Abstract
A subalgebra A(X) of C(X) is said to be a ?-subalgebra if it is closed under bounded inversion and the space of its maximal ideals equipped with the hull-kernel topology is homeomorphic to ?X with a homeomorphism which leaves X pointwise fixed. Kharbhih and Dutta in [Closure formula for ideals in intermediate rings, Appl. Gen. Topol. 21 (2) (2020), 195-200] showed that the closure of every ideal I of an intermediate ring with 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 ?-subalgebras which is shown to be a larger class than intermediate rings. We also study a more extended class of subrings than ?-subalgebras, namely, LBI-subalgebras, and characterize the conditions under which an LBI-subalgebra is a ?-subalgebra. Moreover, some known facts in the context of C(X) and intermediate rings of C(X) are generalized to ?-subalgebras.
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