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

TL;DR

This paper explores the structure of prime ideals in rings of continuous functions C(X,Y), linking algebraic properties of Y with topological features of X and quasi-components, and introduces methods to generate rings with specific prime ideal configurations.

Contribution

It generalizes the concept of prime ideals in C(X,Y) under weak algebraic conditions on Y and establishes conditions for minimal prime ideals and their correspondence with quasi-components.

Findings

Prime ideals of functions vanishing on quasi-components are prime ideals.

When Y has an open zero set, these prime ideals are minimal and max.

The techniques allow constructing rings with prescribed prime ideal structures.

Abstract

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.