A class of ideals in intermediate rings of continuous functions
For any completely regular Hausdorff topological space X, an intermediate ring A(X) of continuous functions stands for any ring lying between C∗(X) and C(X). It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of z◦-ideals. It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X. Interrelation between z-ideals, z◦-ideal and ƷA-ideals in A(X) are examined. It is proved that within the family of almost P-spaces X, each ƷA -ideal in A(X) is a z◦-ideal if and only if each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).
S. K. Acharyya and B. Bose, A correspondence between ideals and z-filters for certain rings of continuous functions-some remarks, Topology Appl. 160 (2013), 1603–1605. https://doi.org/10.1016/j.topol.2013.06.011
H. Azadi,M. Henriksen and E. Momtahan, Some properties of algebra of real valued measurable functions, Acta. Math. Hunger 124 (2009), 15–23. https://doi.org/10.1007/s10474-009-8138-6
F. Azarpanah, O.A.S. Karamzadeh and R. A. Aliabad, On z◦-ideals of C(X), Fund.Math. 160 (1999), 15–25.
F. Azarpanah, O. A. S. Karamzadeh and A. Rezai Aliabad, Onideals consisting entirely of zero Divisors, Communications in Algebra 28 (2000), 1061–1073. https://doi.org/10.1080/00927870008826878
S. Bag, S. K. Acharyya and D. Mandal, z◦-ideals in intermediate rings of ordered field valued continuous functions, communicated.
B. Banerjee, S. K. Ghosh and M. Henriksen, Unions of minimal prime ideals in rings of continuous functions on a compact spaces, Algebra Universalis 62 (2009), 239–246. https://doi.org/10.1007/s00012-010-0051-x
L. H. Byun and S. Watson, Prime and maximals ideal in subrings of C(X) , Topology Appl. 40 (1991), 45–62. https://doi.org/10.1016/0166-8641(91)90057-S
L. Gillman and M. Jerison, Rings of continuous functions, New York: Van Nostrand Reinhold Co., 1960. https://doi.org/10.1007/978-1-4615-7819-2
L. Gilmann and M. Henriksen, Concerning rings of continuous functions, Trans. Amer. Math. Soc. 77 (1954), 340–362. https://doi.org/10.1090/s0002-9947-1954-0063646-5
M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110–130. https://doi.org/10.1090/s0002-9947-1965-0194880-9
W. Murray, J. Sack, S. Watson, P-space and intermediate rings of continuous functions, Rocky Mountain J. Math. 47 (2017), 2757–2775. https://doi.org/10.1216/rmj-2017-47-8-2757
J. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. 13(1963), 31–50. https://doi.org/10.1112/plms/s3-13.1.31
R. Levy, Almost p-spaces, Canad. J. Math. 29 (1977) 284–288. https://doi.org/10.4153/CJM-1977-030-7
G. Mason, Prime ideals and quotient rings of reduced rings, Math. Japon 34 (1989),941–956.
P. Panman, J. Sack and S. Watson, Correspondences between ideals and z-filters for rings of continuous functions between C∗ and C, Commentationes Mathematicae 52,no. 1, (2012) 11–20.
J. Sack and S. Watson, C and C∗ among intermediate rings, Topology Proceedings 43(2014), 69–82.
J. Sack and S. Watson, Characterizing C(X) among intermediate C-rings on X, Topology Proceedings 45 (2015), 301–313.
Metrics powered by PLOS ALM
Cited-By (articles included in Crossref)
This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site
1. Intermediate rings of complex-valued continuous functions
Amrita Acharyya, Sudip Kumar Acharyya, Sagarmoy Bag, Joshua Sack
Applied General Topology vol: 22 issue: 1 first page: 47 year: 2021
Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
Universitat Politècnica de València
e-ISSN: 1989-4147 https://doi.org/10.4995/agt