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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
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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

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