-

a t l e a s t f o r comnu-

i s more natural and also explains i n a b e t t e r way the con-

d i t i o n s i n 181. Ideal means twosided i d e a l .

Unless specified, r i n g s need not have a u n i t y . 1. RINGS GRADED BY A SEMILATTICE

Let

r

r

be a s e m i l a t t i c e , i . e . a commutative semigroup o f idempotents.

i s a p a r t i a l l y ordered s e t by d e f i n i n g

C~,B

E

r.

a 4 B

L e t R be a r-graded r i n g , i.e. R =

i f and o n l y i f Q

r

a €

a

Then

= a8 f o r

R a y a d i r e c t sum o f a d d i t i v e

subgroups R and R R c Rat? f o r a l l a,@ E r. The main d i f f e r e n c e between ra' at? graded r i n g s and group-graded r i n g s i s t h a t each component Ra i s a subring o f

r,

R ! For each a E

r , then E r) i s a

maximal element o f

r E R for a l l B

l e t Ra =

B % cular, f o r a l l

a E

(

1

r,

Q

-

then Ra i s an i d e a l o f R.

R

@ < a @' the map v a :

R

R

:

If a i s a

1

r B wra (where BE^ s u r j e c t i v e ringhomomorphism ( c f . 1131). I n p a r t i -

the map

- which

a

we also denote by ra

-

from R" t o Ra

r ) = ra i s a s u r j e c t i v e homomorphism. B Definition 1.1. : L e t R be a r i n g and J(R) i t s Jacobson r a d i c a l .

defined by

IT

a BGa

R i s semiZocaZ i f and o n l y i f R/J(R) i s A r t i n i a n . R i s semiperfect i f and o n l y i f R i s semilocal and idempotents o f R/J(R) can be l i f t e d t o idempotents t o R. R i s nit-semitocat i f and o n l y i f R i s semilocal and J(R) i s a n i l i d e a l .

i d e a l I o f R i s s a i d t o be left T-nitpotent i f f o r a l l x1,x2 there e x i s t s a k

> 0 such t h a t xlx 2 . . . ~ k

,... ,xk ,...

An i n I,

0.

R i s (left) perfect i f and o n l y i f R i s semilocal and J(R) i s l e f t T - n i l p o t e n t .

R i s primrry (resp. semiprimary) i f and o n l y i f R/J(R) i s simple (resp. semisimple) A r t i n i a n and J(R) i s n i l p o t e n t .

Rings graded by a semilattice

255

The r e l a t i o n between these notions i s as f o l l o w s : primary

3

semiprimary

* ( l e f t ) p e r f e c t * n i l - s e m i l o c a l * semiperfect

-

semilocal. It i s n o t d i f f i c u l t t o see t h a t none o f t h e i m p l i c a t i o n s above i s an equi-

For f u r t h e r (homological) p r o p e r t i e s o f these rings, we r e f e r t o 133.

valence.

L e t us a l s o note t h a t the terminology " n i l - s e m i l o c a l " i s n o t a standard one. For each o f the above mentioned concepts, we w i l l show when a s e m i l a t t i c e graded r i n g s a t i s f i e s these p r o p e r t i e s .

For t h i s purpose we w i l l add some lemmas

i n between which a r e n o t always s t a t e d i n t h e i r greatest g e n e r a l i t y . The f o l l o w i n g observation i s used i n the p r o o f o f Lemna 1.2. i d e a l o f a r i n g R, i f I has a u n i t y e, then e : Let R =

Lerruna 1.2.

Q

a E r

E

the centre o f R.

Z(R),

R a be a r-graded r i n g ,

: l e t I be an

r a f i n i t e semilattice.

I f each Ra has a u n i t y ea, then R has a u n i t y e.

Proof : By i n d u c t i o n on

Irl

Irl,

the case

>2. L e t a be a maximal element of

R = Ra Q Rrl,

where R,,

=

8 B E

r

r,

Irl then

Irl,

has a u n i t y , say e ' .

R,

r ' = r\{a)

Suppose

i s an i d e a l of

r.

Thus

, RB, i s again a s e m i l a t t i c e graded r i n g w i t h

s e m i l a t t i c e h = { a , l " } and m u l t i p l i c a t i o n a = t i o n on

= 1 being t r i v i a l .

a2,

r'

=

Put e = ea t ( e l

-

By induc-

aer'.

,I2

eae').

Using t h e

foregoing observation t h a t e ' E Z(R), i t f o l l o w s t h a t e i s the u n i t y o f R. The foregoing p r o o f shows t h a t we can reduce t o the case

Irl

= 2.

.

This w i l l

be used i n the sequel several times w i t h o u t f u r t h e r d e t a i l . : L e t R = Ra Q RB be a r-graded r i n g , where r = { a , B l and 2 2 a = a , B = 5 = a @ . L e t S be a h e r e d i t a r y r a d i c a l property ( c f . [ZI) such

L e m 1.3.

t h a t S(R) contains a l l n i l p o t e n t i d e a l s f o r a l l r i n g s R.

Proof : Since R S(RB) c S(R).

'II

i s an i d e a l o f R, S(R8) i s an i d e a l o f R ( c f . [2]) and

So S(R/S(RB)) = S(R)/S(R8) and R/S(RB) 2 Ra 8 RB/S(R8) i s s t i l l

a r-graded r i n g . then ra =

B

Then

Therefore we may assume t h a t S(R ) = 0. L e t r = ra t rBE S(R),

8 (r) E na(S(R)) c S(Ra), and R B r c S(R) n RB = S(R8).

Conversely,

256

P. Wauters

let

.

I = {r = ra t rBlraE S(Ra), R g r = r R = 01 B

Then I i s an i d e a l o f R.

We show t h a t I i s an S-radical r i n g , then I C S(R).

Let ~p :

Clearly

~p i

: r = ra t r

I-S(Ra)

B

ra

I-+

.

s a homomorphism which i s i n j e c t i v e , f o r l e t r = ra + rgE k e r q ,

. .

then r = q(r) = 0 and R g r = R B r g = 0 which i m p l i e s t h a t r g = 0 since S(RB) = 0 Therefore I

and by hypothesis. CoroZlary 2 . 4 .

;

~ ( 1 and ) ~ ( 1 i) s S - r a d i c a l .

L e t R and S be as i n Lemma 1.3 and assume t h a t RB has a

Then S(R) = {r = rat r g I r a E S ( R a ) , e g r E S(RB)3.

u n i t y eg.

CoroZZary 1.5.

Then S(R) = {r

-

;

.

L e t R and S be as i n C o r o l l a r y 1.4 and assume t h a t S(RB)=9.

e g r l r E S(Ra)I.

Note t h a t i n C o r o l l a r y 1.5 the map (r - e r ) B

between S(R) and S(Ra)

.

-

r gives an isomorphism

We s t a r t w i t h the f o l l o w i n g basic r e s u l t .

Proposition 2.6. for a l l

CL E

r.

: Let R = a

2

Ra be a r-graded r i n g such t h a t J(RJ

Then R i s semilocal i f and o n l y i f

r

# Ra

i s f i n i t e and each Ra i s

semilocal. S k e t c h of proof : This appears i n [121.

Since t h i s has n o t been published

y e t and since t h i s r e s u l t w i l l f r e q u e n t l y be needed, we include a b r i e f sketch I f R i s semilocal, i t i s e a s i l y checked t h a t each Ra i s semi-

o f the proof. local.

To show t h a t

r

i s f i n i t e , i t s u f f i c e s t o prove t h a t

a. E 1

r.

L e t a1

Since R/J(R) i s A r t i n i a n , from some k

>

s a t i s f i e s descen-

r

does n o t contain an

> a2 >

.. . > an > . . . , each

ding and ascending chain c o n d i t i o n on elements and t h a t i n f i n i t e subset of incomparable elements.

r

0 on we have Rak

+

J(R) =

..., implying t h a t

J(Ra ) = R, a c o n t r a d i c t i o n . Since R/J(R) k k’ i s Noetherian, a s i m i l a r reasoning shows the ascending chain condition, and a

Raktl

t J(R) =

s l i g h t v a r i a t i o n o f i t proves t h a t no f i n i t e subsets o f incomparable elements exist.

Thus

r

i s finite.

Rings graded by a semilattice

257

Conversely, we show t h a t R i s semilocal by i n d u c t i o n on

Irl.

As before, we

Irl

= 2 and R = R Q R as i n Lemma 1.3. Since J(R6) c J(R), a 8 we may suppose t h a t J(RB) = 0. Hence R i s semisimple A r t i n i a n and has a

may assume t h a t

B

unity e

By C o r o l l a r y 1.5

6'

J(R) = I x

-

e g x l x E J(Ra)).

Using t h i s and the

f a c t t h a t Ru i s semilocal, i t f o l l o w s t h a t R/J(R) i s A r t i n i a n . The f o l l o w i n g lemma i s shown i n a s t r a i g h t f o r w a r d way and i f R contains a u n i t y , a p r o o f using p r o j e c t i v e covers i s given i n [ 3 1 . L e m 1.7. : I f R i s a semiperfect r i n g and I an i d e a l o f R, then R / I i s a m

semiperfect r i n g .

Proposition 1.8. : L e t R = a u n i t y ea.

Then R i s semiperfect i f and o n l y i f

:

Suppose t h a t R i s semiperfect.

r.

So R = Ra Q R,

r,

then

r'

r

is

= r \ I a l i s an i d e a l

Q Rx. I n p a r t i c u l a r R,, i s an i d e a l x E r' i s semiRa i s semiperfect by Lemma 1.3. We show now t h a t R,

o f R and R/R,, perfect.

i s f i n i t e and Ra i s semi-

Then R i s semilocal and hence

L e t a be maximal i n

f i n i t e by Proposition 1.6. of

r

r.

perfect f o r a l l a E

roof

R a be a r-graded r i n g such t h a t each Ra has

Q

c t ~ r

where R r l =

An i n d u c t i o n on

Irl

y i e l d s the desired conclusion then.

R,

i s semi-

2 e ) E J(R'). l o c a l because R,, i s an i d e a l of R. L e t e E R r l such t h a t (e Then ( e e 2 ) E J(R) and by semiperfectness o f R, there e x i s t s an element 2 Then f = f E R w i t h ( f e) E J(R). Write f = fa t f ' , fa E Ra, f ' E R,,. 2 f = f2, f e = fa t ( f ' e) E J(R) and so fa = fa = ~ ~ e)( E fJ(Ra).

-

-

-

a

a

-

-

-

Therefore fa = 0 and f E R r , . Conversely, suppose t h a t a E

r.

r

i s f i n i t e and t h a t Ra i s semiperfect f o r a l l

By hypothesis and by Lema 1.2, we may reduce t o the case

Write R = R1

Q

= 2.

R2, R1 and R2 a r e subrings o f R and R1H2 c R2 and R2R1 c R2.

Moreover, R1 and R2 are semiperfect w i t h u n i t y el, x1 E R1,

Irl

x2 E R2, be such t h a t ( x

-

2

resp. e2.

-

L e t x = x1 t x2, 2 xl) E J(R1) and so

x ) E J(R). Then (xl 2 (yl xl) E J(R1) f o r some y1 = yl E R1, because R1 i s semiperfect. On the 2 (e2x)2 = e2(x x ) E J(R) )it R2 = J(R2). Again since R2 o t h e r hand, (e2x)

-

-

-

258

f. Wauters

i s semiperfect, there e x i s t s an element y2 = y$ E R2 w i t h (y2 Put y = y1 e2(y

-

+

y2

x) = y2

- e2y1.

-

e2X

J(R2).

E

-

Then (y-x) E J(R) because (yl

xl)

-

e2x) E J(R2)

E J(R1) and m

F i n a l l y , i t i s easy t o check t h a t y = y2.

It i s n o t known t o the author whether Proposition 1.8 s t i l l holds without the c o n d i t i o n t h a t each R,

has a u n i t y .

Proposition 1.9. : L e t R = for a l l a E R,

r.

Q

r

r

Then R i s nil-semilocal i f and o n l y i f

i s f i n i t e and each

i s nil-semilocal. Proof : Suppose t h a t R i s n i l - s e m i l o c a l .

~1

# R,

Ra be a r-graded r i n g such t h a t J(R,)

E

r.

As before,

r

i s finite.

Let

Then Ra = R, 0 R ' where R ' =

nil-semilocal.

@ R x . Moreover Ra i s c l e a r l y x E ra\Ial Since J ( R ' ) c J(Ra) we have by C o r o l l a r y 1.5 J(Ra)/J(R') =

-

J(Ra/J(R')) = {x Therefore J(R,)

e2xIx E J(R,))

0 U E

t r u e t h a t a l l J(R,)

r

R,

i s a r-graded r i n g and

are n i l i f J(R) i s n i l .

no nonzero n i l i d e a l s b u t J(S) # 0.

.

where e2 i s t h e u n i t y o f R ' / J ( R ' ) .

The converse i s shown i n a s t r a i g h t f o r w a r d way.

i s nil.

I n general, i f R =

J(R,)

r

i s f i n i t e , i t i s not

For example, l e t S be a r i n g w i t h

Put R = S [ X I ,

then R = R1

R1 = S and R2 = X S r X I , i s a s e m i l a t t i c e graded r i n g .

Q

R2, w i t h

C l e a r l y J(R)' = 0 b u t

J(S) = J(R1) i s n o t n i l . Lenunz 1.10. : L e t R be a r i n g .

(1) I f A c B are i d e a l s of R and ifA and B/A a r e l e f t T-nilpotent, then B i s l e f t T-ni l p o t e n t . ( 2 ) I f R i s a l e f t p e r f e c t r i n g and I an i d e a l o f R, then I and R / I are l e f t perfect.

(3) If I i s an i d e a l o f R contained i n J(R) and i f I and R / I are l e f t p e r f e c t , then R i s l e f t p e r f e c t . proof : (1) L e t x1,x2

have yl = xlx 2...x Xkl+l'xkl+2'"'

kl e B.

,...,xk,...

E A f o r some kl

E B.

Since B/A i s l e f t T-nilpotent, we

> 0. Consider then the sequence

As before, y2 = x ~ ~ + ~ . E. A. xf o~r some ~ k2

> kl.

Rings graded by a semilattice

Continuing t h i s process, we get elements y1,y2,

259

... E A.

Hence yl...yn

=

= 0 because A i s l e f t T - n i l p o t e n t . '1' * *'kn ( 2 ) O f course, R / I i s semilocal. Now

Put S = R / I / ( J ( R ) t I/I). Then S 2 ( R / J ( R ) ) / ( J ( R ) J(S) = J(R/I)/(J(R)

t

I/I).

i s A r t i n i a n and

t I/J(R))

Since S i s A r t i n i a n , J(S) i s l e f t T - n i l p o t e n t

i s l e f t T - n i l p o t e n t because R i s l e f t p e r f e c t . and ( J ( R ) t I/I) i s l e f t T - n i l p o t e n t by (1).

J(R/I)

Therefore

F i n a l l y , i t i s obvious t h a t I i s l e f t per-

[ I f R has a u n i t y , i t i s shown i n [31 t h a t R / I i s p e r f e c t using projec-

fect.

t i ve covers . I

(3) Since I C J ( R ) , we have R / J ( R ) 2 ( R / I ) / J ( R / I ) J(R/I)

So the r e s u l t f o l l o w s from (1).

= J(R)/I.

h p o s i t i o n 1.11. : L e t R = for a l l a E

r.

a

Q

~

a E

r

r

i s f i n i t e and Ra i s

r.

Pmof : Suppose t h a t R i s l e f t p e r f e c t .

r.

Moreover

Ra be a r-graded r i n g such t h a t J ( R a ) # R a

Then R i s l e f t p e r f e c t i f and o n l y i f

perfect f o r a l l

a E

.

which i s A r t i n i a n .

By Lemma 1.10

As before,

Ra i s l e f t p e r f e c t and Ra = Ra

r Q

i s finite.

Let

R ' where

Hence Ra Ra/R' i s l e f t p e r f e c t , again by Lemma 1.10. R ~ * For the converse, we may assume t h a t Irl = 2. Write R = R1 Q R2 where R1

R' =

Q

xEra\{a)

and R2 a r e subrings o f R, R1R2 C R2 and R2R1 C R2. R and J ( R 2 ) C J ( R ) .

Now J ( R 2 ) i s l e f t p e r f e c t by hypothesis.

i t s u f f i c e s t o show t h a t

R2/J(R2)

Then J ( R 2 ) i s an i d e a l o f

R

= R/J(R2)

R1

Q

R2/J(R2)

i s l e f t perfect.

i s semisimple A r t i n i a n , i t has a u n i t y , say e2.

1.5, J(R) 2 I x

-

J(R) = J ( R ) / J ( R 2 ) ,

e2xIx

E

J ( R 1 ) I 2 J(R1)

By Lemma 1.10

Since

Then, by C o r o l l a r y

.

which i s l e f t T - n i l p o t e n t .

i t follows that J ( R ) i s l e f t T-nilpotent.

Since

If i n Lemma 1.10, p e r f e c t i s replaced by semiprimary and T - n i l p o t e n t by n i l p o t e n t , then t h i s lemma remains t r u e .

I n p a r t i c u l a r we have

260

P. Wauters

Proposition 1.12. for a l l a E

r.

: Let R =

Q

a €

.

r

Then R i s semiprimary i f and o n l y i f

semiprimary f o r a l l a

r.

E

# Ra

Ra be a r-graded r i n g such t h a t J(R,)

r

i s f i n i t e and Ra i s

We f i n a l l y observe Proposition 1.13. : L e t R =

# Ra f o r a l l

t h a t J(R,)

r.

a E

@

a E r

Ra be a r-graded r i n g w i t h u n i t y , such

Then R i s primary i f and o n l y i f

r

= {a} i s a

singleton and Ra i s primary.

r

Proof : Suppose t h a t

i s not a singleton.

Then I8

This means t h a t R8 i s an i d e a l o f R such t h a t RB # R.

$r

f o r some f3 E

r.

Since R i s primary,

R' C J ( R ) and so RB = J ( R g ) which implies Rf3 = J ( R ) , a c o n t r a d i c t i o n . B Note t h a t even i f we assume t h a t R has a u n i t y , t h e c o n d i t i o n t h a t J(R,) For l e t R = {(:

Ra cannot be dropped.

KI, K a f i e l d , then

i)Ik,k E

' ) I k E K) and R2 = {(: E KI, so R i s o k Moreover R has a u n i t y , R i s primary b u t 1i-l > 1.

R = R1 Q R2 where R1 = graded.

#

r

= {1,2}-

2 . APPLICATIONS TO SEMIGROUP RINGS

We w i l l apply the r e s u l t s o f section 1 t o semigroup r i n g s .

We s t a r t w i t h

Throughout, a l l semigroups are commutative. I f S i s a 1 we denote by S the semigroup obtained by a d j o i n i n g a u n i t y t o S

some g e n e r a l i t i e s . semigroup,

i f S does n o t contain a u n i t y , otherwise S = S 1

.

"s divides t" i f t

E

S's.

I f s,t E S, we say t h a t

A semigroup S i s s a i d t o be Archimedean i f f o r

any two elements o f S, each divides some power o f the other.

L e t n be the

f o l l o w i n g r e l a t i o n on S

Then, as shown i n [ll, (a)

rl

i s the l e a s t s e m i l a t t i c e congruence 0n.S;

(b) each n-class i s an Archimedean subsemigroup o f S; the q-classes are c a l l e d the Archimedean components o f S.

261

Rings grad& by a semilattice

So S =

U

a E r

r

S a , where

=

S/qY

Sa t h e Archimedean components o f S.

Moreover,

t h i s s e m i l a t t i c e decomposition i s unique (111). A semigroup S i s s a i d t o be separative i f and o n l y i f VS,tES

s 2 = s t = t 2 * s = t .

L e t P denote the s e t of a l l prime numbers and l e t p

E

P. A semigroup S i s

s a i d t o be p-separative i f and o n l y i f VS,tES

sP=tP * s = t .

A congruence p on S i s s a i d t o be separative (resp. p-separative) i f S / p i s separative (resp. p-separative).

We denote by 5 (resp. 5 ) the l e a s t separaP t i v e (resp. l e a s t p-separative) congruence;

= {(s,t)

I f (s,t) E 5 and S =

E S x

u

r

CLE

SI 3 n

E

N Vk > n

sk = t k 1

SCL, where the S a ' s are the Archimedean components

o f S, then i t i s obvious t h a t s,t belong t o the same SCL. Therefore S / t ; =

u

/c

S

C L E a~

where 5 denotes the l e a s t separative congruence.

Since S a / c i s

separative and Archimedean, i t f o l l o w s t h a t S a / s i s c a n c e l l a t i v e ( c f . C13). I f SCL/c contains an idempotent, then i t i s a group.

I n particular, i f S i s

torsion, then each S a / s i s a group.

-

I f R i s a r i n g , then there i s an obvious s u r j e c t i v e r i n g homomorphism ~p :

RCSl

RCS/cI :

o f s i n S/c.

(si,ti)

E

c}

1 rSs

1 r,;

where

As shown i n 191, k e r cp = I(R,S,S)

5

denotes t h e equivalence class =

{I r i ( s i -

ti)lri

E

R,

i s a sum o f n i l p o t e n t i d e a l s .

We denote by E(S) the subsemigroup o f idempotents o f S. L e m 2.1.

: L e t S be a semigroup.

The f o l l o w i n g conditions a r e equivalent:

(1) S / c i s a group; (2) 3 e

E

E(S) Se i s a group and S/Se i s a n i l semigroup;

262

P. Waurers

(3) S has an i d e a l G such t h a t G i s a group and S/G i s n i l . Moreover we show t h a t S

(Note : I f S / C i s a group, then S i s Archimedean.

Then t h i s lemma should be compared t o [ I , Ex. 3,

contains an idempotent. p. 1351.)

&oof : (1) * (2) We f i r s t show t h a t E(S) contains p r e c i s e l y one element. i s the u n i t y o f S / c . I n p a r t i c u l a r , a = a', so k I f e,f E E ( S ) , then = (ak)' f o r some k > 0. Hence a E E(S).

L e t a E S be such t h a t ak =

e = 7 since E(S)

= {el.

Let x k

>

E

S.

a

S/c i s a group.

Thus f o r some n

We claim t h a t Se i s a group. For some y E S we have

27

=

so t h a t xn = (ex)n = exn f o r some n (2)

s*

(3)

* (1) Put e

e which

i m p l i e s x(xk-'yk)

= e f o r some

F i n a l l y , i f x E S, then

2

ex

> 0, i . e . S/Se i s n i l .

(3) i s t r i v i a l . = eG where eG i s the u n i t y o f G.

i.e. i f x E S, then xn E Se f o r some n xm = exm = (ex)m f o r a l l m F i n a l l y , l e t x E S. (ex)z = e f o r some CoroZZary 2.2.

G-

Obviously e i s the u n i t y of Se.

I t i s c l e a r now t h a t Se i s a group.

0.

> 0 e = en = fn = f. Therefore

S/c : x c-c ~oroZZaqj2.3.

>n

>

0.

Then G = Se.

Thus xm E Se f o r a l l m

so t h a t 2 = Z, i.e.

By t h e foregoing

z E G and hence 2:

= =

eX

Now S/G i s n i l ,

G

> n. Hence

i s the u n i t y o f S / C .

and ex E G which i s a group.

So

e.

.

: With notations and assumptions as i n Lemma 2.1,

x i s an isomorphism o f groups.

: With notations and assumptions as i n Lemma 2.1,

the map

if

x,y E S, then (x,y) E 5 i f and o n l y i f ex = ey. Proof : Suppose t h a t (x,y) E 5.

So (ex)" = (ey)" f o r a l l n ex = ey.

> k.

For some k

>0

xn = yn f o r a l l n 2 k.

Since ex,ey E eS which i s a group, we have

Conversely, i f ex = ey, then exn = eyn f o r a l l n

i s n i l , -we have xk,yk E eS f o r a l l k b i g enough. f o r a l l k b i g enough.

m

We f i r s t r e c a l l the f o l l o w i n g r e s u l t from [121.

> 0. Because S/eS

Hence xk = exk = eyk = y k

Rings grad& by a semilattice

263

Theorem 2.4. : L e t R be a r i n g such t h a t J(R) # R and l e t S be a semigroup. Then R [ S l i s semilocal i f and o n l y i f t h e f o l l o w i n g conditions h o l d

(1) R i s semilocal; ( 2 ) e i t h e r S/s i s f i n i t e o r e i t h e r S/c

P

i s f i n i t e and char(R/J(R)) = p

The c o n d i t i o n t h a t J(R) # R i s needed t o ensure t h a t J(R[S,l) f o r a l l Archimedean components o f S.

> 0.

.

# R[S,I

For i f R [ S I i s semilocal, then Theorem

2.4 i m p l i e s t h a t S i s t o r s i o n , so i f R = J(R), i t f o l l o w s from 191 t h a t J ( R I S a l ) = R I S a l f o r a l l a.

Proposition 2.5. group.

: L e t R be a r i n g such t h a t J(R)

# R and l e t S be a semi-

The f o l l o w i n g conditions are equivalent :

(1) R [ S l i s semiperfect; ( 2 ) S has o n l y f i n i t e l y many Archimedean components Sa ( a E

r)

and each R I S a l

i s semiperfect; ( 3 ) R [ S / c I i s semiperfect; (4) S / c has o n l y f i n i t e l y many Archimedean components S a / s and each R I S a / c I

i s semiperfect (and S a / s i s a group). Proof:

(1) * (2) and (3)

c l e a r since R[Sl/I(R,S,S)

0

( 4 ) follow from Proposition 1.8.

R [ S / c I and I(R,S,C)

L e t R be a r i n g and G a group.

(1) * (3) i s

i s a nilideal.

I f R C G l i s semiperfect, then R i s semiper-

f e c t and e i t h e r G i s f i n i t e o r e i t h e r G/G

i s f i n i t e and char(R/J(R)) = p P The converse does not, c f . [ I 5 1

(where G

>0

i s the p - t o r s i o n p a r t o f G ) . P where an example i s given o f a f i n i t e (abelian) group G and a semiperfect r i n g R b u t the group r i n g R [ G l i s n o t semiperfect.

As f a r as the author knows, no

obvious necessary and s u f f i c i e n t conditions are known such t h a t a group r i n g i s semiperfect. &~pOSit