| Title:
|
The ring of arithmetical functions with unitary convolution: Divisorial and topological properties (English) |
| Author:
|
Snellman, Jan |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
40 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
161-179 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
unitary convolution |
| Keyword:
|
Schauder Basis |
| Keyword:
|
factorization into atoms |
| Keyword:
|
zero divisors |
| MSC:
|
11A25 |
| MSC:
|
13F25 |
| MSC:
|
13J05 |
| idZBL:
|
Zbl 1122.11004 |
| idMR:
|
MR2068688 |
| . |
| Date available:
|
2008-06-06T22:43:18Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107898 |
| . |
| Reference:
|
[1] Anderson D. D., Valdes-Leon S.: Factorization in commutative rings with zero divisors.Rocky Mountain J. Math. 26(2) (1996), 439–480. Zbl 0865.13001, MR 1406490 |
| Reference:
|
[2] Anderson D. D., Valdes-Leon S.: Factorization in commutative rings with zero divisors. II.In Factorization in integral domains (Iowa City, IA, 1996), Dekker, New York 1997, 197–219. MR 1460773 |
| Reference:
|
[3] Bosch S., Güntzer U., Remmert R.: Non-Archimedean analysis.Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. Zbl 0539.14017, MR 0746961 |
| Reference:
|
[4] Cashwell E. D., Everett C. J.: The ring of number-theorethic functions.Pacific Journal of Mathematics 9 (1959), 975–985. MR 0108510 |
| Reference:
|
[5] Cohen E.: Arithmetical functions associated with the unitary divisors of an integer.Math. Z. Zbl 0094.02601, MR 0112861 |
| Reference:
|
[6] Huckaba, James A.: Commutative rings with zero divisors.Marcel Dekker Inc., New York, 1988. Zbl 0637.13001, MR 0938741 |
| Reference:
|
[7] Narkiewicz W.: On a class of arithmetical convolutions.Colloq. Math. 10 (1963), 81–94. Zbl 0114.26502, MR 0159778 |
| Reference:
|
[8] Schwab, Emil D., Silberberg, Gheorghe: A note on some discrete valuation rings of arithmetical functions.Arch. Math. (Brno) 36 (2000),103–109. Zbl 1058.11007, MR 1761615 |
| Reference:
|
[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. Zbl 1090.13016, MR 1822767 |
| Reference:
|
[10] Sivaramakrishnan R.: Classical theory of arithmetic functions.volume 126 of Pure and Applied Mathematics, Marcel Dekker, 1989. Zbl 0657.10001, MR 0980259 |
| Reference:
|
[11] Vaidyanathaswamy R.: The theory of multiplicative arithmetic functions.Trans. Amer. Math. Soc. 33(2) (1931), 579–662. Zbl 0002.12402, MR 1501607 |
| Reference:
|
[12] Wilson, Richard M.: The necessary conditions for $t$-designs are sufficient for something.Util. Math. 4 (1973), 207–215. Zbl 0286.05005, MR 0325415 |
| Reference:
|
[13] Yocom K. L.: Totally multiplicative functions in regular convolution rings.Canad. Math. Bull. 16 (1973), 119–128. Zbl 0259.10002, MR 0325502 |
| . |
