#### Title

On a Theorem of Gelfand and Kolmogoroff Concerning Maximal Ideals in Rings of Continuous Functions

#### Document Type

Article

#### Department

Mathematics (HMC)

#### Publication Date

6-1954

#### Abstract

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's 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.

#### Rights Information

© 1954 American Mathematical Society

#### Terms of Use & License Information

#### DOI

10.1090/S0002-9939-1954-0066627-6

#### Recommended Citation

Gillman, L., M. Henriksen, and M. Jerison. "On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions." Proceedings of the American Mathematical Society 5.3 (June 1954): 447-455. DOI: 10.1090/S0002-9939-1954-0066627-6

## Comments

Previously linked to as: http://ccdl.libraries.claremont.edu/u?/irw,404.

Pdf created from original.

This article is also available at http://www.ams.org/journals/proc/1954-005-03/S0002-9939-1954-0066627-6/.