Hostname: page-component-5c6d5d7d68-qks25 Total loading time: 0 Render date: 2024-08-14T22:37:47.096Z Has data issue: false hasContentIssue false
  • English
  • Français

Rings which resemble rings of entire functions

Published online by Cambridge University Press:  18 May 2009

David E. Rush
David E. Rush
Affiliation:
University of California Riverside, Ca 92521, USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Since Helmer's 1940 paper [9] laid the foundations for the study of the ideal theory of the ring A(ℂ) of entire functions, many interesting results have been obtained for the rings A(X) of analytic functions on non-compact connected Riemann surfaces. For example, the partially ordered set Spec (A(ℂ) of prime ideals of A(ℂ) has been described by Henrikson and others [2], [10], [11]. Also, it has been shown by Ailing [4] that Spec(A(ℂ))sSpec(A(X)) as topological spaces for any non-compact connected Riemann surface X. Many results on the valuation theory of A(X) have also been obtained [1], [2]. In this note we show that a large portion of the results on the rings A(X) extend to the W-rings with complete principal divisor space which were defined by J. Klingen in [15], [16]. Therefore, many properties of A(ℂ) are shared by its non-archimedian counterparts studied by M. Lazard, M. Krasner, and others [8], [17], [18].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 1 , January 1983 , pp. 7 - 16
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.Ailing, N. L., The valuation theory of meromorphic function fields over open Riemann surfaces, Acta Math. 110 (1963), 79–96.CrossRefGoogle Scholar
2.Ailing, N. L., The valuation theory of meromorphic function fields, Proc. Sympos. Pure Math. Vol. 11, Amer. Math. Soc, Providence, R.I., 1968, pp. 8–29.CrossRefGoogle Scholar
3.Ailing, N. L., Primary ideals in rings of analytic functions, Proc. Amer. Math. Soc. 53 (1975), 423–427.CrossRefGoogle Scholar
4.Ailing, N. L., Global ideal theory of meromorphic function fields, Trans. Amer. Math. Soc. 256 (1979) 241–266.CrossRefGoogle Scholar
5.Aoyama, K., On the structure space of a direct product of rings, J. Sci. Hiroshima Univ. Ser. A-I, 34 (1970), 339–353.Google Scholar
6.Chevalley, C., Introduction to the theory of algebraic functions in one variable, Math. Surveys No. VI, Amer. Math. Soc. 1951.CrossRefGoogle Scholar
7.Gilmer, R., Multiplicative ideal theory (Marcel Dekker, 1972).Google Scholar
8.Güntzer, U., Zur Funtionentheorie einer Veränderlichen über einen vollständigen nicht archimedischen Grunkörper, Arch. Math. 17 (1966), 415–431.CrossRefGoogle Scholar
9.Helmer, O., Divisibility properties of integral functions, Duke Math. J. 6 (1940), 345–356.CrossRefGoogle Scholar
10.Henriksen, M., On the ideal structure of the ring of entire functions, Pacific J. Math. 2 (1952), 179–184.CrossRefGoogle Scholar
11.Henriksen, M., On the prime ideals of the ring of entire functions, Pacific J. Math. 3 (1953), 711–720.CrossRefGoogle Scholar
12.Iss'sa, H., On meromorphic function fields on a Stein variety, Ann. of Math. 83 (1966), 34–46.CrossRefGoogle Scholar
13.Jensen, C. U., Arithmetical rings, Ada Math. Acad. Sci. Hungaricae 17 (1966) 115–123.CrossRefGoogle Scholar
14.Kakutani, S., Rings of analytic functions, in Lectures on functions of a complex variable, edited by Kaplan, W. et al. (Univ. of Michigan, 1955), 71–83.Google Scholar
15.Klingen, J., Eine topolgische Verallgemeinerung von Krullringen, Dissertation Univ. Köln (1975).Google Scholar
16.Klingen, J., Uber eine topologische Verallgemeinerungen von Krullringen, Math. Nachr. 91 (1979), 59–76.CrossRefGoogle Scholar
17.Krasner, M., Rapport sur le prolongement analytique dans les corps valués complets par la methode des éléments analytiques quasi-connexes, Bull. Soc. Math. France Memoire 39–40 (1974), 131–254.CrossRefGoogle Scholar
18.Lazard, M., Les zéros des fonctions analytiques d'une variable sur un corps valué complet., Institut des Hautes. Études Scientifiques 14 (1962), 223–251.Google Scholar
19.Röhrl, H., Unbounded coverings of Riemann surfaces and extensions of rings of meromorphic functions, Trans. Amer. Math. Soc. 107 (1963), 320–346.CrossRefGoogle Scholar
20.Williams, R., Primary ideals in rings of analytic functions, Trans. Amer. Math. Soc, 177 (1973), 37–49.CrossRefGoogle Scholar