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Intersections of Primary Ideals in Rings of Continuous Functions

Published online by Cambridge University Press:  20 November 2018

R. Douglas Williams
R. Douglas Williams*
Affiliation:
Purdue University, Lafayette, Indianna
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Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.

C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 3 , June 1972 , pp. 502 - 519
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).Google Scholar
2. Gillman, L. and Kohls, C. W., Convex and pseudoprime ideals in rings of continuous functions, Math. Z. 72 (1960), 399–409.Google Scholar
3. Kohls, C. W., Prime ideals in rings of continuous functions, Illinois J. Math. 2 (1958), 505–536.Google Scholar
4. Kohls, C. W., Prime ideals in rings of continous functions, II, Duke Math. J. 25 (1958), 447–458.Google Scholar
5. Kohls, C. W., Primary ideals in rings of continuous functions, Amer. Math. Monthly 71 (1964), 980–984.Google Scholar