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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-Pnc*(x) (P a prime ideal of C(X)). 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 21Z of a pm-ring A. The pm-property is strictly related to separation properties of (P and 1Z; specifically, it is equivalent to the normality of (P and implies that M is T2, all three conditions being equivalent if the Jacobson radical and the nilradical of A coincide. In a pm-ring A, let , be the map of (P onto 21Z which sends every prime ideal into the unique maximal ideal containing it. One of our main results is the fact that ,u is continuous; in fact, the pm-rings are also characterized by the presence of a retraction of 6? onto 91; this retraction is unique and is precisely the map p. The preceding results, applied to rings of real valued continuous functions, show that the maximal spectrum of C(X) is a natural model of OX with the consequent possible advantages of introducing 3X in this way; e.g. Stone's theorem and the Gelfand-Kolmogoroff theorem follow easily. The methods used also yield results on the mapping P->Pn C* (X) of the prime spectrum of C(X) into the prime spectrum of C*(X), which extend results obtained in [M ] by different techniques. 1. Algebraic results. 1.1. Throughout this paper, the term ring denotes a commutative ring with an identity, all subrings of a given ring contain the identity of the whole ring, and all ring homomorphisms preserve identities. Let A be a ring, and let J(A) be the Jacobson radical (intersection of Received by the editors October 28, 1970. AMS 1969 subject classifications. Pi imary 1310, 1320; Secondary 5420, 5428.

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