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The ring of arithmetical functions with unitary convolution: Divisorial and topological properties
Archivum mathematicum, Tome 40 (2004) no. 2, pp. 161-179 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We study $(\mathcal {A},+,\oplus )$, the ring of arithmetical functions with unitary convolution, giving an isomorphism between $(\mathcal {A},+,\oplus )$ and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring $(\mathcal {A},+,\cdot )$ of arithmetical functions with Dirichlet convolution and the power series ring $ [\![x_1,x_2,x_3,\dots ]\!]$ on countably many variables. We topologize it with respect to a natural norm, and show that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units.
We study $(\mathcal {A},+,\oplus )$, the ring of arithmetical functions with unitary convolution, giving an isomorphism between $(\mathcal {A},+,\oplus )$ and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring $(\mathcal {A},+,\cdot )$ of arithmetical functions with Dirichlet convolution and the power series ring $ [\![x_1,x_2,x_3,\dots ]\!]$ on countably many variables. We topologize it with respect to a natural norm, and show that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units.
Classification : 11A25, 13F25, 13J05
Keywords: unitary convolution; Schauder Basis; factorization into atoms; zero divisors 
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Snellman, Jan. The ring of arithmetical functions with unitary convolution: Divisorial and topological properties. Archivum mathematicum, Tome 40 (2004) no. 2, pp. 161-179. http://geodesic.mathdoc.fr/item/ARM_2004_40_2_a3/

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