On the algebraic properties of the ring of Dirichlet convolutions

Abstract

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

Theorems & Definitions (42)

• Proposition 1.1
• Remark 1.2
• Proposition 1.3
• proof
• Corollary 1.4
• proof
• Proposition 1.5
• proof
• Proposition 1.6
• proof