Meyer Jerison

The Space of Minimal Prime Ideals of a Commutative Ring

Introduction. Of the various spaces of ideal of rings that have been studied (see [1 ], for example) we are focusing attention on the space of minimal prime ideals because of its special role in the case of rings of continuous functions. For the simplest manifestation of this role, consider the compact space N* obtained from the discrete space of positive integers N by adjoining a single point, o. In the ring C(N*) of all continuous real-valued functions on N*, the maximal ideals are very easy to describe. They are in one-one correspondence with the points of N*, and are given by

On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions

This paper deals with a theorem of Gelfand and Kolmogoroff concerning the ring C= C(X, R) of all continuous real-valued functions on a completely regular topological space X, and the subring C* = C*(X, R) consisting of all bounded functions in C. The theorem in question yields a one-one correspondence between the maximal ideals of C and those of C*; it is stated without proof in [2 ]. Here we supply a proof (?2), and we apply the theorem to three problems previously considered by Hewitt in [5]. Our first result (?3) consists of two simple constructions of the Q-space vX. The second (?4) exhibits a one-one correspondence between the maximal ideals of C and those of C*, in a manner which may be considered qualitatively different from that expressed by Gelfand and Kolmogoroff. In our final application (?5), we confirm Hewitts conjecture that every m-closed ideal of C is the intersection of all the maximal ideals that contain it. In this connection, we also examine the corresponding problem for the ring C*; we find that a necessary and sufficient condition for the theorem to hold here is that every function in C be bounded. The relevant definitions are given below. Further applications of the Gelfand-Kolmogoroff theorem may be found in [3 ].

Minimal projective extensions of compact spaces

Minimal projective extensions of compact spaces.

Rings of Continuous Functions

Contents: Functions of a Topological Space.- Ideals and Z-Filters.- Completely Regular Spaces.- Fixed Ideals. Compact Spaces.- Ordered Residue Class Rings.- The Stone-Czech Compactification.- Characterization of Maximal Ideals.- Realcompact Spaces.- Cardinals of Closed Sets in Beta-x.- Homomorphisms and Continuous Mappings.- Embedding in Products of Real Lines.- Discrete Spaces. Nonmeasurable Cardinals.- Hyper-Real Residue Class Fields.- Prime Ideals.- Uni- form Spaces.- Dimension.- Notes.- Bibliography.- List of Symbols.- Index.

Ordered Residue Class Rings

We saw in the preceding chapter that every residue class field of C or C* modulo a fixed maximal ideal is isomorphic with the real field R. The present chapter initiates the study of residue class fields modulo arbitrary maximal ideals. Each such field has the following properties, as will be shown: it is a totally ordered field, whose order is induced by the partial order in C, and the image of the set of constant functions is an isomorphic copy—necessarily order-preserving—of the real field.

Homomorphisms and Continuous Mappings

We proved in Theorem 8.3 that two realcompact spaces X and Y are homeomorphic if and only if their respective function rings C(X) and C(Y) are isomorphic. The correspondence between the set of all homeomorphisms from X onto Y, and the set of all isomorphisms from C(Y) onto C(X), is one-one; this was not pointed out explicitly at the time, but the information is readily obtainable from an examination of the proof. We shall begin the present chapter by analyzing, in considerable detail, the duality relations expressed by this correspondence. More generally, we describe the relations between arbitrary continuous mappings from X into Y and homomorphisms from C(Y) into C(X). We shall find that, in a sense, every homomorphism from one function ring into another is induced by a continuous mapping.

Fixed Ideals. Compact Spaces

We have seen that in the study of rings of continuous functions there is no need to deal with spaces that are not completely regular. Accordingly, IN THE SEQUEL, ALL GIVEN SPACES ARE ASSUMED TO BE COMPLETELY REGULAR.

Discrete Spaces. Nonmeasurable Cardinals

We have seen many times that the discrete space N is real-compact. More generally, according to 8.18, every discrete space whose cardinal is ≦ c is realcompact. The question arises whether all discrete spaces are realcompact. Since, among discrete spaces, the cardinal is the only significant variable, this is, in fact, a question about cardinal numbers.

Completely Regular Spaces

Up to this point in the text, we have not assumed any separation axioms for the topological space on which our ring of continuous functions is defined. Indeed, separation axioms were irrelevant to most of the subjects discussed. We have now reached the stage where separation properties of the space do enter in an essential way, so that we are forced to make a decision about what class or classes of spaces to consider. We have no desire to become involved in finding the weakest axiom under which each theorem can be proved, but prefer, if possible, to stick to a single class of topological spaces that is wide enough to include all of the interesting spaces, and, at the same time, restrictive enough to admit a significant theory of rings of continuous functions.

Functions on a Topological Space

The set C(X) of all continuous, real-valued functions on a topological space X will be provided with an algebraic structure and an order structure.