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The ring of arithmetical functions with unitary convolution: Divisorial
and topological properties

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*Jan Snellman*

**Address.**

Department of Mathematics, Stockholm University, SE-10691 Stockholm,
Sweden

**E-mail. ** Jan.Snellman@math.su.se

**Abstract.**

We study $(\UNI,+,\oplus)$, the ring of arithmetical functions
with unitary convolution, giving an isomorphismbetween $({\mathcal
A},+,\oplus)$ and a generalized power series ring on infinitely
many variables, similar to theisomorphism of Cashwell-Everett \cite{NumThe}
between the ring $({\mathcal A},+,\cdot)$ of arithmetical functions with
Dirichlet convolution and the power series ring ${\mathbb C} [\![x_1,x_2,x_3,\dots]\!]$
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.

**AMSclassification. **11A25, 13J05, 13F25

**Keywords. ** Unitary convolution, Schauder Basis, factorization
into atoms, zero divisors.