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

Jan Snellman

Archivum Mathematicum (2004)

  • Volume: 040, Issue: 2, page 161-179
  • ISSN: 0044-8753

Abstract

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We study ( 𝒜 , + , ⊕ ) , the ring of arithmetical functions with unitary convolution, giving an isomorphism between ( 𝒜 , + , ⊕ ) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring ( 𝒜 , + , · ) of arithmetical functions with Dirichlet convolution and the power series ring [ [ x 1 , x 2 , x 3 , ⋯ ] ] 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.

How to cite

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Snellman, Jan. "The ring of arithmetical functions with unitary convolution: Divisorial and topological properties." Archivum Mathematicum 040.2 (2004): 161-179. <http://eudml.org/doc/249297>.

@article{Snellman2004,
abstract = {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.},
author = {Snellman, Jan},
journal = {Archivum Mathematicum},
keywords = {unitary convolution; Schauder Basis; factorization into atoms; zero divisors; Schauder basis; factorization into atoms; zero divisors},
language = {eng},
number = {2},
pages = {161-179},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The ring of arithmetical functions with unitary convolution: Divisorial and topological properties},
url = {http://eudml.org/doc/249297},
volume = {040},
year = {2004},
}

TY - JOUR
AU - Snellman, Jan
TI - The ring of arithmetical functions with unitary convolution: Divisorial and topological properties
JO - Archivum Mathematicum
PY - 2004
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 040
IS - 2
SP - 161
EP - 179
AB - 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.
LA - eng
KW - unitary convolution; Schauder Basis; factorization into atoms; zero divisors; Schauder basis; factorization into atoms; zero divisors
UR - http://eudml.org/doc/249297
ER -

References

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  9. Schwab, Emil D., Silberberg, Gheorghe, The valuated ring of the arithmetical functions as a power series ring, Arch. Math. (Brno) 37(1) (2001), 77–80. Zbl1090.13016MR1822767
  10. Sivaramakrishnan R., Classical theory of arithmetic functions, volume 126 of Pure and Applied Mathematics, Marcel Dekker, 1989. (1989) Zbl0657.10001MR0980259
  11. Vaidyanathaswamy R., The theory of multiplicative arithmetic functions, Trans. Amer. Math. Soc. 33(2) (1931), 579–662. (1931) Zbl0002.12402MR1501607
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