A minimal regular ring extension of $C(X)$

Volume 172 / 2002

M. Henriksen, R. Raphael, R. G. Woods Fundamenta Mathematicae 172 (2002), 1-17 MSC: Primary 54G10; Secondary 46E25, 16E50. DOI: 10.4064/fm172-1-1


Let $G(X)$ denote the smallest (von Neumann) regular ring of real-valued functions with domain $X$ that contains $C(X)$, the ring of continuous real-valued functions on a Tikhonov topological space $(X, \tau )$. We investigate when $G(X)$ coincides with the ring $C(X, \tau _\delta )$ of continuous real-valued functions on the space $(X, \tau _\delta )$, where $\tau _\delta $ is the smallest Tikhonov topology on $X$ for which $\tau \subseteq \tau _\delta $ and $C(X, \tau _\delta )$ is von Neumann regular. The compact and metric spaces for which $G(X) = C(X, \tau _\delta )$ are characterized. Necessary, and different sufficient, conditions for the equality to hold more generally are found.


  • M. HenriksenDepartment of Mathematics
    Harvey Mudd College
    Claremont, CA 91711, U.S.A.
  • R. RaphaelDepartment of Mathematics
    Concordia University
    Montreal, Québec
    Canada H4B 1R6
  • R. G. WoodsDepartment of Mathematics
    University of Manitoba
    Winnipeg, Manitoba
    Canada R3T 2N2

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