In 1964 Frink defined a
normal base. He hypothesized that every Hausdorff compactification of a
Tychonoff space X may be realized as a compactification w(ℱ) of Wallman type
obtained from a normal base ℱ on X, where ℱ is the family of zero sets for
some subring of C(X). Later Biles formally defined a Wallman Ring on a
Tychonoff space to be a subring of C(X) whose zero sets form a normal base on
X.
The problem in this paper is to study examples of Wallman Rings and
develop properties of Wallman Rings. For a locally compact space with a given
compactification and a certain type of retract map, a Wallman Ring is defined which
induces the given compactification.
General algebraic and topological properties of Wallman Rings are considered.
Among the results obtained are “Every Wallman Ring is equivalent to one which
contains all rational constant functions” and “An ideal of a Wallman Ring which is
itself a Wallman Ring is equivalent to the superring.”
