Digital Commons Network^™

A Simple Characterization Of Commutative Rings Without Maximal Ideals, Melvin Henriksen

All HMC Faculty Publications and Research

In a course in abstract algebra in which the instructor presents a proof that each ideal in a ring with identity is contained in a maximal ideal, it is customary to give an example of a ring without maximal ideals.

Sums Of Kth Powers In The Ring Of Polynomials With Integer Coefficients, Ted Chinburg, Melvin Henriksen

All HMC Faculty Publications and Research

A working through of two theorems.

Suppose R is a ring with identity element and k is a positive integer. Let J(k, R) denote the subring of R generated by its kth powers. If Z denotes the ring of integers, then G(k, R) = {a ∈ Z: aR ⊂ J(k, R)} is an ideal of Z.

Is Mathematical Truth Time-Dependent?, Judith V. Grabiner

Pitzer Faculty Publications and Research

Another such mathematical revolution occurred between the eighteenth and nineteenth centuries, and was focused primarily on the calculus. This change was a rejection of the mathematics of powerful techniques and novel results in favor of the mathematics of clear definitions and rigorous proofs. Because this change, however important it may have been for mathematicians themselves, is not often discussed by historians and philosophers, its revolutionary character is not widely understood. In this paper, I shall first try to show that this major change did occur. Then, I shall investigate what brought it about. Once we have done this, we can …

Fields Medals And Nevanlinna Prize Presented At Icm-94 In Zurich, Nicholas Pippenger, J. Lindenstrauss, L.C. Evans, A. Douady, A. Shalev

All HMC Faculty Publications and Research

The Notices solicited the following five articles describing the work of the Fields Medalists and Nevanlinna Prize winner.

Toeplitz Operators On Locally Compact Abelian Groups, Henry A. Krieger, C.A. Schaffner

All HMC Faculty Publications and Research

The problem of global optimization of M incoherent phase signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for $N = 2$, and the optimal signal sets are determined for $M = 2,3,4,6$ and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ${\bf s}_j $, $j = 1,2, \cdots ,M$, on the unit sphere $S_1 $ in $C^N $. If $W_{jk} $ is the half space determined by ${\bf s}_j …

The Global Optimization Of Incoherent-Phase Signals, Henry A. Krieger, Charles Albert Schaffner

All HMC Faculty Publications and Research

No abstract provided.

Calculus And The Computer: A Conservative Approach, Melvin Henriksen

All HMC Faculty Publications and Research

This paper describes a program for making the use of numerical methods an integral part of the freshman college course in single variable calculus.

The Eigenvalue Set Of A Class Of Equimodular Matrices, Gerald L. Bradley

CMC Faculty Publications and Research

The paper is divided into four sections, the first of which is basically introductory and contains definitions, notational conventions, and well-known results. The second section contains a fundamental theorem on multilinear polynomials which forms the crux of all our mail results, and the final two sections are devoted to discussions of the combinatorial and boundary properties of Y(A), respectively.

The Space Of Minimal Prime Ideals Of A Commutative Ring, Melvin Henriksen, Meyer Jerison

All HMC Faculty Publications and Research

The present paper is devoted to the space of minimal prime ideals of a more-or-less arbitrary commutative ring. Rings C(X) of continuous functions on topological spaces X appear only in §5 where they serve largely to provide significant examples.

Minimal Projective Extensions Of Compact Spaces, Melvin Henriksen, Meyer Jerison

All HMC Faculty Publications and Research

A compact space E is called projective if for each mapping ψ of E into a compact space X, and each continuous mapping τ of a compact space Y onto X, there is a continuous mapping φ of E into Y such that ψ = τ o φ.

Averages Of Continuous Functions On Countable Spaces, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

Let X = {x1, x2, ...} be a countably infinite topological space; then the space C*(X) of all bounded real-valued continuous functions f may be regarded as a space of sequences (f(x1), f(x2), ...). It is well known [7, p. 54] that no regular (Toeplitz) matrix can sum all bounded sequences. On the other hand, if (x1, x2, ...) converges in X (to x_m), then every regular matrix sums all f in C*(X) (to f(x_m)).

The main result of this paper is that if a regular matrix sums all f in C*(X) then it sums f …

On The Structure Of A Class Of Archimedean Lattice-Ordered Algebras, Melvin Henriksen, D. G. Johnson

All HMC Faculty Publications and Research

By a Φ-algebra A, we mean an Archimedean lattice-ordered algebra over the real field R which has an identity element 1 that is a weak order unit. The Φ-algebras constitute the class of the title. It is shown that every ф-algebra is isomorphic to an algebra of continuous functions on a compact space X into the two-point compactification of the real line R, each of which is real-valued on an (open) everywhere dense subset of X. Under more restrictive assumptions on A, ropresentations of this sort have long been known. An (incomplete) history of them …

Lattice-Ordered Rings And Function Rings, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

This paper treats the structure of those lattice-ordered rings which are subdirect sums of totally ordered rings -- the f-rings of Birkhoff and Pierce [4]. Broadly, it splits into two parts, concerned respectively with identical equations and with ideal structure; but there is an important overlap at the beginning.

Some Properties Of Compactifications, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

A compactification of a topological space X is a compact (Hausdorff) space containing a dense subspace homeomorphic with X. Since only completely regular spaces have compactifications, all spaces mentioned here will be completely regular unless the contrary is assumed explicitly. This paper is a study of properties of the sets of points which may be added to a space in compactifying it.

Local Connectedness In The Stone-Cech Compactification, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

This is a study of when and where the Stone-Čech compactification of a completely regular space may be locally connected. As to when, Banaschewski [1] has given strong necessary conditions for βX to be locally connected, and Wallace [19] has given necessary and sufficient conditions in case X is normal. We show below that Banaschewski's necessary conditions are also sufficient and may be restated as follows: βX is locally connected if and only if X is locally connected and pseudo-compact (Corollary 2.5). Moreover, the requirement that βX be locally connected is so strong that it implies that every completely regular …

Some Remarks On A Paper Of Aronszajn And Panitchpakdi, Melvin Henriksen

All HMC Faculty Publications and Research

In the paper of the title [1], a number of problems are posed. Negative solutions of two of them (Problems 2 and 3) are derived in a straightforward way from a paper of L. Gillman and the present author [2]. Motivation will not be supplied since it is given amply in [1], but enough definitions are given to keep the presentation reasonably self contained.

On Minimal Completely Regular Spaces Associated With A Given Ring Of Continuous Functions, Melvin Henriksen

All HMC Faculty Publications and Research

Let C(X) denote the ring of all continuous real-valued functions on a completely regular space X. If X and Y are completely regular spaces such that one is dense in the other, say X is dense in Y, and every f ε C(X) has a (unique) extension f E C(Y), then C(X) and C(Y) are said to be strictly isomorphic. In a recent paper [2], L. J. Heider asks if it is possible to associate with the completely regular space X a dense subspace μX minimal with respect to the property that C(μX) and C(X) are strictly isomorphic.

On The Equivalence Of The Ring, Lattice, And Semigroup Of Continuous Functions, Melvin Henriksen

All HMC Faculty Publications and Research

A large number of papers have been published that are devoted to showing that certain algebraic objects obtained by defining operations on the set of all continuous real-valued functions on a suitably restricted topological space determine the space. We mention but a few of them in this article.

On Rings Of Bounded Continuous Functions With Values In A Division Ring, Ellen Correl, Melvin Henriksen

All HMC Faculty Publications and Research

Let C*(X, A) denote the ring of bounded continuous functions on a (Hausdorff) topological space X with values in a topological division ring A. If, for every maximal (two-sided) ideal M of C*(X, A), we have C*(X, A)/M is isomorphic with A, we say that Stone's theorem holds for C*(X, A). It is well known [9; 6] that Stone's theorem holds for C*(X, A) if A is locally compact and connected, or a finite field. In giving a partial answer to a question of Kaplansky [7], Goldhaber and Wolk showed in [5] that, with restriction on X, and if A …

Some Remarks About Elementary Divisor Rings, Leonard Gillman, Melvin Henriksen

All HMC Faculty Publications and Research

By a slight modification of Kaplansky's argument, we find that the condition on zero-divisors can be replaced by the hypothesis that S be an Hermite ring (i.e., every matrix over S can be reduced to triangular form). This is an improvement, since, in any case, it is necessary that S be an Hermite ring, while, on the other hand, it is not necessary that all zero-divisors be in the radical. In fact, we show that every regular commutative ring with identity is adequate. However, the condition that S be adequate is not necessary either.

We succeed in obtaining a necessary …

Rings Of Continuous Functions In Which Every Finitely Generated Ideal Is Principal, Leonard Gillman, Melvin Henriksen

All HMC Faculty Publications and Research

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, σ-compact space (e.g., the reals), then βX-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 …

An Isomorphism Theorem For Real-Closed Fields, P. Erdös, L. Gillman, Melvin Henriksen

All HMC Faculty Publications and Research

A classical theorem of Steinitz states that the characteristic of an algebraically closed fields, together with its absolute degree of transcendency, uniquely determine the field (up to isomorphism). It is easily seen that the word real-closed cannot be substituted for the words algebraically closed in this theorem. It is therefore natural to inquire what invariants other than the absolute transcendence degree are needed in order characterize a real-closed field.

Some Remarks On Elementary Divisor Rings Ii, Melvin Henriksen

All HMC Faculty Publications and Research

A commutative ring S with identity element 1 is called an elementary divisor ring (resp. Hermite ring) if for every matrix A over S there exist nonsingular matrices P, Q such that PAQ (resp. AQ) is a diagonal matrix (resp. triangular matrix). It is clear that every elementary divisor ring is an Hermite ring, and that every Hermite ring is an F-ring (that is, a commutative ring with identity in which all finitely generated ideals are principal).

Concerning Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen

All HMC Faculty Publications and Research

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-spaces -- those 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 …

On A Theorem Of Gelfand And Kolmogoroff Concerning Maximal Ideals In Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen, Meyer Jerison

All HMC Faculty Publications and Research

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. …

On The Continuity Of The Real Roots Of An Algebraic Equation, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

It is well known that the root of an algebraic equation is a continuous multiple-valued function of its coefficients [5, p. 3]. However, it is not necessarily true that a root can be given by a continuous single-valued function. A complete solution of this problem has long been known in the case where the coefficients are themselves polynomials in a complex variable [3, chap. V]. For most purposes the concept of the Riemann surface enables one to bypass the problem. However, in the study of the ideal structure of rings of continuous functions, the general problem must be met directly. …

On Rings Of Entire Functions Of Finite Order, Melvin Henriksen

All HMC Faculty Publications and Research

In an earlier paper, the author showed that if M is any maximal ideal of R, the residue class field R/M is isomorphic with the complex field K. In this paper, under some restrictions, this theorem is extended to the ring R_λ of all entire functions of order no greater than λ, and hence to R*.

On The Prime Ideals Of The Ring Of Entire Functions, Melvin Henriksen

All HMC Faculty Publications and Research

Let R be the ring of entire functions, and let K be the complex field. In an earlier paper [6], the author investigated the ideal structure of R, particular attention being paid to the maximal ideals. In 1946, Schilling [9, Lemma 5] stated that every prime ideal of R is maximal. Recently, I. Kaplansky pointed out to the author (in conversation) that this statement is false, and constructed a non maximal prime ideal of R (see Theorem 1(a), below). The purpose of the present paper is to investigate these nonmaximal prime ideals and their residue class fields. The author is …

On The Ideal Structure Of The Ring Of Entire Functions, Melvin Henriksen

All HMC Faculty Publications and Research

Let R be the ring of entire functions, and let K be the complex field. The ring R consists of all functions from K to K differentiable everywhere (in the usual sense).

The algebraic structure of the ring of entire functions seems to have been investigated extensively first by O. Helmer [1].

The ideals of R are herein classified as in [2]: an ideal I is called fixed if every function in it vanishes at at least one common point; otherwise, I is called free. The structure of the fixed ideals was determined in [1]. The structure of the …