On the ideal structure of $C(X)$

Abstract:

The ideal structure of $C(X)$, the algebra of continuous functions from a completely regular Hausdorff space $X$ to the scalars is analyzed by examining for fixed $A \subset \beta X$ (the Stone-Čech compactification of $X$) the structure of the quotient ${I^A}/{F^A}$, where ${I^A}[{F^A}]$ is the ideal of maps $f \in C(X)$ for which \[ A \subset {\text {cl} _{\beta X}}Z(f)\quad [A \subset {\operatorname {int} _{\beta X}}{\text {cl} _{\beta X}}Z(f)].\] Unless it vanishes, ${I^A}/{F^A}$ has no minimal or maximal ideals, and its Krull dimension is infinite. If $J$ is an ideal of $C(X)$ strictly between ${F^A}$ and ${I^A}$, there are ideals $\underline {J}$ and $\bar J$ of $C(X)$ for which ${F^A} \subset \underline {J} \subset J \subset \bar J \subset {I^A}$ with all inclusions proper. For $K \subset C(X)$, let $Z(K) = \cap \{ {\text {cl} _{\beta X}}Z(f):f \in K\}$. If $J \subsetneqq I$ are ideals of $C(X)$ with $Z(J) = Z(I)$ and if $I$ is semiprime, there is an ideal $K$ strictly between $J$ and $I$. If $I$ and $J$ are $Z$-ideals, $K$ can be chosen to be of the form $P \cap I, P$ a prime ideal of $C(X)$. The maximal ideals of a semiprime ideal $I$ of $C(X)$ are of the form ${I^q} \cap I,q \in \beta X - Z(I)$. If $A \subset \beta X$ is closed, ${I^A}$ is a finitely generated ideal iff $A$ is open.