Leonard Gillman

Rings of Continuous Functions in Which Every Finitely Generated Ideal is Principal

An abstract ring in which all finitely generated ideals are principal will be called an F-ring. Let C(X) denote the ring of all continuous real-valued functions defined on a completely regular (Hausdorff) space X. This paper is devoted to an investigation of those spaces X for which C(X) is an F-ring. In any such study, one of the problems that arises naturally is to determine the algebraic properties and implications that result from the fact that the given ring is a ring of functions. Investigation of this problem leads directly to two others: to determine how specified algebraic conditions on the ring are reflected in topological properties of the space, and, conversely, how specified topological conditions on the space are reflected in algebraic properties of the ring. Our study is motivated in part by some purely algebraic questions concerning an arbitrary F-ring S-in particular, by some problems involving matrices over S. Continual application will be made of the results obtained in the preceding paper [4]. This paper will be referred to throughout the sequel as GH. We wish to thank the referee for the extreme care with which he read both this and the preceding paper, and for making a number of valuable suggestions. The outline of our present paper is as follows. In ?1, we collect some preliminary definitions and results. ?2 inaugurates the study of F-rings and F-spaces (i.e., those spaces X for which C(X) is an F-ring). The space of reals is not an F-space; in fact, a metric space is an F-space if and only if it is discrete. On the other hand, if X is any locally compact, a-compact space (e.g., the reals), then fX-X is an F-space. Examples of necessary and sufficient conditions for an arbitrary completely regular space to be an F-space are: (i) for every f C(X), there exists k E C(X) such that f k Jf J; (ii) for every maximal ideal M of C(X), the intersection of all the prime ideals of C(X) contained in M is a prime ideal. In ??3 and 4, we study Hermite rings and elementary divisor rings(2).

Concerning Rings of Continuous Functions

The present paper deals with two distinct, though related, questions, concerning the ring C(X, R) of all continuous real-valued functions on a completely regular topological space X. The first of these, treated in ? ?1-7, is the study of what we call P-spacesthose spaces X such that every prime ideal of the ring C(X, R) is a maximal ideal. The background and motivation for this problem are set forth in ?1. The results consist of a number of theorems concerning prime ideals of the ring C(X, R) in general, as well as a series of characterizations of P-spaces in particular. The second problem, discussed in ??8-10, is an investigation of what Hewitt has termed Q-spaces-those spaces X that cannot be imbedded as a dense subset of any larger space over which every function in C(X, R) can be continuously extended. An introduction to this question is furnished in ?8. Our discussion of Q-spaces is confined to the class of linearly ordered spaces (introduced in ?6). We are able to settle the question as to when an arbitrary linearly ordered space is or is not a Q-space. The concept of a paracompact space turns out to be intimately related to these considerations. We also derive a characterization of linearly ordered paracompact spaces, and we find in particular that every linearly ordered Q-space is paracompact. A result obtained along the way is that every linearly ordered space is countably paracompact.

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 ].

Selecta : expository writing

A selection of the mathematical writings of Paul R. Halmos (1916 - 2006) is presented in two Volumes. Volume I consists of research publications plus two papers of a more expository nature on Hilbert Space. The remaining expository articles and all the popular writings appear in this second volume. It comprises 27 articles, written between 1949 and 1981, and also a transcript of an interview.

An axiomatic approach to the integral

This invariably gets a good laugh at a lecture. Maybe instead it deserves applause for being so efficient and logical. Concentrating on a single segment permits circumventing the notational paraphernalia of all those subscripts. Calling the width dx permits skipping the summation sign and going directly to the integral sign. (Some concerned teachers will argue that it is better to remind the student of the sums on which the integral is based. Others will liken that to solving quadratics by completing the square.) The usual way is to pick an arbitrary Zk in the kth segment of a partition, form the corresponding Riemann sum, and proclaim that since arbitrary Riemann sums approach the integral (as the norm goes to 0), so then do these arbitrary sums. But then so do particular sums. The practicing scientist picks Zk = Xkl (It is assumed in all this that f is continuous.)

Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis

(2002). Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis. The American Mathematical Monthly: Vol. 109, No. 6, pp. 544-553.

On Intervals of Ordered Sets

The present discussion stems from the following specific problem in the theory of (linearly) ordered sets: let m be any infinite cardinal; is it true that every ordered set of power m has a family of mutually disjoint intervals which is also of power m? This problem is solved, completely or partially, for all except those m > No which are strongly inaccessible.2 The solution is negative for all regular cardinals which are not strongly inaccessible, and is positive for some singular cardinals. Whether it is positive for all singular cardinals we do not know; the statement that it is (proposition Q) proves to be equivalent to a certain hypothesis from the domain of cardinal arithmetic (proposition P) which somewhat resembles, but is weaker than, the generalized hypothesis of the continuum. In particular the solution is negative for every cardinal which is a power of 2. For strongly inaccessible cardinals > No the problem remains open. These results are presented in ?2. The results in this section have been obtained jointly with Alfred Tarski and are included here with his kind permission. Some related problems, and some questions concerning the inaccessible numbers, are discussed in ?3. ?4 is devoted to a few somewhat less closely related theorems on decompositions of sets. 1. In this section we recall some definitions and notation and make a few preliminary remarks.3 The cardinal number of an arbitrary set M is denoted by M; the cardinal number of the set of all ordinals less than any fixed ordinal 4 is given the special symbol A. For every ordinal a, ;f(a) denotes the least cardinal p such that Na can be expressed as the sum of p cardinals each K,,, holds for every a. It follows that if ,, = no (for any n and d) then d < cf(a) (since (01)I n=b). In particular, as deduced by J. Konig a half-century ago, we have that 2NO K, (since cf(cw) = 0). But it is

Paul Halmos’s Expository Writing

Paul Halmos is a redoubtable expositor. This essay presents mini-reviews of some of his works. It is a natural sequel to my article in Selecta—Expository writing, Springer, 1983.

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.