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In the set of continuous functions C(X,Y) where Y has a topology close tobeing discrete, there is an equivalence relation on X which characterizes thequasi-components of X. If Y satisfies weak algebraic conditions with a singlebinary operation then a stable set of functions forms an object generalizing anideal 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 ofC(X,Y). If Y has a zero set that is open then these prime ideals are min- maximplying that when Y is a ring all of the prime ideals of C(X,Y) are of thisform and min-max. However this is a study of C(X,Y) and its ideals beginningwith a few algebraic hypothesis on Y and adding to them as needed. So there areconditions when a prime ideal is minimal, when the set of quasi-components isin bijective correspondence with the set of prime ideals of functions whichvanish 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 disconnectedtopological space. Example of results are; if a quasi-component of x is openthen the prime ideal of functions vanishing on it is minimal independent of thezero set of Y being open; if the zero set of Y is open then the set of allideals of functions vanishing on quasi-components is the set of minimal primeideals irrespective of any quasi-component being open in X. These results implythe same for the ideals of the ring C(X,Y) when Y is a ring. The techniquesdeveloped give a method of generating unlimited number of rings with prescribedsets of prime ideals and minimal prime ideals.

Generalized prime ideals of the rings C(X,Y) and the quasi-components of X