Generalized prime ideals of the rings C(X,Y) and the quasi-components of   X
  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. 
Harvey J. Charlton
math.RA math.GN