Let X be a Tychonoff space. Let and denote, respectively, the ring of all real-valued continuous functions and the ring of all bounded real-valued continuous functions on X. For a minimal ideal  of  we prove that if set of rings  such that  is finite, then so is X. Let  where 1 is the identity of  Then it is shown that the ring extension  satisfies FIP if and only if the extension  satisfies FIP if and only if X is finite. Further,  where  is the nth Bell number and n is the cardinality of X.

subring, intermediate ring, Bell numbers, completely regular space, Tychonoff space, rings and algebras of continuous functions, Banach algebras of continuous functions.

Received: November 8, 2023; Accepted: December 20, 2023; Published: January 10, 2024

How to cite this article: Abdullah Assiry, Noômen Jarboui and Mohamed Mabrouk, On the number of unital subrings of C(X), JP Journal of Algebra, Number Theory and Applications 63(1) (2024), 55-64. http://dx.doi.org/10.17654/0972555524003

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] D. D. Anderson, D. E. Dobbs and B. Mullins, The primitive element theorem for commutative algebras, Houston J. Math. 25(4) (1999), 603-623.
[2] M. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, New York, 1969.
[3] H. L. Byun and S. Watson, Prime and maximal ideals in subrings of Topology Appl. 40(1) (1991), 45-62.
[4] D. E. Dobbs, G. Picavet and M. P. L’Hermitte, Characterizing the ring extensions that satisfy FIP or FCP, J. Algebra 371 (2012), 391-429.
[5] D. E. Dobbs, P. Gabriel and M. Picavet-L’Hermitte, Transfer results for the FIP and FCP properties of ring extensions, Comm. Algebra 43 (2015), 1279-1316.
[6] E. Hewitt, Rings of real-valued continuous functions. I, Trans Amer. Math. Soc. 64 (1948), 45-99.
[7] D. Ferrand and J. P. Olivier, Homomorphismes minimaux d’anneaux, J. Algebra 16(3) (1970), 461-471.
[8] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, New York, 1978.
[9] A. Jabaallah and N. Jarboui, From topologies of a set to subrings of its power set, Bull. Austral. Math. Soc. 102(1) (2020), 15-20.