In this paper we compare the
Gelfand and Wallman methods of constructing a compactification for a Tychonoff
space X from a suitable ring of continuous real-valued functions on X. Every
Hausdorff compactification T of X is Gelfand constructable; in particular,
T is equivalent, as a compactification of X, to the structure space of all
maximal ideals of the ring of all continuously extendable functions from X to
T. However, Wallman’s method applied to this ring may not yield T. We
thus inquire into some relationships that exist between the Wallman and
Gelfand compactification of X constructed from a suitable ring of functions on
X.
