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In this article, we investigate new topological descriptions for two well-known mappings ${𝒵}_A$ and ${\Im }_A$ defined on intermediate rings $A\left(X\right)$ of $C\left(X\right)$. Using this, coincidence of each two classes of $z$-ideals, ${𝒵}_A$-ideals and ${\Im }_A$-ideals of $A\left(X\right)$ 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].

Let $A\left(X\right)$ denote a subalgebra of $C\left(X\right)$ which is closed under local bounded inversion, briefly, an $LBI$-subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A\left(X\right)$, we generalize the notion of $z_A^{\beta }$-ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C\left(X\right)$ containing...