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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
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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

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