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The Minimal Prime Spectrum of a Commutative Ring

Published online by Cambridge University Press:  20 November 2018

M. Hochster
M. Hochster*
Affiliation:
University of Minnesota, Minneapolis, Minnesota
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We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note that if A has an identity, is a subspace of Spec A (as defined in [1, p. 124]). It is well known that a minspectral space is Hausdorff and has a clopen basis (and hence is completely regular). We give here a topological characterization of the minspectral spaces, and we show that all minspectral spaces can actually be obtained from rings with identity and that open (but not closed) subspaces of minspectral spaces are minspectral (Theorem 1, Proposition 5).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 749 - 758
Copyright
Copyright © Canadian Mathematical Society 1971

Footnotes

This research was partially supported by NSF grant GP-8496.

References

1. Bourbaki, N., Algèbre commutative (Hermann, Paris, 1961).Google Scholar
2. Henriksen, M. and Jerison, M., The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110–130.Google Scholar
3. Hochster, M., Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 1/+2 (1969), 43–60.Google Scholar
4. Kelley, J. L., General topology (Van Nostrand, Princeton, 1955).Google Scholar
5. Roy, P., Failure of equivalence of dimension concepts for metric spaces, Bull. Amer. Math. Soc. 68 (1962), 609–613.Google Scholar