Rings which resemble rings of entire functions

Rush, David E.

Glasgow mathematical journal, Tome 24 (1983) no. 1, pp. 7-16

Résumé

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].

Détail

DOI : 10.1017/S0017089500005000

Bibliographie

[1] 1.Ailing, N. L., The valuation theory of meromorphic function fields over open Riemann surfaces, Acta Math. 110 (1963), 79–96. Google Scholar | DOI

[2] 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. Google Scholar | DOI

[3] 3.Ailing, N. L., Primary ideals in rings of analytic functions, Proc. Amer. Math. Soc. 53 (1975), 423–427. Google Scholar | DOI

[4] 4.Ailing, N. L., Global ideal theory of meromorphic function fields, Trans. Amer. Math. Soc. 256 (1979) 241–266. Google Scholar | DOI

[5] 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] 6.Chevalley, C., Introduction to the theory of algebraic functions in one variable, Math. Surveys No. VI, Amer. Math. Soc. 1951. Google Scholar | DOI

[7] 7.Gilmer, R., Multiplicative ideal theory (Marcel Dekker, 1972). Google Scholar

[8] 8.Güntzer, U., Zur Funtionentheorie einer Veränderlichen über einen vollständigen nicht archimedischen Grunkörper, Arch. Math. 17 (1966), 415–431. Google Scholar | DOI

[9] 9.Helmer, O., Divisibility properties of integral functions, Duke Math. J. 6 (1940), 345–356. Google Scholar | DOI

[10] 10.Henriksen, M., On the ideal structure of the ring of entire functions, Pacific J. Math. 2 (1952), 179–184. Google Scholar | DOI

[11] 11.Henriksen, M., On the prime ideals of the ring of entire functions, Pacific J. Math. 3 (1953), 711–720. Google Scholar | DOI

[12] 12.Iss'sa, H., On meromorphic function fields on a Stein variety, Ann. of Math. 83 (1966), 34–46. Google Scholar | DOI

[13] 13.Jensen, C. U., Arithmetical rings, Ada Math. Acad. Sci. Hungaricae 17 (1966) 115–123. Google Scholar | DOI

[14] 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] 15.Klingen, J., Eine topolgische Verallgemeinerung von Krullringen, Dissertation Univ. Köln (1975). Google Scholar

[16] 16.Klingen, J., Uber eine topologische Verallgemeinerungen von Krullringen, Math. Nachr. 91 (1979), 59–76. Google Scholar | DOI

[17] 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. Google Scholar | DOI

[18] 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] 19.Röhrl, H., Unbounded coverings of Riemann surfaces and extensions of rings of meromorphic functions, Trans. Amer. Math. Soc. 107 (1963), 320–346. Google Scholar | DOI

[20] 20.Williams, R., Primary ideals in rings of analytic functions, Trans. Amer. Math. Soc, 177 (1973), 37–49. Google Scholar | DOI