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Zbl 1090.13016
Schwab, Emil Daniel; Silberberg, Gheorghe
The valuated ring of the arithmetical functions as a power series ring. (English)
[J] Arch. Math., Brno 37, No. 1, 77-80 (2001). ISSN 0044-8753; ISSN 1212-5059/e

In this paper the ring $A$ of the arithmetical functions is investigated. This ring $A$ is isomorphic to the domain $A'$ of formal power series over the field of complex numbers in a countable set of indeterminates. On the ring $A'$ a discrete nonarchimedian valuation of rank $1$ is introduced, then a nonarchimedian norm and an ultrametric. These notions are transferred from $A'$ to $A$. \par By means of the domain $A'$ it is shown that $A$ is a complete ultrametric space and the domain $A$ is a quasi-noetherian ring (every ideal is quasi-finite, Def.\ 2). At the conclusion all continuous endomorphisms are described.
[Ladislav Skula (Brno)]

MSC 2000:: *13F25 Formal power series rings

Keywords: arithmetical function; valuated ring; ring of formal power series; complete ultrametric space; quasi-noetherian ring

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