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)


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 ).


Local Ring Maximal Ideal Cauchy Sequence Inverse Limit Noetherian Ring 
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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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