On the algebraic properties of the ring of Dirichlet convolutions

TL;DR

This paper investigates the algebraic structure of rings formed by Dirichlet convolutions over a commutative ring and monoid, introducing extensions and analyzing their properties, especially for the case of positive rationals.

Contribution

It constructs a natural extension of the Dirichlet convolution ring for cancellative monoids and explores its algebraic properties, including an explicit isomorphism for the positive rationals case.

Findings

Extension ring $ ext{F}^f(G( ext{Gamma}), R)$ constructed

Isomorphism $ ext{F}^f( ext{Q}^*_+, R) o R[[x_1,x_2,\

Abstract

Let $R$ be a commutative ring and $\Gamma$ a commutative monoid of finite type. We study algebraic properties of modules and derivations over the associated ring $\mathcal F(\Gamma,R)$ of Dirichlet convolutions. If $\Gamma$ is cancellative and $G(\Gamma)$ is its associated Grothendieck group, we construct a natural extension $\mathcal F^f(G(\Gamma),R)$ of $\mathcal F(\Gamma,R)$ and we study its basic properties. Further properties are discussed in the case $\Gamma=\mathbb N^*$ and $G(\Gamma)=\mathbb Q^*_+$. In particular, we show that $\mathcal F^f(\mathbb Q^*_+,R)\cong R[[x_1,x_2,\ldots,]][x_1^{-1},x_2^{-1},\ldots]$.