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About this title Synopsis: Part I eleven chapters of this text for graduate students provides a Survey of topological fields, while Part II five chapters provides a relatively more idiosyncratic account of valuation theory. No exercises but a good number of examples; appendices support the author in his intent, which ha "About this title" may belong to another edition of this title. To learn more, view our Privacy Policy. Log In Sign Up. Minimal non-totally minimal topological rings. To appear in: Rendiconti del Seminario Mathematico della Universita di Padova, vol 97, Minimal non-totally minimal topological rings M.

We establish the existence of minimal non-totally minimal topological rings with a unit answering a question of Dikranjan. A Hausdorff topological ring R is called minimal if its topology is minimal in the sense of Zorn among all Hausdorff ring topologies on R. The induced topology of a nontrivial valuation on a field is totally minimal see [10, 6]. Some generalizations and related results in the context of fields or divisible rings may be found in [11, 13, 14].

For more general cases we refer to [1, 2, 3, 9]. Recall [2, 3] for instance that the class of all minimal rings with a unit is closed under forming topological products, direct sums and matrix rings. If P is a non-zero prime ideal of finite index in a Dedekind ring, then the P -adic topology is minimal. The question about existence of minimal non-totally minimal rings with a unit is discussed by Dikranjan in [2, 3]. Conventions and preliminaries. As usual N, Z, R denote the set of all natural, integer and real numbers, respectively.

All rings are assumed to be associative. A ring R is unital if it has a unit. The zero-element will be denoted by 0. Key words and phrases. Minimal ring, locally compact ring, Pontryagin duality. The following useful result is well known. Main results. Recall a construction from [12].

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Let R be a topological ring and X a topological R, R -bimodule. Now we generalize this construction in two directions. The first change is mi- nor.

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### Topological Fields and Near Valuations (Chapman & Hall/CRC Pure and Applied Mathematics)

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## Topological Fields and Near Valuations by Niel Shell

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