Closure formula for ideals in intermediate rings

[EN] In this paper, we prove that the closure formula for ideals in C(X) under m topology holds in intermediate ring also, i.e. for any ideal I in an intermediate ring with m topology, its closure is the intersection of all the maximal ideals containing I.Kharbhih, JPJ.; Dutta, S. (2020). Closure formula for ideals in intermediate rings. Applied General Topology. 21(2):195-200. https://doi.org/10.4995/agt.2020.11903OJS195200212S. K. Acharyya and B. Bose, A correspondence between ideals and z-filters for certain rings of continuous functions - some remarks, Topology and its Applications 160, no. 13 (2013), 1603-1605. https://doi.org/10.1016/j.topol.2013.06.011S. K. Acharyya, K. C. Chattopadhyay and D. P. Ghosh, A class of subalgebras of C(X) and the associated compactness, Kyungpook Math. J. 41, no. 2 (2001), 323-324.S. K. Acharyya and D. De, An interesting class of ideals in subalgebras of C(X) containing C*(X), Comment. Math. Univ. Carolin. 48, no. 2 (2007), 273-280.S. K. Acharyya and D. De, Characterization of function rings between C*(X) and C(X), Kyungpook Math. J. 46 (2006), 503-507.H. L. Byun and S. Watson, Prime and maximal ideals in subrings of C(X), Topology and its Applications 40 (1991), 45-62. https://doi.org/10.1016/0166-8641(91)90057-SJ. M. Domínguez and J.-Gómez Pérez, Intersections of maximal ideals in algebras between C*(X) and C(X), Topology and its Applications 98 (1999), 149-165. https://doi.org/10.1016/S0166-8641(99)00043-7L. Gillman, M. Henriksen and M. Jerison, On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. 5 (1954), 447-455. https://doi.org/10.1090/S0002-9939-1954-0066627-6L. Gillman and M. Jerison, Rings of continuous functions, Univ. Ser. Higher Math, D. Van Nostrand Company, Inc., Princeton, N. J., 1960. https://doi.org/10.1007/978-1-4615-7819-2E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64, no. 1 (1948), 45-99. https://doi.org/10.1090/S0002-9947-1948-0026239-9D. Plank, On a class of subalgebras of C(X) with applications to $beta X setminus X$, Fund. Math. 64 (1969), 41-54. https://doi.org/10.4064/fm-64-1-41-54L. Redlin and S. Watson, Maximal ideals in subalgebras of C(X), Proc. Amer. Math. Soc. 100, no. 4 (1987), 763-766. https://doi.org/10.2307/2046719T. Shirota, On ideals in rings of continuous functions, Proc. Japan Acad. 30, no. 2 (1954), 85-89. https://doi.org/10.3792/pja/119552617