Methods in Ring Theory

Methods in Ring Theory

NATO ASI Series Advanced Science Institutes Series A series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenull'\' Publishing Corporation B Physics London and New York C Mathematical D. Reidel Publishing Company and Physical Sciences Dordrecht, Boston and Lancaster D Behavioural and Social Sciences Martinus Nijhoff Publishers E Engineering and The Hague, Boston and Lancaster Materials Sciences F Computer and Systems Sciences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York and Tokyo Series C: Mathematical and Physical Sciences Vol. 129

Methods in Ring Theory edited by F. van Oystaeyen Department of Mathematics, University of Antwerp, Antwerp, Belgium D. Reidel Publishing Company Dordrecht / Boston / Lancaster Published in cooperation with NATO Scientific Affairs Division

Proceedings of the NATO Advanced Study Institute on Methods in Ring Theory Antwerp August 2-12, 1983 Library of Congress Cataloging in Publication Data NATO Advanced Study Institute on Methods in Ring Theory (1983: Antwerp, Belgium) Methods in ring theory. (NATO ASI series. Series C, Mathematical and physical sciences, vol, 129) "Proceedings of the NATO Advanced Study Institute on Methods in Ring Theory, Antwerp, August 2-12,1983" - T.p. verso. 1. Rings (Algebra)-Congresses. I. Oystaeyen, F. van, 1947-. II. Title. III. Series: NATO advanced study institutes series. Series C, Mathematical and physical sciences, v. 129. QA247.N37 1983 512'.4 84-9876 ISBN-13: 978-94-009-6371-9 e-isbn-13: 978-94-009-6369-6 001: 10.1007/978-94-009-6369-6 Published by D. Reidel Publishing Company PO. Box 17,3300 AA Dordrecht, Holland Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 190 Old Derby Street, Hingham, MA 02043, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, PO Box 322, 3300 AH Dordrecht, Holland D. Reidel Publishing Company is a member of the Kluwer Academic Publishers Group All Rights Reserved 1984 by D. Reidel Publishing Company, Dordrecht, Holland. Softcover reprint of the hardcover 1 st edition 1984 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.

TABLE OF CONTENTS Acknowledgement viii J. ALAJBEGOVIC I R-Prufer Rings and Approximation Theorems T. ALBU I Certain Artinian Lattices Are Noetherian. Applications to the Relative Hopkins-Levitzki Theorem J. L. BUESO and P. JARA I A Generalization of Semisimple Modules 53 S. CAENEPEEL I Graded Complete and Graded Henselian Rings 67 B. FEIN and M. SCHACHER I Cyclic Classes in Relative Brauer Groups K. R. GOODEARL I Simple Noetherian Non-Matrix Rings R. GORDON I Group-Gradings of Categories W. HABOUSH I Brauer Groups of Homogeneous Spaces I. M. HARADA I Simple Submodules in a Finite Direct Sum of Uniform Modules 145 R. T. HOOBLER I Functors of Graded Rings 161 A. HUDRY I Sur une Classe d'algebres Filtrees 171 Y. IWANAGA I Some Special Class of Artin Rings of Finite Type 181 E. JESPERS and P. F. SMITH I Group Rings and Maximal Orders 185 T. KANZAKI I A Note on Infinite Torsion Primes of a Commutative Ring I. KERSTEN and J. MICHALICEK I Applications of Kummer Theory Without Roots of Unity A. A. KLEIN I The Index of Nility of a Matrix Ring Over a Ring With Bounded Index J. KREMPA and J. MATCZUK I On Algebraic Derivations of Prime Rings 37 81 87 89 111 197 201 207 211

VI TABLE OF CONTENTS L. LE BRUYN / Smooth Maximal Orders in Quaternion Algebras I M. LORENZ / Group Rings and Division Rings L. MAKAR-LIMANOV / On Free Subobjects of Skew Fields 281 W. S. MARTINDALE, 3rd and C. R. MIERS / Herstein's Lie and Jordan Theory Revisited 287 H. MARUBAYASHI / Divisorially Graded Orders in a Simple Artinian Ring 291 K. MASAIKE /6. -Injective Modules and QF-3 Endomorphism Rings 317 S. MONTGOMERY / Group Actions on Rings: Some Classical Problems 327 B. J. MULLER / Links Between Maximal Ideals in Bounded Noetherian Prime Rings of Krull Dimension One 347 I. N. MUSSON / Noetherian Subrings of Quotient Rings 379 C. NASTASESCU and S. RAIANU / Stability Conditions for Cummutative Rings with Krull Dimension 391 D. S. PASSr-1AN / Cancellative Group-Graded Rings 403 M. PITTALUGA / The Automorphism Group of a Polynomial Algebra 415 K. W. ROGGENKAMP / Auslander-Reiten Quivers for Some Artinian Torsion Theories and Integral Representations 433 K. W. ROGGENKAMP / Automorphisms and Isomorphisms of Integral Group Rings of Finite Groups 451 H. SATO / Self-Injective Dimension of Serial Rings 455 W. F. SCHELTER / Smooth Affine PI Algebras 483 A. H. SCHOFIELD / Questions on Skew Fields 489 S. K. SEHGAL / Torsion Units in Group Rings 497 J.-P. TIGNOL / On the Length of Decompositions of Central Simple Algebras in Tensor Products of Symbols 505 M. VAN DEN BERGH / A Duality Theorem for Hopf Algebras 517 F. VAN OYSTAEYEN / Note on Central Class Groups of Orders Over Krull Domains A. VERSCHOREN / On the Picard Group of a Quasi-Affine Scheme J. M. ZELMANOWITZ / Duality Theory for Quasi-Injective Modules List of Participants Index 231 265 523 541 551 567 570

LECTURES (one-hour) 2-8-83 M. ARTIN, Algebras of global dimension two. E. FRIEDLANDER, Foundations. of algebraic K-Theory. H. BASS, Automorphisms of polynomial rings. 3-8-83 M. LORENZ, Group rings and division rings. S.A. AMITSUR, Generic methods I. E. FORMANEK. The invariants of nxn matrices. 4-8-83 K. ROGGENKAMP, Isomorphisms and automorphisms of group rings. L. SMALL, Bergman-SmaIl's theorem revisited. E. FRIEDLANDER, Topological methods in K-Theory. H. BASS, The Jacobian conjecture. 5-8-83 S. MONTGOMERY, Group actions on rings; classical problems. D. PASSMAN, Infinite crossed products. E. FORMANEK, Noncommutative invariant theory. M. HARADA. Extending property of simple modules. 8-8-83 S.A. AMITSUR, Generic methods II. S. MONTGOMERY, Free algebras and generic matrix rings. K. GOODEARL, Simple Noetherian non-matrix rings. 9-8-83 K. ROGGENKAMP. Auslander-Reiten quivers of some torsion theories. M. AUSLANDER. Almost split exact sequences and rational singularities. D. SALTMAN. Brauer-Severi varieties and units in orders. M. HARADA, Simple submodules in a direct sum of uniform modules. 11-8-83 T. STAFFORD, Stable range and projective modules. A. VERSCHOREN. Relative invariants of rings. 12-8-83 S.P. SMITH, Sheaves of twisted differential operators on the flag variety. F. VAN OYSTAEYEN, Clifford systems and generalized crossed products.

ACKNOWLEDGEMENT We are grateful to the NATO scientific committee for making the Advanced Study Institute "Methods in Ring Theory" possible. Further financial support towards the organization of this meeting has been supplied to us by : the Belgian Foundation of Scientific Research N.F.W.D., the University of Antwerp U.I.A., the Belgian Friends of the Hebrew University of Jerusalem, the Ministry of Education. The "Kredietbank" (KB) and "Algemene Spaar en Lijfrente Kas" (ASLK) and also the "Herfurth" travel agency provided some partial assistence in the organization of the meeting and the entertainment program. Finally the A.S.I. owes all of its success to the cooperation of all lecturers, authors of papers for the proceedings and the hard labour of the local committee. Allow me to single out Alain Verschoren from the latter, not only because he organized the book-exhibit, but also because he promised to help me out with some typing work (if this line does not appear in print, it means that he did not stick to his promise). viii