Some results on the intermediate rings of complex-valued continuous functions
Submitted: 2024-08-10
|Accepted: 2025-06-12
|Published: 2025-10-01
Copyright (c) 2025 Yangersenba Jamir, Longjaijai Kharkamni, Sanghita Dutta
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Downloads
Keywords:
ring of complex-valued continuous functions, z-ideals, absolutely convex ideals, $z^{\beta}_{[A(X)]_c}$-ideals, hull-kernel topology, structure space
Supporting agencies:
Council of Scientific and Industrial Research, India
Abstract:
Let X be a Tychonoff space and C(X), C ( X , ℂ ) be the rings of all real-valued and complex-valued continuous functions defined on X respectively. For each intermediate subring A(X) of C(X), Acharyya and De have introduced the notion of zβA -ideals in A(X) and zβA-filters on β X. For each A(X), we extend this notion to zβ[A(X)]c -ideals and zβ[A(X)]c-filters for an intermediate subring [A(X)]c of C ( X , ℂ ). We establish a correspondence between the collection of zβ[A(X)]c-ideals in the subrings [A(X)]c of C ( X , ℂ ) and the collection of zβ[A(X)]c-filters on β X. We study the properties of the zβ[A(X)]c-ideals. We also deduce that the structure space of the subrings [A(X)]c is homeomorphic to β X.
References:
A. Acharyya, S. K. Acharyya, S. Bag, and J. Sack, Intermediate rings of complex-valued continuous functions, Applied General Topology 22, no. 1 (2021), 47-65. https://doi.org/10.4995/agt.2021.13165
S. 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), 323-334.
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.
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-S
L. Gillman and M. Jerison, Rings of Continuous Functions, Univ. Ser. Higher Math, D. Van Nostrand Company, Inc., Princeton, N. J., 1960. https://doi.org/10.1007/978-1-4615-7819-2
G. Mason, z-ideals and prime ideals, Journal of Algebra 26 (1973), 280-297. https://doi.org/10.1016/0021-8693(73)90024-0
P. Panman, J. Sack, and S. Watson, Correspondences between ideals and z-filters for rings of continuous functions between C* and C, Comment. Math. 52, no. 1 (2012), 11-20.
M. Parsinia, Remarks on LBI-subalgebras of C(X), Comment. Math. Univ. Carolin. 57, no.2 (2016), 261-270. https://doi.org/10.14712/1213-7243.2015.158
M. Parsinia, On the mappings ℤ_A and _A in intermediate rings of C(X), Comment. Math. Univ. Carolin. 59, no. 3 (2018), 383-390. https://doi.org/10.14712/1213-7243.2015.249
D. Plank, On a class of subalgebras of C(X) with applications to βX X, Fund. Math. 64 (1969), 41-54. https://doi.org/10.4064/fm-64-1-41-54
L. Redlin and S. Watson, Maximal ideals in subalgebras of C(X), Proc. Amer. Math. Soc. 100, no. 4 (1987), 763-766. https://doi.org/10.1090/S0002-9939-1987-0894451-2
