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Keywords:
$z$-ideal; ${\mathcal Z}_A$-ideal; $\Im_A$-ideal; $z$-filter; ${\mathcal Z}_A$-filter; $\Im_A$-filter; intermediate ring
$z$-ideal; ${\mathcal Z}_A$-ideal; $\Im_A$-ideal; $z$-filter; ${\mathcal Z}_A$-filter; $\Im_A$-filter; intermediate ring
Summary:
In this article, we investigate new topological descriptions for two well-known mappings ${\mathcal Z}_A$ and $\Im_A$ defined on intermediate rings $A(X)$ of $C(X)$. Using this, coincidence of each two classes of $z$-ideals, ${\mathcal Z}_A$-ideals and $\Im_A$-ideals of $A(X)$ is studied. Moreover, we answer five questions concerning the mapping $\Im_A$ raised in [J. Sack, S. Watson, {$C$ and $C^*$ among intermediate rings}, Topology Proc. {43} (2014), 69--82].
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