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 CiX) to obtain Stone's theorem and the Gelfand-Kolmogoroff theorem. As a side result, the methods give new information on the mapping P-*PC\C*iX) (P a prime ideal of CiX)). Introduction. A ring with an identity possessing the properties of the title will be called a pm-ring. The purpose of this paper is to study some properties of the prime spectrum (P and of the maximal spectrum 3TC of a pm-ring A. The pm-property is strictly related to separation properties of (P and 9TC; specifically, it is equivalent to the normality of (P and implies that 9ÎI is T2, all three conditions being equivalent if the Jacobson radical and the nilradical of A coincide. In a pm-ring A, let u be the map of G> onto 9TC which sends every prime ideal into the unique maximal ideal containing it. One of our main results is the fact that p is continuous ; in fact, the pm-rings are also characterized by the presence of a retraction of
P(~\C*iX) of the prime spectrum of CiX) into the prime spectrum of C*(X), which extend results obtained in [M ] by different techniques.
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