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The valuated ring of the arithmetical functions as a power series ring

Emil Daniel Schwab; Gheorghe Silberberg — 2001

Archivum Mathematicum

The paper examines the ring $A$ of arithmetical functions, identifying it to the domain of formal power series over $𝐂$ in a countable set of indeterminates. It is proven that $A$ is a complete ultrametric space and all its continuous endomorphisms are described. It is also proven that $A$ is a quasi-noetherian ring.

A note on some discrete valuation rings of arithmetical functions

Emil Daniel Schwab; Gheorghe Silberberg — 2000

Archivum Mathematicum

The paper studies the structure of the ring A of arithmetical functions, where the multiplication is defined as the Dirichlet convolution. It is proven that A itself is not a discrete valuation ring, but a certain extension of it is constructed,this extension being a discrete valuation ring. Finally, the metric structure of the ring A is examined.

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