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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 .
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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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