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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Commutative rings in which every prime ideal is contained in a unique maximal ideal


Authors: Giuseppe De Marco and Adalberto Orsatti
Journal: Proc. Amer. Math. Soc. 30 (1971), 459-466
MSC: Primary 13.20; Secondary 46.00
MathSciNet review: 0282962
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Abstract: The class of rings with 1 satisfying the properties of the title is characterized by some separation properties on the prime and maximal spectra, and, in such rings, the map which sends every prime ideal into the unique maximal ideal containing it, is continuous. These results are applied to $ C(X)$ to obtain Stone's theorem and the Gelfand-Kolmogoroff theorem. As a side result, the methods give new information on the mapping $ P \to P \cap {C^ \ast }(X)$ (P a prime ideal of $ C(X)$).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0282962-0
PII: S 0002-9939(1971)0282962-0
Keywords: Prime spectrum, maximal spectrum, retraction, Jacobson radical, nilradical
Article copyright: © Copyright 1971 American Mathematical Society