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  • Abelian Groups and Modules

  • Mathematics and Its Applications

    Managing Editor:

    M. HAZEWINKEL

    Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

    Volume 343

  • Abelian Groups and Modules Proceedings of the Padova Conference, Padova, Italy, June 23-July 1, 1994

    edited by

    Alberto Facchini Department of Maţhematics and Infonnatics, University of Udille, Udine, Italy

    and

    Claudia Menini Department of Mathematics, University of Ferrara, Ferrara,ltaly

    .... " 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) DOI 10.1007/978-94-011-0443-2

    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

    Preface Acknowledgements Conference Information

    T. Albu and P.F. Smith Dual relative Krull dimension of modules over commutative rings.

    P. N. Anh

    xi xiii xv

    1

    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

    L. Bican 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

    D. Dikranjan 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. . . . . . .

    R. Gilmer Zero-dimensional subrings of commutative rings . . . . . . . . .

    F. Girolami and S. E. Kabbaj The dimension of the tensor product of two particular pullbacks .

    73

    95

    123

    135

    145

    163

    185

    193

    209

    221

  • 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

    D. Happel 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

    P. Keef Representable preradicals with enough projectives .

    J. Krempa Rings with periodic unit groups . . . . . . . . . .

    F. V. Kuhlmann Abelian groups with contractions II: Weak o-minimality

    A. Mader Almost completely decomposable torsion-free abelian groups

    S. Mohamed and B. J. Muller On the exchange property for quasi-continuous modules

    o. Mutzbauer Endomorphism rings of Reid groups. . . . . . . . . . .

    L. G. Nongxa and C. Vinsonhaler Completely decomposable subgroups and factors of finite rank completely

    301

    313

    323

    343

    367

    373

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

    P. Schultz

    395

    419

    The near endomorphism ring of an almost completely decomposable group 441

    N. Snashall Reflexivity of left serial algebras . . . . . . 453

    A. Tonolo On a class of minimal topological modules . . . . . . . . . . . 459

    J. Trlifaj Steady rings may contain large sets of orthogonal idempotents 467

  • TABLE OF CONTENTS ix

    P. Vamos 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

  • PREFACE

    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 Algebra meeting.

    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 recent interest.

    Alberto Facchini and Claudia Menini

  • ACKNOWLEDGEMENTS

    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

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