Abelian Groups and Modules
Mathematics and Its Applications
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Abelian Groups and
Proceedings of the Padova Conference,
Padova, Italy, June 23-July 1, 1994
Department of Maţhematics and Infonnatics,
University of Udille,
Department of Mathematics,
University of Ferrara,
" SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4198-0 ISBN 978-94-011-0443-2 (eBook)
Printed on acid-free paper
AU Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint ofthe hardcover lst edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Dedicated to Laszlo Fuchs
on his 70th birthday
TABLE OF CONTENTS
T. Albu and P.F. Smith
Dual relative Krull dimension of modules over commutative rings.
P. N. Anh
Morita duality, linear compactness and AB5*: a survey . . . . . . 17
D. M. Arnold
Near isomorphism of Butler groups and representations of finite posets . 29
D. M. Arnold and M. Dugas
Locally free finite rank Butler groups and near isomorphism . . . . . . . .. 41
Subgroups of Butler groups. . . . . 49
A. Blass, J. Irwin and G. Schlitt
Some abelian groups with free duals . . . . . . . . . . . . . . . . . . . . 57
W. D. Burgess
Confirmation of a conjecture of Dauns and Hofmann on biregular rings 67
S. Caenepeel and e. Raianu
Induction functors for the Doi-Koppinen unified Hopf modules
N. Cimen, R. Wiegand and S. Wiegand
One-dimensional rings of finite representation type
Uniqueness of dualities . . . . . . . . . . . . . . . .
P. C. Eklof
Classification and non-classification results for abelian groups .
A. Facchini and G. Puninski
'E-pure-injective modules over serial rings. . . . . . . .
T. G. Faticoni
Modules over endomorphism rings as homotopy classes
A. Fomin and W. J. Wickless
Categories of mixed and torsion-free finite rank abelian groups
K. R. Fuller
Reflexive representations of algebras. . . . . . .
Zero-dimensional subrings of commutative rings . . . . . . . . .
F. Girolami and S. E. Kabbaj
The dimension of the tensor product of two particular pullbacks .
viii TABLE OF CONTENTS
R. Gobel and S. Shelah
On the existence of rigid Nl -free abelian groups of cardinality Nl 227
H. P. Goeters
Warfield duality and module extensions over a noetherian domain 239
K. R. Goodearl
Von Neumann regular rings and direct sum decomposition problems 249
Selforthogonal modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
w. J. Heinzer
Subrings of finite-dimensional rings . . . . . . . . . . . . . . . . . . . .. 277
P. Hill and C. Megibben
Primary abelian groups whose countable subgroups have countable closure 283
P. Hill and W. Ullery
Isotype separable subgroups of totally projective groups. . . . . . . . . . . . 291
Representable preradicals with enough projectives .
Rings with periodic unit groups . . . . . . . . . .
F. V. Kuhlmann
Abelian groups with contractions II: Weak o-minimality
Almost completely decomposable torsion-free abelian groups
S. Mohamed and B. J. Muller
On the exchange property for quasi-continuous modules
Endomorphism rings of Reid groups. . . . . . . . . . .
L. G. Nongxa and C. Vinsonhaler
Completely decomposable subgroups and factors of finite rank completely
decomposable groups . . . . . . . . . . . . . . . . . . . . . . . . . 385
A. Orsatti and N. Rodino
On the endomorphism ring of an infinite dimensional vector space
C. M. Ringel
Some algebraically compact modules I . . . . . . . . . . . . . . . .
The near endomorphism ring of an almost completely decomposable group 441
Reflexivity of left serial algebras . . . . . . 453
On a class of minimal topological modules . . . . . . . . . . . 459
Steady rings may contain large sets of orthogonal idempotents 467
TABLE OF CONTENTS ix
The Holy Grail of Algebra: Seeking complete sets of invariants 475
B. C. Vinsonhaler
Invariants for almost completely decomposable groups 485
J. M. Zelmanowitz
A class of modules with semisimple behavior . . . . . 491
B. Zimmermann Huisgen
The jinitistic dimension conjectures - a tale of 3.5 decades 501
On the 26th of November 1992 the organizing committee gathered together, at
Luigi Salce's invitation, for the first time. The tradition of abelian groups and
modules Italian conferences (Rome 77, Udine 85, Bressanone 90) needed to be kept
up by one more meeting.
Since that first time it was clear to us that our goal was not so easy. In fact the
main intended topics of abelian groups, modules over commutative rings and non
commutative rings have become so specialized in the last years that it looked
really ambitious to fit them into only one meeting. Anyway, since everyone of us
shared the same mathematical roots, we did want to emphasize a common link.
So we elaborated the long symposium schedule: three days of abelian groups
and three days of modules over non commutative rings with a two days' bridge
of commutative algebra in between.
Many of the most famous names in these fields took part to the meeting. Over
140 participants, both attending and contributing the 18 Main Lectures and 64
Communications (see list on page xv) provided a really wide audience for an
Now that the meeting is over, we can say that our initial feeling was right.
Mathematicians in different fields got the opportunity to gather together and learn
what was going on "in their neighbour's yard". We do hope that in spite of the
unusually hot weather we had to deal with (a very important mathematician was
shouting aloud that it was "as hot as hell") all the participants got a profitable
experience for their "own backyard".
Most of the articles appearing in this volume reflect the authors' contribution
to the conference. All the papers, except for some of the Main Lectures, were
subjected to a judgement by Referees appointed by the Editors. Unfortunately,
due to the restriction on the number of pages, we had to be really selective and
asked the Referees to be rather strict: sometimes we had to make painful choices,
but the number of pages available was too small compared to all the papers we
got. The reader will find some useful reports, by some of the Main Lectures, on
the development of a particular subject. All the other papers are completely
original and, in many cases, of an outstanding quality. To the Mathematical
community we are sure to present on the one hand the surveys of some basic
topics in Algebra and on the other hand very new results on some problems of
Alberto Facchini and Claudia Menini
The conference on Abelian Groups and Modules was held in Padova on June 23-
July 1, 1994 at the Dipartimento di Matematica Pura e Applicata of the University
of Padova. We wish to thank the Director of the Department for supplying us with
all the structures and technical supports we needed.
The opening session took place at the prestigious Palazzo del Bo (1542-1601),
main seat of the University, founded in 1222. We wish to thank the Administration
of the University of Padova for this generous concession.
We are grateful to the following organizations for their financial support: the
M. U.R.S. T., Gruppo 40% "Teoria dei Gruppi e Algebra Non Commutativa", the
Dipartimento di Matematica dell'UniversitO. di Ferrara, the Dipartimento di Mate-
matica dell'UniversitO. di Napoli, the Dipartimento di Matematica dell'Universitd di
Padova, the University of Udine, the Cassa di Risparmio di Padova e Rovigo, the
Banca Popolare Veneta. In particular we are grateful to Consiglio Nazionale del-
le Ricerche and to Gruppo 40% Algebra non commutativa whose financial support
allowed the publication of these Proceedings.
We would especially like to thank Luigi Salce for having asked to join him in this
trip and leading us out of all organizing mazes we met. Special thanks also are due
to Enrico Gregorio who took care of all the 1EX-D-1EX problems. Finally we wish to
thank all the colleagues that submitted us their contributions and those who have
accepted to serve as referees.
In the typesetting of some of the