Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-10T15:23:32.396Z Has data issue: false hasContentIssue false
  • English
  • Français

Transferring Results From Rings of Continuous Functions to Rings of Analytic Functions

Published online by Cambridge University Press:  20 November 2018

Andrew Adler  and
R. Douglas Williams
Andrew Adler
Affiliation:
University of British Columbia
R. Douglas Williams
Affiliation:
Vancouver, British Columbia
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.

Let C(X) be the ring of all real-valued continuous functions on a completely regular topological space X, and let A﹛Y) be the ring of all functions analytic on a connected non-compact Riemann surface F. The ideal theories of these two function rings have been extensively studied since the fundamental papers of E. Hewitt on C﹛X)[12] and of M. Henriksen on the ring of entire functions [10; 11]. Despite the obvious differences between these two rings, it has turned out that there are striking similarities between their ideal theories. For instance, non-maximal prime ideals of A (F) [2; 11] behave very much like prime ideals of C﹛X)[13; 14], and primary ideals of A(Y) which are not powers of maximal ideals [19] resemble primary ideals of C(X) [15]. In this paper we show that there are very good reasons for these similarities. It turns out that much of the ideal theory of A (Y) is a special case of the ideal theory of rings of continuous functions. We develop machinery that enables one almost automatically to derive results about the ideal theory of A(Y) from corresponding known results of ideal theory for rings of continuous functions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 75 - 87
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Adler, A. and Williams, R. D., Restricted ideals in rings of analytic functions (to appear in Bull. Amer. Math. Soc).Google Scholar
2. Ailing, N. L., The valuation theory of meromorphic function fields over open Riemann surfaces, Acta Math. 110 (1962), 79–96.Google Scholar
3. Ailing, N. L., The valuation theory of meromorphic function fields, Proc. Sympos. Pure Math., vol. 11 (Amer. Math. Soc, Providence, R.I., 1968), 8–29.Google Scholar
4. De Marco, G., On the countably generated z-ideals of C(X), Proc. Amer. Math. Soc. 31 (1972), 574–576.Google Scholar
5. Dietrich, W. E., Jr., On the ideal structure of C(X), Trans. Amer. Math. Soc. 152 (1970), 61–77.Google Scholar
6. Florack, H., Reguldre und meromorphe Funktionen auf nicht geschlossenen Riemannschen Fldchen, Schr. Math. Inst. Univ. Munster 1 (1948).Google Scholar
7. Gillman, L., Countably generated ideals in rings of continuous functions, Proc. Amer. Math. Soc. 11 (1960), 660–666.Google Scholar
8. Gillman, L. and Jerison, M., Rings of continuous functions, The University Series in Higher Math. (Van Nostrand, Princeton, N.J., 1960).Google Scholar
9. Helmer, O., Divisibility properties of integral functions, Duke Math. J. 6 (1940), 345–356.Google Scholar
10. Henriksen, M., On the ideal structure of the ring of entire functions, Pacific J. Math. 2 (1952), 179–184.Google Scholar
11. Henriksen, M., On the prime ideals of the ring of entire functions, Pacific J. Math. 3 (1953), 711–720.Google Scholar
12. Hewitt, E., Rings of real valued continuous functions, I, Trans. Amer. Math. Soc. 64 (1948), 54–99.Google Scholar
13. Kohls, C. W., Prime ideals in rings of continuous functions, Illinois J. Math. 2 (1958), 505–536.Google Scholar
14. Kohls, C. W., Prime ideals in rings of continuous functions, II, Duke Math. J. 25 (1958), 447–458.Google Scholar
15. Kohls, C. W., Primary ideals in rings of continuous functions, Amer. Math. Monthly 71 (1964), 980–984.Google Scholar
16. Kohls, C. W., A note on countably generated ideals in rings of continuous functions, Proc. Amer. Math. Soc. 12 (1961), 744–749.Google Scholar
17. Rudd, D., On two sum theorems for ideals of C(X), Michigan Math. J. 17 (1970), 139–141.Google Scholar
18. Williams, R. D., Intersections of primary ideals in rings of continuous functions, Can. J. Math. 24 (1972), 502–519.Google Scholar
19. Williams, R. D., Primary ideals in rings of analytic functions, Trans. Amer. Math. Soc. 178 (1973), 37–49.Google Scholar