Intermediate rings of complex-valued continuous functions


  • Amrita Acharyya University of Toledo
  • Sudip Kumar Acharyya University of Calcutta
  • Sagarmoy Bag University of Calcutta
  • Joshua Sack California State University



z-ideals, zâ—¦-ideals, algebraically closed field, C-type rings, zero divisor graph


For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on X, let C ∗ (X, C) be its subring of bounded functions, and let Σ(X, C) denote the collection of all the rings that lie between C ∗ (X, C) and C(X, C). We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z â—¦ -ideals in the rings P(X, C) in Σ(X, C) and in their real-valued counterparts P(X, C) ∩ C(X). These correlations culminate to the fact that the structure space of any such P(X, C) is βX. For any ideal I in C(X, C), we observe that C ∗ (X, C)+I is a member of Σ(X, C), which is further isomorphic to a ring of the type C(Y, C). Incidentally these are the only C-type intermediate rings in Σ(X, C) if and only if X is pseudocompact. We show that for any maximal ideal M in C(X, C), C(X, C)/M is an algebraically closed field, which is furthermore the algebraic closure of C(X)/M ∩C(X). We give a necessary and sufficient condition for the ideal CP (X, C) of C(X, C), which consists of all those functions whose support lie on an ideal P of closed sets in X, to be a prime ideal, and we examine a few special cases thereafter. At the end of the article, we find estimates for a few standard parameters concerning the zero-divisor graphs of a P(X, C) in Σ(X, C).


Download data is not yet available.

Author Biographies

Amrita Acharyya, University of Toledo

Department of Mathematics and Statistics

Sudip Kumar Acharyya, University of Calcutta

Department of Pure Mathematics

Sagarmoy Bag, University of Calcutta

Department of Pure Mathematics

Joshua Sack, California State University

Department of Mathematics and Statistics


S. K. Acharyya, S. Bag, G. Bhunia and P. Rooj, Some new results on functions in C(X) having their support on ideals of closed sets, Quest. Math. 42 (2019), 1017-1090.

S. K. Acharyya and S. K. Ghosh, On spaces X determined by the rings Ck(X) and C∞(X), J. Pure Math. 20 (2003), 9-16.

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.

S. K. Acharyya and S. K. Ghosh, Functions in C(X) with support lying on a class of subsets of X, Topology Proc. 35 (2010), 127-148.

S. K. Acharyya and S. K. Ghosh, A note on functions in C(X) with support lying on an ideal of closed subsets of X, Topology Proc. 40 (2012), 297-301.

S. K. Acharyya, K. C. Chattopadhyay and P. Rooj, A generalized version of the rings CK(X) and C∞(X)-an enquery about when they become Noetheri, Appl. Gen. Topol. 16, no. 1 (2015), 81-87.

N. L. Alling, An application of valuation theory to rings of continuous real and complexvalued functions, Trans. Amer. Math. Soc. 109 (1963), 492-508.

F. Azarpanah, O. A. S. Karamzadeh and A. R. Aliabad, On Zâ—¦-ideal in C(X), Fundamenta Mathematicae 160 (1999), 15-25.

F. Azarpanah and M. Motamedi, Zero-divisor graph of C(X), Acta Math. Hungar. 108, no. 1-2 (2005), 25-36.

F. Azarpanah, Algebraic properties of some compact spaces. Real Anal. Exchange 25, no. 1 (1999/00), 317-327.

F. Azarpanah and T. Soundararajan, When the family of functions vanishing at infinity is an ideal of C(X), Rocky Mountain J. Math. 31, no. 4 (2001), 1133-1140.

S. Bag, S. Acharyya and D. Mandal, A class of ideals in intermediate rings of continuous functions, Appl. Gen. Topol. 20, no. 1 (2019), 109-117.

L. H. Byum and S. Watson, Prime and maximal ideals in subrings of C(X), Topology Appl. 40 (1991), 45-62.

R. E. Chandler, Hausdorff Compactifications, New York: M. Dekker, 1976.

D. De and S. K. Acharyya, Characterization of function rings between C∗(X) and C(X), Kyungpook Math. J. 46, no. 4 (2006) , 503-507.

J. M. Domínguez, J. Gómez and M.A. Mulero, Intermediate algebras between C∗ (X) and C(X) as rings of fractions of C∗ (X), Topology Appl. 77 (1997), 115-130.

L. Gillman and M. Jerison, Rings of Continuous Functions, New York: Van Nostrand Reinhold Co., 1960.

M. Mandelkar, Supports of continuous functions, Trans. Amer. Math. Soc. 156 (1971), 73-83.

W. Wm. McGovern and R. Raphael, Considering semi-clean rings of continuous functions, Topology Appl. 190 (2015), 99-108.

W. Murray, J. Sack and S. Watson, P-space and intermediate rings of continuous functions, Rocky Mountain J. Math. 47 (2017), 2757-2775.

D. Plank, On a class of subalgebras of C(X) with applications to βX X, Fund. Math. 64 (1969), 41-54.

L. Redlin and S. Watson, Maximal ideals in subalgebras of C(X), Proc. Amer. Math. Soc. 100, no. 4 (1987), 763-766.

L. Redlin and S. Watson, Structure spaces for rings of continuous functions with applications to real compactifications, Fundamenta Mathematicae 152 (1997), 151-163.




How to Cite

A. Acharyya, S. K. Acharyya, S. Bag, and J. Sack, “Intermediate rings of complex-valued continuous functions”, Appl. Gen. Topol., vol. 22, no. 1, pp. 47–65, Apr. 2021.