Commutative Rings in Which Every Prime Ideal is Contained in a Unique Maximal Ideal

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 $C(X)$).