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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the ideal structure of $ C(X)$


Author: William E. Dietrich
Journal: Trans. Amer. Math. Soc. 152 (1970), 61-77
MSC: Primary 46.55
MathSciNet review: 0265941
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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

$\displaystyle A \subset {\text{cl} _{\beta X}}Z(f)\quad [A \subset {\operatorname{int} _{\beta X}}{\text{cl} _{\beta X}}Z(f)].$

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 $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{J} $ and $ \bar J$ of $ C(X)$ for which $ {F^A} \subset \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{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.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0265941-2
PII: S 0002-9947(1970)0265941-2
Keywords: Completely regular space, algebra of continuous functions, Stone-Čech compactification, prime ideal, $ Z$-ideal, maximal ideal, minimal ideal, Krull dimension, maximal algebra, minimal algebra
Article copyright: © Copyright 1970 American Mathematical Society