Chain Conjectures in Ring Theory An Exposition of Conjectures on Cat
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647
Louis J. Ratliff, Jr.
Chain Conjectures in Ring Theory An Exposition of Conjectures on Catenary Chains
Springer-Verlag Berlin Heidelberg New York 1978
Author Louis J. Ratliff, Jr. Department of Mathematics University of California Riverside, CA 92521
U.S.A.
AMS Subject Classifications (1970): 13A15, 13B20, 13C15, 13H99 Secondary: 13B 25, 13 J 15
ISBN 3-540-08758-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08758-3 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 64 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-643210
To
my Mother and Step-Father Ruth and Earl McCracken
ACKNOWLEDGMENTS
I want to thank Steve McAdam and David E. Rush for many helpful and stimulating conversations concerning the chain conjectures.
Their
help shows up in many (frequently indiscernab1e) ways in these notes. I want to express my sincere gratitude to the National Science Foundation for their support of my research on these problems (Grants 28939, 28939-1, 28939-2, MPS71-02929-A03, MCS76-06009, and MCS77-00951, and to the University of California, Riverside for my Sabbatical year in residence (1976-77) in which I essentially completed writing these notes. Finally, I am very grateful to Mrs. Jane Scully for her very excellent typing and her constant good humor and patience.
PREFACE These notes are concerned with a number of open problems, the catenary chain conjectures, some of which are of quite long-standing. In the hope of attracting some new people to do research on theseproblems, an attempt has been made to make these notes understandable to non-experts in this area.
This shows up partly in the history given
for the terminology (Chapter 1), the discussion of the recently solved problems (Chapter 2), the explanation of where the conjectures come from and their history (Chapters 4 - 13), the examples (Chapter 14), and the discussion of the related open problems (Chapter 15).
Also,
numerous references have been included in the proofs, and many of the referenced results are summarized in Appendix A to help decrease the amount of time the reader must spend searching for a result in research journals.
For those more familiar with this area, there are a number
of new conjectures given, many implications between the conjectures are shown to hold, numerous equivalences of the more important conjectures are proved, and an extensive bibliography is included.
The chain con-
jectures are interesting and important problems, and it is my hope that the material in t
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