A class of ideals in intermediate rings of continuous functions

Sagarmoy Bag, Sudip Kumar Acharyya, Dhananjoy Mandal


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).


P-space; almost P-space; UMP-space; z-ideal; z◦-ideal; ƷA-ideal

Subject classification

Primary 54C40; Secondary 46E25.

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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.

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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  
doi: 10.4995/agt.2021.13165

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