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Topological Rings Satisfying Compactness Conditions Mathematics and Its Applications Topological Rings Satisfying Compactness Conditions Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam , The Netherlands Volume 549 Topological Rings Satisfying Compactness Conditions by Mihail Ursul Universitatea Din Oradea, Oradea, Romania ~ "SPRINGER SCIENCE+BUSINESS MEDIA, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-3946-8 ISBN 978-94-010-0249-3 (eBook) DOI 10.1007/978-94-010-0249-3 Printed on acid-free paper AH Rights Reserved © 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint ofthe hardcover Ist edition 2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form Of by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specificaHy for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Contents Introduction vii Notation ix Chapter 1. Elements of topological groups 1 1. The definition of a topological group 1 2. Neighborhoods of elements of a topological group 5 3. Subgroups of a topological group 8 4. Morphisms and quotients of topological groups 11 5. The axioms of separation in topological groups 15 6. Initial topologies. Products of topological groups 19 7. The co-product topology on the algebraic direct sum 28 8. Semi-direct products of topological groups 31 9. The embedding of totally bounded groups in pseudo-compact ones 34 10. Metrization of topological groups 35 11. The connected component of a topological group 38 12. Quasi-components of topological groups 45 13. Complete topological groups 59 14. Minimal topological groups 71 15. Free topological groups 75 16. The finest precompact topology on an Abelian group 77 17. Ordered topological groups 78 18. Topological groups of the second category 80 19. Inverse limits of topological groups 80 Chapter 2. Topological rings 83 1. The notion of a topological ring 83 2. Neighborhoods of zero of a topological ring 90 3. Subrings of topological rings 93 4. Compact right topological rings 94 5. The local structure of locally compact rings 113 6. Structure of compact rings 124 7. The separated completion of a topological ring 143 8. Trivial extensions 149 9. Nil and nilpotence in the class of locally compact rings 155 10. The Wedderburn-Mal'cev theorem for compact rings 170 vi CONTENTS 11. Topological products of primary compact rings 172 12. Zero divisors in topological rings 177 13. The group of units of a topological ring 179 14. Boundedness in locally compact rings 186 15. Simple topological rings 200 16. Homological dimension of a compact ring 203 17. Local direct sums of locally compact rings 211 18. Radicals in the class of locally compact rings 211 19. Endc(RM) 230 20. Locally compact division rings 243 21. Non-metrizable compact domains 253 22. Open subrings of topological division rings 262 23. Tensor products of compact rings 267 24. Pseudo-compact topologies on the ring of polynomials 272 25. The Lefschetz duality 274 26. The uniqueness of compact ring topologies 287 27. Totally bounded topological rings 291 28. Representations of locally compact rings 305 29. Open questions in topological groups and rings 307 Bibliography 311 Index 325 INTRODUCTION vii Introduction In the last few years a few monographs dedicated to the theory of topolog­ ical rings have appeared [Warn27], [Warn26], [Wies 19], [Wies 20], [ArnGM]. Ring theory can be viewed as a particular case of Z-algebras. Many general results true for rings can be extended to algebras over commutative rings . In topological algebra the structure theory for two classes of topological algebras is well developed: Banach algebras; and locally compact rings . The theory of Banach algebras uses results of Banach spaces, and the theory of locally compact rings uses the theory of LCA groups. As far as the author knows, the first papers on the theory of locally compact rings were [Pontr1]' [J1], [J2], [JT], [An], lOt], [K1]' [K2]' [K3], [K4], [K5], [K6]. Later two papers, [GS1,GS2] appeared, which contain many results concerning locally compact rings. This book can be used in two w.ays . It contains all necessary elementary results from the theory of topological groups and rings. In order to read these parts of the book the reader needs to know only elementary facts from the theories of groups, rings, modules, topology. The book consists of two parts. The first part also contains some results published earlier only in journals. We expound some results about iterated quasi-components in topological Abelian groups. It is constructed an example of a planar group whose quasi-component is not equal to the component. Using this example and some constructions we prove that each Abelian topological group can be realized as a quasi-component of a given rank of another topological Abelian group. In the second part of the monograph we shall develop the theory of rings which are closed to that of compact rings. The class of compact rings is closed to the class of finite rings . Kaplansky mentioned that "A noteworthy feature is the extent to which compactness serves as a substitute for the classical chain conditions". The nearness of the class of compact rings to finite rings is empha­ sized, for example, by the following result of the author [U7]: every compact nilring is a nilring of bounded degree. The study of the class of compact rings has an influence on the study of classes near to the class of compact rings: pseudo-compact, countably compact, sequentially compact. In topological algebra the notion of a linearly compact module was introduced (Zelinsky, Chevalley, Leptin). The notion of a linearly compact ring generalizes in the case of rings with a local base of left ideals the notion of a compact ring. But the class of discrete linearly compact rings is very broad. These rings are important in the duality theory (see, e.g., [Mu2]). We have also included in this book the proof of the existence of a locally topologically nilpotent radical in the class of locally compact rings. This radical can be applied to the proof of results about the topological local nilpotence for locally compact rings. viii An important notion in topology is Urysohn-Menger dimension of a topo­ logical space. We will construct examples of locally totally bounded topological fields of any dimension n . NOTATION ix Notation The set of all natural numbers is denoted by N and the set of all positive natural numbers by N+. By JR will be denoted the topological group of real numbers and by <p the canonical homomorphism of JR on T = JR /Z. For each e > °denote by Dc the subset {x: x E JR,lxl < c:} . Canonical neighborhoods of zero of T are subsets of the form <p( Dc), e > 0, where <p is the canonical homomorphism of JR on 'lI'. Rings are not assumed to be associative. The closure of a subset A of a topological space X is denoted by cl(A) or A. A surjective mapping is denoted by -» . The subgroup generated by a subset 8 of the additive group R( +) of a ring R is denoted by (8)+. An ideal of a ring generated by a subset 8 is denoted by (8). Topological rings and groups are not assumed to be Hausdorff. If 8 is a groupoid and A, B two subsets, then A.B := {ab : a E A, bE B}. If n is a natural number and A a subset of it then denote A[n] := A.··· .A (n times) . For any Abelian group and each subset A,B put A+B := {a+b: a E A, bE B}. If 8 is a ring, n a natural number and A a subset, then denote [nJA := A + ... + A(n times). The sub ring (subgroup, subspace) generated by a subset A of a ring R (group G, linear space V) will be denoted by (A). The additive group of a ring R will be denoted by R( +) and its multiplicative groupoid by R( ·). For a ring R and two of its subsets A, B denote by AB := {I:~=l aibi : n E N+,ai E A, b, E B}. The dual of a LCA group X is denoted by X·. The canonical isomorphism of X on X·· is denoted by w. Remind that for every x E X,w(x)(~) = ~(x) for every ~ EX·. The end of a proof will be denoted by •. Acknowledgements. I'd like to thank Mitrofan Ciobanu for his interest for the book. I also thank Viktor Bovdi for careful reading of the manuscript and Dan Erzse for technical assistance..
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