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-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 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

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

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 fЄC(X), there exists kЄC(X) such that f=k|f|; (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. A necessary and sufficient condition that C(X) be an Hermite ring is that for all f, gЄC(X), there exist k, lЄC(X) such that f=k|f|, g=l|g|, and (k, l) = (1). We also construct an F-ring that is not an Hermite ring, and an Hermite ring that is not an elementary divisor ring. To produce these examples, we translate the algebraic conditions on C(X) into topological conditions on X, as indicated above. The construction of a ring having one algebraic property but not the other is then accomplished by finding a space that has the topological properties corresponding to the one, but not to the other. In §§5 and 6, we investigate some further special classes of F-rings, including regular rings and adequate rings. Appendices (§§7 and 8) touch upon various related questions. A diagram is included to show the implications that exist among the principal classes of spaces that have been considered

Some Remarks About Elementary Divisor Rings

By a slight modification of Kaplansky\u27s 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 and sufficient condition that S be an elementary divisor ring. Along the way, we obtain a necessary and sufficient condition that S be an Hermite ring. In the paper that follows [2], we make constant use of these results. In particular, we construct examples of rings that satisfy F but are not Hermite rings, and examples of Hermite rings that are not elementary divisor rings. However, all these examples contain zero-divisors; therefore, the question as to whether there exist corresponding examples that are integral domains is left unsettled

Meyer Jerison, 1922-1995

Memories and biography of mathematician Meyer Jerison (b. November 28, 1922 - d. March 19, 1995)

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 Hewitt\u27s 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

Birth cohorts in asthma and allergic diseases: Report of a NIAID/NHLBI/MeDALL joint workshop

Population-based birth cohorts on asthma and allergies increasingly provide new insights into the development and natural history of the diseases. Over 130 birth cohorts focusing on asthma and allergy have been initiated in the last 30 years. A NIAID (National Institute of Allergy and Infectious Diseases), NHLBI (National Heart Lung and Blood Institute), MeDALL (Mechanisms of the Development of Allergy, Framework Programme 7 of the European Commission) joint workshop was held in Bethesda, MD, USA September 11–12, 2012 with 3 objectives (1) documenting the knowledge that asthma/allergy birth cohorts have provided, (2) identifying the knowledge gaps and inconsistencies and (3) developing strategies for moving forward, including potential new study designs and the harmonization of existing asthma birth cohort data. The meeting was organized around the presentations of 5 distinct workgroups: (1) clinical phenotypes, (2) risk factors, (3) immune development of asthma and allergy, (4) pulmonary development and (5) harmonization of existing birth cohorts. This manuscript presents the workgroup reports and provides web links (AsthmaBirthCohorts.niaid.nih.gov or www.medall-fp7.eu) where the reader will find tables describing the characteristics of the birth cohorts included in this report, type of data collected at differing ages, and a selected bibliography provided by the participating birth cohorts

Calculus : Solutions Manual

New York225 p.: illus.; 23 c