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Completions of filtered groups, rings and modules. Applications to m-adic topologies

  • Silvio Greco
  • Paolo Salmon
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 58)

Abstract

Let G be a filtered abelian group (i.e. a filtered ℤ-module, see § 1), and let (G n ) be its filtration. We can define a mapping
$$ v:G \to \mathbb{N} \cup \{ \infty \} $$
in the following way: \( v(x) = \sup \{ n \in \mathbb{N}\} |x \in {G_n} \). It is clear that ν(x) = ∞ if and only if \( x \in \cap {G_n},i.e.x \in \mathop 0\limits^ - \) (lemma 1.1). The mapping allows us to define a pseudometric in G: let
$$ d:G \times G \to G $$
be the mapping defined by d(x, y) = e (x-y) (we agree that e-∞ = 0). Then it is easy to see that
$$ d(x,y) \leqslant \sup \{ d(x,z),d(y,z)\} $$
and that d defines in G the topology induced by the filtration (G n ).

Keywords

Local Ring Maximal Ideal Cauchy Sequence Inverse Limit Noetherian Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • Silvio Greco
    • 1
  • Paolo Salmon
    • 1
  1. 1.Università di Genova Istituto di MatematicaItaly

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