##
**Topological rings satisfying compactness conditions.**
*(English)*
Zbl 1041.16037

Mathematics and its Applications (Dordrecht) 549. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0939-9/hbk). ix, 327 p. (2002).

Although it is a book on topological rings, the first chapter deals with topological groups and is a good introduction to their theory. It contains a thorough discussion on basic notions in topological group theory like subgroups, quotient groups, continuous homomorphisms, connected component …as well as on basic contructions like completions, coproducts, semi-direct products, inverse limits and free topological groups, etc.

The second chapter, the aim of the book, focuses on topological rings satisfying certain compactness conditions. The theory of topological rings, beginning with D. van Dantzig’s thesis [“Studien über topologische Algebra”, H. J. Paris, Amsterdam (1931; Zbl 0006.10201)], can be divided into three cases. Since the connected component of 0 is a closed two-sided ideal, the first two cases concern connected and totally connected rings and the third one is just the extension problem that arises. The connected case is a vast area including, for example, all Banach algebras. The totally connected case contains all linearly topologized rings which become an important part of torsion theory in Grothendieck categories. In connection with torsion theory one can find good books on linearly topologized rings and modules, for instance, B. Stenström’s book [“Rings of quotients”, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer Verlag, Berlin (1975; Zbl 0296.16001)], but there are in fact no books on the structure of locally compact rings. Ursul’s book fills this gap and provides a systematic description of results on locally (linearly) compact rings. Most of the results date from the sixties of the last century, but there are some older and some more recent results as well. Numerous examples, remarks and exercises help to understand better concepts, methods and results presented in the book. The book ends with a useful section listing some open questions, problems in both topological groups and rings giving the readers impetus, challenge for their own research. Some parts of the book are appropriate for both introductory and advanced courses in topological groups and rings. The book is well-organized and clearly written such that the reader can study and become fond of it. I highly recommend it to both beginners and researchers who are interested in topological groups and rings.

The second chapter, the aim of the book, focuses on topological rings satisfying certain compactness conditions. The theory of topological rings, beginning with D. van Dantzig’s thesis [“Studien über topologische Algebra”, H. J. Paris, Amsterdam (1931; Zbl 0006.10201)], can be divided into three cases. Since the connected component of 0 is a closed two-sided ideal, the first two cases concern connected and totally connected rings and the third one is just the extension problem that arises. The connected case is a vast area including, for example, all Banach algebras. The totally connected case contains all linearly topologized rings which become an important part of torsion theory in Grothendieck categories. In connection with torsion theory one can find good books on linearly topologized rings and modules, for instance, B. Stenström’s book [“Rings of quotients”, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer Verlag, Berlin (1975; Zbl 0296.16001)], but there are in fact no books on the structure of locally compact rings. Ursul’s book fills this gap and provides a systematic description of results on locally (linearly) compact rings. Most of the results date from the sixties of the last century, but there are some older and some more recent results as well. Numerous examples, remarks and exercises help to understand better concepts, methods and results presented in the book. The book ends with a useful section listing some open questions, problems in both topological groups and rings giving the readers impetus, challenge for their own research. Some parts of the book are appropriate for both introductory and advanced courses in topological groups and rings. The book is well-organized and clearly written such that the reader can study and become fond of it. I highly recommend it to both beginners and researchers who are interested in topological groups and rings.

Reviewer: Ánh Pham Ngoc (Budapest)

### MSC:

16W80 | Topological and ordered rings and modules |

22A05 | Structure of general topological groups |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |