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ON THE NUMBER OF UNITAL SUBRINGS OF $C(X)$
Keywords:
subring, intermediate ring, Bell numbers, completely regular space, Tychonoff space, rings and algebras of continuous functions, Banach algebras of continuous functions.DOI:
https://doi.org/10.17654/0972555524003Abstract
Let $X$ be a Tychonoff space. Let $\mathfrak{A}:=C(X)$ and $\mathfrak{A}^*:=C^*(X)$ denote, respectively, the ring of all real-valued continuous functions and the ring of all bounded real-valued continuous functions on $X$. For a minimal ideal $\mathfrak{B}$ of $\mathfrak{A}^*$, we provethat if $\left[\mathfrak{B}, \mathfrak{A}^*\right]$ (the set of rings $\Gamma$ such that $\mathfrak{B} \subseteq \Gamma \subseteq \mathfrak{A}^*$ ) is finite, then so is $X$. Let $\mathcal{R}=\mathbb{R} \mathbf{1}$, where $\mathbf{1}$ is the identity of $\mathfrak{A}$. Then it is shown that the ring extension $\mathcal{R} \subset \mathfrak{A}$ satisfies FIP if and only if the extension $\mathcal{R} \subset \mathfrak{A}^*$ satisfies FIP if and only if $X$ is finite. Further, $|[\mathcal{R}, \mathfrak{A}]|=B_n$, where $B_n$ is the $n$th Bell number and $n$ is the cardinality of $X$.
Received: November 8, 2023
Accepted: December 20, 2023
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