On the algebraic properties of the ring of Dirichlet convolutions
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arXiv
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866914775210917888 |
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| author | Cimpoeaş, Mircea |
| author_facet | Cimpoeaş, Mircea |
| contents | 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 $\mathcal F^f(\mathbb Q^*_+,R)\cong R[[x_1,x_2,\ldots,]][x_1^{-1},x_2^{-1},\ldots]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_02700 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On the algebraic properties of the ring of Dirichlet convolutions Cimpoeaş, Mircea Commutative Algebra 13G05, 13N15, 13F25 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 $\mathcal F^f(\mathbb Q^*_+,R)\cong R[[x_1,x_2,\ldots,]][x_1^{-1},x_2^{-1},\ldots]$. |
| title | On the algebraic properties of the ring of Dirichlet convolutions |
| topic | Commutative Algebra 13G05, 13N15, 13F25 |
| url | https://arxiv.org/abs/1808.02700 |