CONTEMPORARY 124

Azumaya Algebras~ Actions~ and Modules Proceedings of a Conference in Honor of Goro Azumaya's Seventieth Birthday http://dx.doi.org/10.1090/conm/124

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Volume 58.1 The Lefschetz centennial conference. 74 Geometry of group representations, Proceedings on algebraic geometry, William M. Goldman and Andy R. Magid, D. Sundararaman, Editor Editors 58.11 The Lefschetz centennial conference. 75 The finite calculus associated with Proceedings on algebraic topology, Bessel functions, Frank M. Cholewinski S. Gitler, Editor 76 The structure of finite algebras, 58.111 The Lefschetz centennial conference. David C. Hobby and Ralph Mckenzie Proceedings on differential equations, 77 Number theory and its applications in A. Verjovsky, Editor China, Wang Yuan, Yang Chung-chun, and Pan Chengbiao, Editors 59 Function estimates, J. S. Marron, Editor 78 Braids, Joan S. Birman and Anatoly 60 Nonstrictly hyperbolic conservation Libgober, Editors laws, Barbara Lee Keyfitz and Herbert C. Kranzer, Editors 79 Regular differential forms, Ernst Kunz and Rolf Waldi 61 Residues and traces of differential 80 Statistical inference from stochastic forms via Hochschild homology, processes, N. U. Prabhu, Editor Joseph Lipman 81 Hamiltonian dynamical systems, 62 Operator algebras and mathematical Kenneth R. Meyer and Donald G. Saari, physics, Palle E. T. Jorgensen and Editors Paul S. Muhly, Editors 82 Classical groups and related topics, 63 Integral geometry, Robert L. Bryant, Alexander J. Hahn, Donald G. James, Victor Guillemin, Sigurdur Helgason, and and Zhe-xian Wan, Editors R. 0. Wells, Jr., Editors 83 Algebraic K-theory and algebraic 64 The legacy of Sonya Kovalevskaya, number theory, Michael R. Stein and , Editor R. Keith Dennis, Editors 65 Logic and combinatorics, Stephen G. 84 Partition problems in topology, Simpson, Editor Stevo Todorcevic 66 Free group rings, Narian Gupta 85 theory, Bor-Luh Lin, Editor 67 Current trends in arithmetical algebraic geometry, Kenneth A. Ribet, Editor 86 and number theory in connection with the local 68 Differential geometry: The interface Langlands conjecture, J. Ritter, Editor between pure and applied mathematics, 87 Abelian group theory, Laszlo Fuchs, Mladen Luksic, Clyde Martin, and RUdiger Gobel, and Phillip Schultz, William Shadwick, Editors Editors 69 Methods and applications of 88 Invariant theory, R. Fossum, mathematical logic, Walter A. Carnielli W. Haboush, M. Hochster, and and Luiz Paulo de Alcantara, Editors V. Lakshmibai, Editors 70 Index theory of elliptic operators, 89 Graphs and algorithms, R. Bruce foliations, and operator algebras, Richter, Editor Jerome Kaminker, Kenneth C. Millett, 90 Singularities, Richard Randell, Editor and Claude Schochet, Editors 91 Commutative harmonic analysis, 71 Mathematics and general relativity, David Colella, Editor James A. Isenberg, Editor 92 Categories in computer science and 72 Fixed point theory and its applications, logic, John W. Gray and Andre Scedrov, R. F. Brown, Editor Editors 73 Geometry of random motion, Rick 93 Representation theory, group rings, and Durrett and Mark A. Pinsky, Editors coding theory, M. Isaacs, A. Lichtman, Titles in This Series

Volume D. Passman, S. Sehgal, N. J. A. Sloane, 111 Finite geometries and combinatorial and H. Zassenhaus, Editors designs, Earl S. Kramer and Spyros S. 94 Measure and measurable dynamics, Magliveras, Editors R. Daniel Mauldin, R. M. Shortt, and 112 Statistical analysis of measurement Cesar E. Silva, Editors error models and applications, 95 Infinite algebraic extensions of finite Philip J. Brown and Wayne A. Fuller, fields, Joel V. Brawley and George E. Editors Schnibben 113 Integral geometry and tomography, 96 Algebraic topology, Mark Mahowald Eric Grinberg and Eric Todd Quinto, and Stewart Priddy, Editors Editors 97 Dynamics and control of multibody 114 Mathematical developments arising systems, J. E. Marsden, P. S. from linear programming, Jeffrey C. Krishnaprasad, and J. C. Simo, Editors Lagarias and Michael J. Todd, Editors 115 Statistical multiple integration, 98 Every planar map is four colorable, Nancy Kenneth Appel and Wolfgang Haken Flournoy and Robert K. Tsutakawa, Editors 99 The connection between infinite 116 Algebraic geometry: Sundance 1988, dimensional and finite dimensional Brian Harbourne and Robert Speiser, dynamical systems, Basil Nicolaenko, Editors Ciprian Foias, and Roger Temam, Editors 117 Continuum theory and dynamical systems, Morton Brown, Editor 100 Current progress in hyperbolic systems: Riemann problems and computations, 118 Probability theory and its applications in W. Brent Lindquist, Editor China, Yan Shi-Jian, Wang Jiagang, and Yang Chung-chun, Editors 101 Recent developments in geometry, 119 Vision geometry, Robert A. Melter, S.-Y. Cheng, H. Choi, and Robert E. Azriel Rosenfeld, and Prabir Greene, Editors Bhattacharya, Editors 102 Primes associated to an ideal, 120 Selfadjoint and nonselfadjoint operator Stephen McAdam algebras and operator theory, 103 Coloring theories, Steve Fisk Robert S. Doran, Editor 104 Accessible categories: The foundations 121 Spinor construction of vertex operator of categorical model theory, Michael algebras, triality, and E~ 1 }, Alex J. Makkai and Robert Pare Feingold, Igor B. Frenkel, and John F. X. 105 Geometric and topological invariants Ries of elliptic operators, Jerome Kaminker, 122 Inverse scattering and applications, Editor D. H. Sattinger, C. A. Tracy, and S. 106 Logic and computation, Wilfried Sieg, Venakides, Editors Editor 123 Enumerative algebraic geometry, 107 Harmonic analysis and partial Steven L. Kleiman and Anders Thorup, differential equations, Mario Milman Editors and Tomas Schonbek, Editors 124 Azumaya algebras, actions, and 108 Mathematics of nonlinear science, modules, Darrell Haile and James Melvyn S. Berger, Editor Osterburg, Editors 109 Combinatorial group theory, Benjamin Fine, Anthony Gaglione, and Francis C. Y. Tang, Editors 11 0 Lie algebras and related topics, and J. Marshall Osborn, Editors Azumaya Algebras, Actions, and Modules Photof?raph courtesy of James 6sterhu~rR ·· Goro Azumaya CoNTEMPORARY MATHEMATICS

124

Azumaya Algebras, Actions, and Modules Proceedings of a Conference in Honor of Goro Azumaya's Seventieth Birthday

Proceedings of a Conference held May 23-27, 1990 with support from the National Science Foundation, the Institute of Mathematics and its Applications, Indiana University, and the University of Cincinnati

Darrell Haile James Osterburg Editors

American Mathematical Society Providence, Rhode Island EDITORIAL BOARD

Richard W. Beals, managing editor Sylvain E. Cappell Linda Preiss Rothschild Craig Huneke Michael E. Taylor The Conference in Honor of Goro Azumaya's Seventieth Birthday was held at Indiana University, Bloomington, Indiana on May 23-27, 1990 with support from the National Science Foundation, Grant DMS-8918779, Institute of Mathematics and its Applications, Indiana University, and University of Cincinnati. 1991 Mathematics Subject Classification. Primary: 13C1 0, 14A22, 16K20, 16W30, 17B35; Secondary: 13A18, 13A20, 13C05, 16P20, 16S35, 16S36, 16S40, 20C10.

Library of Congress Cataloging-in-Publication Data Azumaya algebras, actions, and modules: proceedings of a conference in honor of Goro Azumaya's seventieth birthday/Darrell Haile and James Osterburg, editors. p. cm.-(Contemporary mathematics; 124) "Proceedings of a conference held May 23-27, 1990, with support from the National Science Foundation, Institute of Mathematics and Its Applications, Indiana University, and University of Cincinnati." Includes bibliographical references. ISBN 0-8218-5132-2 (alk. paper) 1. Azumaya algebras-Congresses. I. Azumaya, Goro, 1920- . II. Haile, Darrell, 1949-. Ill. Osterburg, James, 1944-. IV. Series: Contemporary mathematics (American Mathematical Society); 124. QA251.5.A98 1992 91-41788 512 1.4-dc20 CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this pub- lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Ser- vices, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid ·directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright @1992 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. § This publication was typeset by the authors using ANtS-Tf:X, the American Mathematical Society's Tf:X macro system. 10 9 8 7 6 54 3 2 1 97 96 95 94 93 92 This volume is dedicated to Professor Goro Azumaya on the occasion of his seventieth birthday Contents

PREFACE Xlll

BIOGRAPHY OF GORO AZUMAYA XV

LIST OF PARTICIPANTS xvii

Geometry of Quantum Planes 1 Locally Pure-Projective Modules GORO AZUMAYA 17 On the Blattner-Montgomery Duality Theorem for Hop£ Algebras MARGARET BEATTIE 23 Actions of Solvable Algebraic Groups on Noncommutative Rings WILLIAM CHIN 29 Nontrivial, Locally Trivial Azumaya Algebras F. R. DEMEYER AND T. J. FORD 39 The Cartan Determinant and Global Dimension of Artinian Rings KENT R. FULLER 51 Integral Representations of Finite Groups with Finite Representation Type JEREMY HAEFNER AND LEE KLINGLER 73 Homomorphisms Between Ore Extensions T. Y. LAM AND A. LEROY 83 Differential Identities of Prime Rings, Kharchenko's Theorem, and Appli- cations CHARLES LANSKI 111 Hopf Galois Extensions S. MONTGOMERY 129 Enveloping Algebras of Lie Superalgebras: A Survey IAN M. MUSSON 141

xi xii CONTENTS

Constructing Nonstandard Uniserial Modules over Valuation Domains BARBARA L. 0SOFSKY 151 What Makes A Skew Group Ring Prime? JAMES OSTERBURG AND D. S. PASSMAN 165 Wedderburn's Method and Algebraic Elements of Simple Artinian Rings LOUIS H. ROWEN 179 Finite Dimensional Division Algebras DAVID J. SALTMAN 203 Gubeladze's Proof of Anderson's Conjecture RICHARD G. SWAN 215 On Azumaya Projective Group Rings GEORGE SZETO AND YUEN-FAT WONG 251 Some Ring Themes in the Theory of Varieties of Group Representations SAMUEL M. VOVSI 257 Valuations on Tensor Products of Symbol Algebras ADRIAN R. WADSWORTH 275 Torsion-Free Modules over Regular Domains of Dimension Two ROGER WIEGAND 291 Preface

A conference in honor of Goro Azumaya's seventieth birthday was held at Indiana University from May 23 to May 27, 1990. The scientific purpose of the conference was to stimulate interaction among three areas of Professor Azumaya's interests: Azumaya algebras, group and Hopf algebra actions, and module theory. The principal speakers were Shimshon Amitsur, Michael Artin, Kent Fuller, Susan Montgomery, Donald Passman, David Saltman, Lance Small, Richard Swan and Goro Azumaya himself. In addition there were six special sessions, two in each of the three areas. These sessions were organized by Lawrence Levy, Patrick Morandi and Declan Quinn. Participants were encouraged to contribute papers. The submissions were refereed and those accepted are the contents of this volume. They are in final form and no version of them will be submitted for publication elsewhere. We owe thanks to many people and organizations. First we thank our finan- cial sponsors, the National Science Foundation, the Institute for Mathematics and its Applications, Indiana University, and the University of Cincinnati. We also thank Indiana University for providing the excellent physical facilities and Charlotte Klamer of the Indiana University Conference Bureau for all her cheer- ful work. The work of putting the individual papers into a standardized form and compiling the volume was done by Elena Fraboschi, to whom we are very grateful. Thanks too to Mary Jane Wilcox and Vicki Botos for their help with both the conference and the proceedings.

Darrell Haile James Osterburg

xiii Biography of Goro Azumaya

Professor Goro Azumaya was born in Yokohama, Japan, and raised in Osaka. He obtained his undergraduate degree from Tokyo University and his doctorate in mathematics from Nagoya University in 1949. His thesis advisor was the number theorist Iyanaga. He was an assistant professor at Nagoya University until1953, when he moved to Hokkaido University to become a full professor. From 1956 to 1959 he visited in the United States, for two years at Yale and then one year at Northwestern. He then returned to Hokkaido University until 1964, when he visited at the University of Massachusetts. The following year he visited at Indiana University. He returned to Japan for one more year before accepting a permanent position at Indiana in 1968. Professor Azumaya has made many important contributions to modern ab- stract algebra. In 1951 in a fundamental paper entitled "Maximally central algebras", he introduced the notion of what have come to be called Azumaya algebras and investigated their important basic properties. He then went on to classify such algebras over henselian rings. From these foundations the sub- ject of such rings and algebras has been developed greatly with applications to geometry and number theory. Professor Azumaya also has made fundamental contributions to the theory of group actions on rings and to module theory. Of particuar importance is his work on what has come to be called the Krull-Schmidt-Azumaya theorem. Of equal importance is his work in 1959 on a duality theory of modules, an early version of Morita duality theory. In addition to the United States and Japan, Professor Azumaya has given in- vited addresses in , Hungary, Belgium, Switzerland, France and China. In 1975-76 he was a visiting professor at the University of Munich and in 1983-84 he was a research member at the Mathematische Institut ETH. Professor Azumaya has had over twenty doctoral students. In 1971 seven students obtained their degrees from Indiana University under him. Among the honors bestowed on him was the Chunichi prize in 1949. This was a prize given annually to five people living in the Nagoya area for excellence in the sciences and arts. Professor Azumaya received the prize jointly with T. Nakayama for research on infinite dimensional algebras. His sixtieth birthday was celebrated in Japan with a special volume of papers dedicated to him.

XV

List of Participants

AKIRA ABE JOHN BEACHY Department of Mathematics Department of Mathematics Indiana University Northern Illinois University Bloomington, IN 47405 DeKalb, IL 60115 [email protected]

GENE ABRAMS MARGARET BEATTIE Department of Mathematics Department of Mathematics University of Colorado Mt. Allison University Colorado Springs, CO 80933 Sackville (EOA 3CO) [email protected] New Brunswick, CANADA [email protected]

ELI ALJADEFF ERIC BEHR Department of Mathematics Department of Mathematics Technion Illinois State University Institute of Technology Normal, IL 61761 Haifa 32000, ISRAEL [email protected]

SHIMSHON AMITSUR ESTHER BENEISH Department of Mathematics 850 N. State Hebrew University Chicago, IL 60610 , ISRAEL

EFRAIM ARMENDARIZ JEFFREY BERGEN Department of Mathematics Department of Mathematics University of Texas DePaul University Austin, TX 78713 Chicago, IL 60614 MAT [email protected]

MICHAEL ARTIN WILLIAM BLAIR Department of Mathematics Department of Mathematics MIT Northern Illinois University Cambridge, MA DeKalb, IL 60115 [email protected] [email protected] .EDU

GORO AZUMAYA AMI BRAUN Department of Mathematics Department of Mathematics Indiana University Haifa University Bloomington, IN 47405 Haifa, ISRAEL

xvii xviii LIST OF PARTICIPANTS

WALTER BURGESS CAROLYN DEAN Department of Mathematics Department of Mathematics University of Ottawa University of Michigan Ottawa (KIN 6N5) Ann Arbor, MI 48109 Ontario, Canada [email protected]. UMICH.EDU [email protected]

DOUGLAS BURKHOLDER FRANK DEMEYER Department of Mathematics Department of Mathematics Wichita State University Colorado State University Wichita, KS 67208 Ft. Collins, CO 80523

VICTOR CAMILLO VINAY DEODHAR Department of Mathematics Department of Mathematics University of Iowa Indiana University Iowa City, lA 52240 Bloomington, IN 47405 [email protected]

LINDSAY CHILDS DANIEL FARKAS Department of Mathematics Department of Mathematics SUNY at Albany VPI and SU Albany, NY 12222 Blacksburg, VA 24061 [email protected]

WILLIAM CHIN BURT FEIN Department of Mathematics Department of Mathematics DePaul University Oregon State U. Chicago, IL 60614 Corvallis, OR 97331 [email protected] [email protected]

BARBARA CORTZEN JOE FISHER Department of Mathematics Department of Mathematics DePaul University University of Cincinnati Chicago, IL 60614 Cincinnati, OH 45221

DAVID CusiCK ADELE FRASER Department of Mathematics Department of Mathematics Marshall University University of Western Ontario Huntington, WV 25755-2560 London, Ontario N6A 5B7 [email protected]

ROMUALD DABROWSKI KENT FULLER Department of Mathematics Department of Mathematics Indiana University University of Iowa Bloomington, IN 47405 Iowa City, lA 52242 [email protected] [email protected]

JAMES DAVIS JONATHAN GOLAN Department of Mathematics Department of Mathematics Indiana University University of Haifa Bloomington, IN 47405 31999 Haifa, ISRAEL [email protected] [email protected] LIST OF PARTICIPANTS xix

KENNETH GOODEARL S. K. JAIN Department of Mathematics Department of Mathematics University of Utah Ohio University Salt Lake City, UT 84112 Athens, OH 45701 MA.GOODEARL@SCIENCE. UTAH.EDU

EDWARD GREEN JOHN KAUTA Department of Mathematics Department of Mathematics VPI and SU Indiana University Blacksburg, VA 24061 Bloomington, IN 47405 [email protected]

GARY GREENFIELD JEANNE KERR Department of Mathematics Department of Mathematics University of Richmond Michigan State University Richmond, VA 23173 East Lansing, MI 48823 [email protected] 21144JWK@MSU .BITNET

WILLIAM GusTAFSON SOUMAYA KHURI Department of Mathematics Department of Mathematics Texas Tech University East Carolina University Lubbock, TX 79409 Greenville, NC 27858 [email protected]

JEREMY HAEFNER ELLEN KIRKMAN Department of Mathematics Department of Mathematics University of Colorado University of California at San Diego Colrado Springs, CO 80933 La Jolla, CA 92093 [email protected] [email protected] .EDU

DARRELL HAILE JEE KoH Department of Mathematics Department of Mathematics Indiana University Indiana University Bloomington, IN 47405 Bloomington, IN 47405 [email protected] [email protected]

ALLAN HEINICKE KARL KOSLER Department of Mathematics Department of Mathematics University of Western Ontario University of Wisconsin London, Ontario N6A 5B7 Waukesha, WI 53188 HEINICKE@UWOVAX. UWO.CA

TIMOTHY HODGES GUENTER KRAUSE Department of Mathematics Department of Mathematics University of Cincinnati University of Manitoba Cincinnati, OH 45221-4090 Winnipeg (R3T 2N2) [email protected] Manitoba, CANADA [email protected]

NATHAN JACOBSON JAMES KUZMANOVICH Department of Mathematics Department of Mathematics Wake Forest University New Haven, CT 06520 Winston-Salem, NC 27109 [email protected] XX LIST OF PARTICIPANTS

TSIT-YUEN LAM MARTIN LORENZ Department of Mathematics Department of Mathematics University of California at Berkeley Temple University Berkeley, CA 94720 Philadelphia, PA 19122 [email protected]

CHARLES LANSKI PHILIPPE LOUSTAUNAU Department of Mathematics Department of Mathematics University of Southern California George Mason University Los Angeles, CA 90089 Fairfax, VA 22030 [email protected]

WILLIAM LEAVITT PETER MALCOLMSON Department of Mathematics Department of Mathematics University of Nebraska Wayne State University Lincoln, NE 68588 Detroit, MI 48202 [email protected]

DAVID LEEP JERRY MARTINDALE Department of Mathematics Department of Mathematics University of Kentucky University of Massachusetts Lexington, KY 40506 Amherst, MA 01003 LEEP@MS. UKY.EDU

ANDRE LEROY YOICHI MIYASHITA Department of Mathematics Department of Mathematics University of Valenciennes University of Tsukuba 42 rue Dorey (7500) Tsukuba-Shi Ibaraki, 305 JAPAN Tournai, BELGIUM

GAIL LETZTER SUSAN MONTGOMERY Department of Mathematics Department of Mathematics Wayne State University University of Southern California Detroit, MI 48202 Los Angeles, CA 90089 [email protected] [email protected]

LAWRENCE LEVY PATRICK MORANDI Department of Mathematics Department of Mathematics University of Wisconsin New Mexico State University Madison, WI 53706 Las Cruces, NM 88003 [email protected]

ALEXANDER LICHTMAN BRUNO MUELLER Department of Mathematics Department of Mathematics University of Wisconsin McMaster University Parkside, WI 53141 Hamilton (L8S 4K1) Ontario, CANADA

PETER LINNELL IAN MussoN Department of Mathematics Department of Mathematics VPI and SU University of Wisconsin Blacksburg, VA 24061 Milwaukee, WI 53201 [email protected] [email protected]. UWM .EDU

SERGIO LOPEZ-PERMOUTH DAN NAKANO Department of Mathematics Department of Mathematics Ohio University Yale University Athens, OH 45701 New Haven, CT 06520 [email protected] LIST OF PARTICIPANTS xxi

JOHN O'NEILL ALEX ROSENBERG Department of Mathematics Department of Mathematics University of Detroit University of California Detroit, MI 48221 Santa Barbara, CA 93106

CHARLES ODENTHAL LOUIS ROWEN Department of Mathematics Department of Mathematics University of Toledo Bar Ilan University Toledo, Ohio 43606 Ramat Gan, ISRAEL [email protected]

FRANK 0KOH DAVID SALTMAN Department of Mathematics Department of Mathematics Wayne State University University of Texas Detroit, MI 48202 Austin, TX 78712 [email protected]

BARBARA 0SOFSKY MURRAY SCHACHER Department of Mathematics Department of Mathematics Rutgers University University of California at Los Angeles New Brunswick, NJ 08903 Los Angeles, CA 90024 [email protected] MMS@MATH. UCLA.EDU

JAMES OsTERBURG DANIEL SHAPIRO Department of Mathematics Department of Mathematics University of Cincinnati Ohio State University Cincinnati, Ohio 45221 Columbus, OH 43210 [email protected] [email protected]

DONALD PASSMAN JAY SHAPIRO Department of Mathematics Department of Mathematics University of Wisconsin George Mason University Madison, WI 53706 Fairfax, VA 22030 [email protected] [email protected]

DECLAN QUINN LANCE SMALL Department of Mathematics Department of Mathematics Syracuse University University of California at San Diego Syracuse, NY 13244 La Jolla, CA 92093 [email protected] [email protected]

PATRICK RABAU RoBERT SNIDER Department of Mathematics Department of Mathematics Ohio State University VPI and SU Columbus, OH 42310 Blacksburg, VA 24061 [email protected] [email protected]

ZINOVY REICHSTEIN J. T. STAFFORD Department of Mathematics Department of Mathematics University of Pennsylvania University of Michigan Philadelphia, PA 19104 Ann Arbor, MI 48109 ZINOVY@PENNDRLS. UPENN .EDU [email protected]. UMICH.EDU

S. T. Rrzvr MICHAEL R. STEIN Department of Mathematics Department of Mathematics Ohio State University at Lima Northwestern U. Lima, OH 45804 Evanston, IL 60208-2730 [email protected] .EDU xxii LIST OF PARTICIPANTS

RICHARD SWAN MICHAEL WESTMORELAND Department of Mathematics Department of Mathematics University of Chicago Denison University Chicago, IL 60637 Granville, OH 43023 [email protected]

GEORGE SZETO SYLVIA WIEGAND Department of Mathematics Department of Mathematics Bradley University University of Nebraska Peoria, IL 61625 Lincoln, NE 68588 SYLVIA@HOSS. UNL.EDU

STEVEN TESSER ROGER WIEGAND Department of Mathematics Department of Mathematics Indiana State University University of Nebraska Terre Haute, IN 47809 Lincoln, NE 68588 [email protected] RW@HOSS. UNL.EDU

OWEN THOMAS BRIAN WINKEL Department of Mathematics Department of Mathematics Indiana University Rose-Hulman Institute Bloomington, IN 47405 Terre-Haute, IN 47803

ANNIE VOGEL MARY WRIGHT Department of Mathematics Department of Mathematics University of Nantes Southern Illinois University 44072 Nantes, Cedex 03 Carbondale, IL 62901 FRANCE

NIKOLAUS VONESSEN M. YousiF Department of Mathematics Department of Mathematics usc Ohio State University Los Angeles, CA 90089 Lima, OH 45804 VONESSEN@MATH. USC.EDU

SAMUEL VOVSI DANIEL ZELINSKY Department of Mathematics Department of Mathematics Rutgers University New Brunswick, NJ 08903 Evanston, IL 60208-2730 [email protected] [email protected]

ADRIAN WADSWORTH BIRGE ZIMMERMAN Department of Mathematics Department of Mathematics University of California at San Diego University of Utah La Jolla, CA 92093 Salt Lake City, UT 84112 [email protected] [email protected]

JOSEPH WEHLEN YI-MING Zou Computer Science Corporation Department of Mathematics 304 W. Rte. 38, P.O Box N University of Wisconsin Moorestown, NJ 08057 Milwaukee, WI 53201

DAVID WEINRAUB Department of Mathematics SUNY at Albany Albany, NY 12222 [email protected] / /

ISBN 0-8218-5132-2

ISBN 978- 0 - 8218- 5132- 6

9 78082 1 851326 CONM/124