On m-closure of ideals in a class of subrings of C(X)

Abstract

A subalgebra $A(X)$ of $C(X)$ is said to be a $\beta$-subalgebra if the space of maximal ideals of $A(X)$ equipped with the hull-kernel topology is homeomorphic to $\beta X$. Kharbhih and Dutta in [Closure formula for ideals in intermediate rings, {\em Appl. Gen. Topol}. {\bf 21} (2) (2020), 195-200] showed that the closure of every ideal $I$ of an intermediate ring under the $m$-topology, briefly, the $m$-closure of $I$, equals the intersection of all maximal ideals in $A(X)$ containing $I$. In this paper, we extend this fact to the class of $\beta$-subalgebras which is shown to be a larger class than intermediate rings. We also study a more extended class of subrings than $\beta$-subalgebras, namely, $LBI$-subalgebras, and characterize the conditions under which an $LBI$-subalgebra is a $\beta$-subalgebra. Moreover, some known facts in the context of $C(X)$ and intermediate rings of $C(X)$ are generalized to $\beta$-subalgebras.