Regularity and Substructures of Hom
Regular rings were originally introduced by John von Neumann to clarify aspects of operator algebras ([33], [34], [9]). A continuous geometry is an indecomposable, continuous, complemented modular lattice that is not ?nite-dimensional ([8, page 155], [32,
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Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta, USA) Benoît Perthame (Ecole Normale Supérieure, Paris, France) Laurent Saloff-Coste (Cornell University, Rhodes Hall, USA) Igor Shparlinski (Macquarie University, New South Wales, Australia) Wolfgang Sprössig (TU Bergakademie, Freiberg, Germany) Cédric Villani (Ecole Normale Supérieure, Lyon, France)
Friedrich Kasch Adolf Mader
Regularity and
Substructures of
Hom
Birkhäuser Verlag Basel . Boston . Berlin
Authors: Friedrich Kasch Mathematisches Institut Universität München Theresienstrasse 39 D-80333 München e-mail: [email protected]
Adolf Mader Department of Mathematics University of Hawaii 2565 McCarthy Mall Honolulu, HI 96822 USA e-mail: [email protected]
2000 Mathematical Subject Classification: 08A05, 08A35, 13A10, 13C10, 13C11, 13E99, 16D10, 16D40, 16D50, 16S50, 20K15, 20K20, 20K25
Library of Congress Control Number: 2008939516
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 978-3-7643-9989-4 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All 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. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Birgit Blohmann, Zürich, Switzerland Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-9989-4
e-ISBN 978-3-7643-9990-0
987654321
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Contents Preface
vii
I Notation and Background 1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II Regular Homomorphisms 1 Definition and Characterization . . . . . . . . . . . . . . . . . . . 2 Partially Invertible Homomorphisms and Quasi-Inverses . . . . . 3 Regular Homomorphisms Generate Projective Direct Summands 4 Existence and Properties of Reg(A, M ) . . . . . . . . . . . . . . . 5 The Transfer Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Inherited Regularity . . . . . . . . . . . . . . . . . . . . . . . . . 7 Appendix: Various Formulas . . . . . . . . . . . . . . . . . . . . .
1 1 2 6
. . . . . . .
11 11 15 19 21 25 26 39
III Indecomposable Modules 1 Reg(A, M ) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Structure Theorems . . . . . . . . . . . . . . . . . . . . . . . . . .
41 41 42
IV
. . . . . . .
49 49 52 52 53 54 56 57
V Regularity in HomR (A, M ) as a One-sided Module 1 Itera
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