  In the set of continuous functions C(X,Y) where Y has a topology close to
being discrete, there is an equivalence relation on X which characterizes the
quasi-components of X. If Y satisfies weak algebraic conditions with a single
binary operation then a stable set of functions forms an object generalizing an
ideal of a ring. Calling such sets ideals there is a concept of a prime ideal.
The ideal of functions vanishing on a quasi-component are prime ideals of
C(X,Y). If Y has a zero set that is open then these prime ideals are min- max
implying that when Y is a ring all of the prime ideals of C(X,Y) are of this
form and min-max. However this is a study of C(X,Y) and its ideals beginning
with a few algebraic hypothesis on Y and adding to them as needed. So there are
conditions when a prime ideal is minimal, when the set of quasi-components is
in bijective correspondence with the set of prime ideals of functions which
vanish on them, when C(X,Y) can not be like a local ring, when the prime (nil)
radical is trivial, and when the corresponding Spec(C(X,Y)) is a disconnected
topological space. Example of results are; if a quasi-component of x is open
then the prime ideal of functions vanishing on it is minimal independent of the
zero set of Y being open; if the zero set of Y is open then the set of all
ideals of functions vanishing on quasi-components is the set of minimal prime
ideals irrespective of any quasi-component being open in X. These results imply
the same for the ideals of the ring C(X,Y) when Y is a ring. The techniques
developed give a method of generating unlimited number of rings with prescribed
sets of prime ideals and minimal prime ideals.
