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Abelian Groups

and Modules International Conference in

Dublin, August 10-14, 1998

Paul C. Eklof Rüdiger Göbel Editors

Springer Basel AG

Editors' addresses:

Paul C. Eklof Mathematics Department University of California at Irvine Irvine, CA 92697-3875 USA

Rudiger Gbbel Fachbereich 6, Mathematik und Informatik Universităt GH Essen 45117 Essen Germany

1991 Mathematical Subject Classification 20-06, 13-06

A CIP catalogue record for this book is available fram the Library of Congress, Washington D.C., USA

Abelian groups and modules : international conference in Dublin, August 10 - 14, 1998 I Paul C. Eklof ; Rudiger Gbbel ed. - Basel ; Boston; Berlin : Birkhăuser, 1999

(Trends in mathematics)

ISBN 978-3-0348-7593-6 ISBN 978-3-0348-7591-2 (eBook) DOI 10.1007/978-3-0348-7591-2

This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is con­cerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 1999 Springer Basel AG Originally published by Birkhiiuser Verlag, Basel, Switzerland in 1999

Printed on acid-free paper produced from chlorine-free pulp. TCF =

Softcover reprint of the hardcover 1 st edition 1999

987654321

CONTENTS

Preface iii

Authors and conference participants v

Ross Allen Beaumont (In Memoriam) J. D. Reid and W. J. Wickless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Modular group algebras and simply presented groups P. Hill .......................................................... 7

Abelian automorphism groups of countable rank W. May ............ ..... ........ ........... ....... ............. 23

Transitivity and full transitivity over subgroups of abelian p-groups G. Hennecke ........................................ ....... ..... 43

Subgroups of p5-bounded groups F. Richman and E. A. Walker

Groups acting on modules

55

P. Schultz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Some mixed abelian groups as modules over the ring of pseudo-rational numbers A. A. Fomin .................................................... 87

The Baer-Kaplansky theorem for direct sums of self-small mixed groups W. J. Wickless

Finite rank Butler groups with small typesets

101

D. M. Arnold and M. Dugas .................................... 107

Normal forms of matrices with applications to almost completely decomposable groups O. Mutzbauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Admissible matrices as base changes of B(lLgroups: a realizing algorithm C. De Vivo and C. Metelli ...................................... 135

Butler modules over I-dimensional Noetherian domains H. P. Goeters ........... ................................... ..... 149

Completely decomposable summands of almost completely decomposable groups A. Mader and 1. G. Nongxa ., ....... ........................... 167

Some matrix rings associated with ACD groups J. D. Reid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

11

Stacked bases for a pair of homogeneous completely decomposable groups with bounded quotient M. A. Ould-Beddi and L. Strüngmann ............ .. ............ 199

Separability conditions for vector R-modules U. Albrecht, A. Giovanitti and H. P. Goeters

Almost disjoint pure subgroups of the Baer-Specker group

211

O. Kolman and S. Shelah ............... .................... .... 225

Abelian groups mapping onto their endomorphism rings S. Feigelstock, J. Hausen and R. Raphael . . . . . . . . . . . . . . . . . . . . . . . 231

Purity and Reid's theorem A. Blass and J. Irwin ........................................... 241

Basic subgroups and a freeness criterion for torsion-free abelian groups A. Blass and J. Irwin

Absolutely rigid systems and absolutely indecomposable groups

247

P. C. Eklof and S. Shelah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Around nondassifiability for countable torsion free abelian groups G. Hjorth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

On the compact-open topology of Ext(C,A) C. Leopold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Direct decompositions of LCA groups P. Loth ......................................................... 301

Realizing automorphism groups of metabelian groups R. Gäbel and A. T. Paras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 309

On the dass semigroups of Prüfer domains L. Fuchs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Uniform modules, f-invariants, and Ziegler spectra of regular rings J. Trlifaj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Locally simple objects I. Herzog .......................... ........................ ..... 341

On purely extending modules J. Clark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

The number of submodules A. Ecker ........... .................................... ......... 359

Index 369

PREFACE

This volume contains the refereed proceedings of the INTERNATIONAL CON­FERENCE ON ABELIAN GROUPS AND MODULES held at the Dublin Institute of Technology in Dublin, Ireland from August 10th until August 14th, 1998. The meeting brought together more than 50 researchers and graduate stu­dents from 14 countries around the world. In aseries of eight invited survey talks, experts in the field presented some active areas of research inc1uding:

• Almost completely decomposable abelian groups, Butler groups and almost free groups - the c1assification problem, and invariants of special c1asses of torsion-free abelian groups.

• Totally projective groups, their automorphism groups and their group rings - questions about unique passage between these categories.

• Radicals commuting with products.

• The Ziegler spectra of Neumann regular rings and the c1ass (semi-) groups of Prüfer domains.

• The Krull-Schmidt property for valuation domains.

These main talks were accompanied by many other presentations of cur­rent research on abelian groups and modules. Methods from model theory, category theory, infinite combinatorics, representation theory, c1assical alge­bra and geometry were applied to the study of abelian groups and modules; on the other hand, results and methods from abelian group theory were applied to general module theory and non-commutative groups.

All this is reftected in the thirty articles in this volume, which introduce the reader to an active and attractive part of algebra that over the years has gained much from its position at crossroads of mathematics. Lively discussions at the conference inftuenced the final work on the presented papers; the editors hope that the papers will convey some sense of the intellectual ferment they generated and stimulate the reader to consider and actively investigate the topics and problems contained therein.

The papers inc1uded in this volume, each of which passed the critical scrutiny of two referees, deal with the topics of the survey lecture and also with the following:

• If the type set of a pure subgroup of a direct sum of finitely many subgroups of the rationals is small and not complicated, then we have a chance to c1assify these (Butler) groups: where is the borderline for 'not complicated'?

• A similar problem, also related to representation theory, arises for valuated p-groups of exponent p5: is 5 the critical number?

iv

• The countable rank automorphism groups of torsion-free groups are determined and those of arbitrary rank almost determined.

• Indecomposable abelian groups are constructed that remain indecom­posable when the uni verse of sets is extended.

• In several papers, the 'wild behavior' of torsion-free abelian groups or modules is described, even for restricted subclasses like those with large free subgroups, those of finite rank, or modules over rings close to fields.

• A proof that classifying countable torsion-free groups is as difficult as classifying arbitrary countable structures; and that any classification scheme for finite rank torsion-free groups is inherently complex.

• An application of Corner's ideas for realizing countable rings as endo­morphism rings to non-commutative groups, for investigations of auto­morphisms of metabelian groups.

Other papers investigate, among other topics: generalized notions of purity; generalized E-rings; vector groups; mixed self-small groups; and methods for studying general module categories.

These proceedings are headed by a memorial to Ross Beaumont, who was Reinhold Baer's first Ph.D. student. We would like to thallk Jim Reid and Bill Wickless for their summary of Ross' important contributions to algebra, which include the deep theory of torsion-free groups of rank two, developed jointly with R. S. Pierce.

The Dublin conference is one link in a chain of meetings dealing with abelian groups and modules which have taken place in re cent years in the United States, Italy, Germany, Australia and in Cura<;ao; the next link will be added in Perth, Australia in the year 2000. However, it was the first conference of this kind in Ireland, which gave Ray Mines the opportunity, in a special talk, to draw connections between James Joyce's great Ulysses and mathematics. This and other special events made this meeting a particularly pleasant and memorable one. We would like to thank the two organizers Brendall Goldsmith, President of DIT, and Simone Pabst, post-doctoral fellow at DIT, for runlling a conference which we all enjoyed very much. We also would like to thank DIT for financial support and for an exciting trip on an antique bus which explored the countryside and history of Ireland and gave many of us the chance to observe closely how sheep eat.

Paul Eklof and Rüdiger Göbel

Authors and Conference Participants

u. Albrecht, Department of Mathematics, Auburn University, Auburn, Alabama 36849, USA; albreuf@mail.auburn.edu D. Arnold, Department of Mathematics, Baylor University, Waco, TX 76798, USA; arnoldd@baylor.edu M. A. Avino, Department of Mathematics, Fac de Ciencias, Univ Nat Autonoma de Mexico, Circuito Exterior, DF CP 04360, CU Mexico; maad@hp.fciencias.unam.mx L. Bican, Katedra Algebry MFF UK, Sokolovska 83, 186 75 Praha 8 Karlin, Czech Republic; bican@karlin.mff.cuni.cz E. Blagoveshchenskaya, Department of Mathematics, St. Petersburg State Technical University, Polytechnicheskaya 29, St Petersburg 195251, Russia; kat@osipenko.stu.neva.ru A. Blass, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA; ablass@umich.edu J. Clark, Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, New Zealand; jclark@maths.otago.ac.nz A. L. S. Corner, Worcester College, Oxford OX1 2HB, England, UK; tony.comer@worcester.oxford.ac.uk C. De Vivo, Dipartimento di Matematica e Applicazioni, Universita. Federico II di Napoli, 80100 Napoli, Italy; devivo@matna2.dma.unina.it M. Dugas, Department of Mathematics, Baylor University, Waco, TX 76798, USA; dugasm@baylor.edu A. Ecker, Mathematisches Institut, Universität Tübingen, Auf der Mor­gen stelle 10, 72076 Tübingen, Germany; andreas.ecker@uni-tuebingen.de P. Eklof, Department of Mathematics, University of California at Irvine, CA 92697-3875, USA; peklof@math.uci.edu S. Feigelstock, Department of Mathematics, Bar-Han University, 52900 Ramat-Gan, Israel; feigel@macs.biu.ac.il A. A. Fomin, Department of Mathematics, Moscow Pedagogical State University, Krasnopudnaja ul., d. 14. Moscow 107140, Russia; afomin@ipc.ru L. Fuchs, Department of Mathematics, Tulane University, New Orleans, LA 70118, USA; fuchs@mailhost.tcs.tulane.edu A. Giovannitti, Department of Mathematics, State University of West Georgia, Carrollton, Georgia 30118, USA; agiovann@westga.edu H. P. Goeters, Department of Mathematics, Auburn University, Auburn, Alabama 36849, USA; goetehp@mail.auburn.edu R. Göbel, Fachbereich Mathematik und Informatik, Universität GH Essen, 45117 Essen, Germany; r.goebel@uni-essen.de B. Goldsmith, Dublin Institute of Technology, 30 Upper Pembroke Street, Dublin 2, Ireland; bgoldsmith@dit.ie

VI

J. Hausen, Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA; hausen@uh.edu G. Hennecke, Fachbereich Mathematik und Informatik, Universität GH Essen, 45117 Essen, Germany; georg.hennecke@uni-essen.de 1. Herzog, Department of Mathematics, The Ohio State University at Lima, Lima, OH 45804, USA; herzog.23@osu.edu P. HilI, Department of Mathematics, Auburn University, Auburn, AL 36849, USA; hillpad@mail.auburn.edu G. Hjorth Department of Mathematics, University of California at Los Angeles, CA 90095-1555, USA; greg@math.ucla.edu J. Irwin, 9001 Starmount, Las Vegas, Nevada 89134, USA; drjmirwin@lvcm.com P. Keef, Whitman College, Walla Walla, WA 99362, USA; keef@whitman.edu T. Kelly, 40 Wilfied Road, Sandymount, Dublin 4, Ireland; tka@tinet.ie O. Kolman, c/o London School of Jewish Studies, Schaller House, Albert Road, London NW4 2SJ, UK; okolman@e-math.ams.org T. Koyama, 5-24-21-703 Koishikawa, Bunkyo-ku, Tokyo 112-0002, Japan; koyama@is.ocha.ac.ip K. P. Krog, Department of Mathematics, Marist College, 290 North Road, Poughkeepsie, NY 12601, USA; jz9g@musicb.marist.edu C. LeopoId, Mathematisches Institut der FAU Erlangen, Bismarckstr. l~ , 91054 Erlangen, Germany; leopold@mi.uni-erlangen.de W. Liebert, Technische Universität München, Zentrum Mathematik, 80290 München, Germany; liebert@mathematik.tu-muenchen.de P. Loth, Department of Mathematics, Sacred Heart University, Fairfield, Connecticut 06432, USA; lothp@sacredheart.edu A. Mader, Department of Mathematics, University of Hawaii, 2565 The Mall, Honolulu, HI 96822, USA; adolf@math.hawaii.edu W. May, Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA; may@math.arizona.edu C. Meehan, 18 Dodder Court, Firhouse, Dublin 24, Ireland C. Metelli, Dipartimento di Matematica e Applicazioni, Universita Federico II di Napoli, 80100 Napoli, Italy; cmetelli@math.unipd.it R. Mines, Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003, USA; ray@nmsu.edu O. Mutzbauer, Mathematisches Institut, Universität Würz burg, Am Hubland, 97074 Würzburg, Germany; mu tz bauer@mathematik.uni-wuerzburg.de L. G. Nongxa, Department of Mathematics, University of the Western Cape, Private Bag x17, Belville 7535, South Africa; gnongxa@math.uwc.ac.za J. O'Hogain, 3 Meath Square, Dublin 8, Ireland

T. Okuyama, Toba National College of Maritime, Technology 1-1, Ikegami-Cho, Toba-Shi, Mie-Ken 5178501, Japan; olcuyamat@toba-cmt.ac.jp

vii

A. Opdenhövel, Fachbereich Mathematik und Informatik, Universität GH Essen, 45117 Essen, Germany; ansgar.opdenhoevel@uni-essen.de N. E. O'Sullivan, Department of Mathematics, NUI Galway, Galway, Ireland, niamh.eleanor.osullivan@ucg.ie M. Ouldbeddi, Faculte des Sciences et Techniques, Universite de Nouak­chott, Nouakchott B.P. 5026, Mauritania; beddima@univ-nkc.mr S. L. Pabst, Fachbereich Mathematik und Informatik, Universität GH Essen, 45117 Essen, Germany; simone.pabst@uni-essen.de A. T. Paras, Department of Mathematics, University of the Philippines at Diliman, 1101 Quezon City, Philippines; agnes@mathOl.cs.upd.edu.ph P. Plaumann, Mathematisches Institut der FAU Erlangen, Bismarckstr. 1~, 91054 Erlangen, Germany; peter.plaumann@mi.uni-erlangen.de K. M. Rangaswamy, Department of Mathematics, University of Colorado, PO Box 7150, Colorado Springs, Colorado 80933-7150, USA; ranga@math.uccs.edu R. Raphael, Department of Mathematics, Concordia University, Montreal PQ, H3G IM8, Canada; raphael@alcor.concordia.ca J. D. Reid, Department of Mathematics, Wesleyan University, Middle­town, CT 06459, USA; jreid@wesleyan.edu F. Richman, Florida Atlantic University, Boca Raton, FL 33431, USA; richman@fau.edu L. Salce, Dipartimento di Matematica Pura e Applicata, Universita, Via Belzoni 7, 35131 Padova, Italy; salce@math.unipd.it P. Schultz, Department of Mathematics, University of Western Australia, Nedlands 6907, Australia; schultz@maths.uwa.edu.au A. Scott, "Les Beaux" , Knockmark, Drumree, Co. Meath, Ireland; lscott@iol.ie N. A. Serdiukova, UI. Marshala Tukhachevskogo, 17 Kovp. 1, kv. 30, 123423 Moscow, Russia; abic@academ-mf.rosmaiI.com S. Shelah, Institute of Mathematics, Hebrew University, Jerusalem, Israel; shelah@math.huji.ac.il M. Siddoway, Department of Mathematics, Colorado College, Colorado Springs, CO 80903, USA; msiddoway@cc.colorado.edu L. Strüngmann, Fachbereich Mathematik und Informatik, Universität GH Essen, 45117 Essen, Germany; lutz.struengmann@uni-essen.de E. Toubassi, Department of Mathematics, 617 N Santa Rita, PO Box 210089, Tucson, AZ 85721, USA; elias@math.arizona.edu J. Trlifaj, Katedra Algebry MFF UK, Sokolovska 83, 18675 Praha 8, Czech Republic; trlifaj@karlin.mff.cuni.cz C. Vinsonhaler, Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA; vinson@math.uconn.edu

viii

E. A. Walker, Department of Mathematics, New Mexico State University, Las Cruces, NM 88003, USA; elbert@nmsu.edu W. J. Wickless, Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA; wickless@math.uconn.edu A. Yakovlev, Math-mech Faculty, Bibliotechllaya PI. 2, Stary Peterhof, St. Petersburg 198904, Russia; yakovlev@yakpdmi.ras.ru P. Zanardo, Dipartimento di Matematica Pura e Applicata, Ulliversita, Via Belzoni 7, 35131 Padova, Italy; pzanardo@math.unipd.it