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manuscripta mathematica

, Volume 8, Issue 1, pp 33–38 | Cite as

Kahl bewertete Ringe in der nichtarchimedischen Analysis

  • Ulrich Güntzer
  • Reinhold Remmert
Article

Abstract

An integral domain R provided with a non-archimedean valuation | | is called “bald” (kahl), if there exists a real number η, 0<η<1, such that the value set |R| ∼∔ does not meet the open interval (η, 1). Bald rings are important in non-archimedean analysis because the method of iteration (classical and well known for fields with discrete valuation) is convergent in these rings. In this note it is shown that each valuated field contains big bald subrings, more precisely:

Let K be a completely valuated field and let\(\mathop K\limits^ \circ \) denote the valuation ring. Let {aν}ν≥1 be a sequence in\(\mathop K\limits^ \circ \) converging to zero. Then the smallest complete local subring of\(\mathop K\limits^ \circ \) containing all aν is bald.

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Literatur

  1. [1]
    BOSCH, S.: Orthonormalbasen in der nichtarchimedischen Funktionentheorie. manuscripta math.1, 35–57 (1969).Google Scholar
  2. [2]
    GRAUERT, H. u. REMMERT, R.: Über die Methode der diskret bewerteten Ringe in der nichtarchimedischen Analysis. Inventiones math.2, 87–133 (1966).Google Scholar
  3. [3]
    WAERDEN, B.L. van der: Algebra I, 5. Auflage, Berlin 1960.Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • Ulrich Güntzer
    • 1
  • Reinhold Remmert
    • 2
  1. 1.II. Mathematisches Institut der FUBerlin 33
  2. 2.Math. Institut der UniversitätMünster

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