Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 1-1-1956 Rings of Continuous Functions in Which Every Finitely Generated Ideal is Principal Leonard Gillman University of Texas at Austin Melvin Henriksen Harvey Mudd College Recommended Citation Gillman, L., and M. Henriksen. "Rings of continuous functions in which every finitely generated ideal is principal." Transactions of the American Mathematical Society 82:2 (1956): 366–391. DOI: 10.1090/S0002-9947-1956-0078980-4 This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact
[email protected]. RINGS OF CONTINUOUS FUNCTIONS IN WHICH EVERY FINITELY GENERATED IDEAL IS PRINCIPAL(l) BY LEONARD GILLMAN AND MELVIN HENRIKSEN 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 func tions 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 deter mine 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 prop erties of the ring.