On algebraic subrings of rings of continuous functions
DOI:
https://doi.org/10.24193/subbmath.2026.3.1Keywords:
Algebraic subring, absolutely convex subring, rings of continuous functions, saturated subring, weakly algebraic subringAbstract
A subring \(S\) of \(C(X)\) is called {\it algebraic} if it satisfies the following conditions: \(\mathbb{R}\subseteq S\), and for all \(f\in C(X)\), if \(f^2\in S\), then \(f\in S\). This notion of algebraic subrings is further explored and characterized in the paper. Moreover, the paper provides examples to illustrate the results and introduces two generalizations of algebraic subrings, which are examined in detail.
References
H. Al-Ezeh, Exchange PF-rings and almost PP-rings, Internat. J. Math. Math. Sci. 12 (1989), 725-728. DOI: https://doi.org/10.1155/S016117128900089X
F. Azarpanah, When is $C(X)$ a clean ring?. Acta Math. Hungar. 94(1-2) (2002), 53-58. DOI: https://doi.org/10.1023/A:1015654520481
F. Azarpanah and E. Ghashghaei, Norm-closed and norm-reflecting ideals in rings of continuous functions, J. Algebra Appl. (2026) 2650073.
F. Azarpanah, E. Ghashghaei, and M. Ghoulipour, $C(X)$: Something old and something new, Commun. Algebra 49(2020), 185-206. DOI: https://doi.org/10.1080/00927872.2020.1797070
F. Azarpanah, O.A.S. Karamzadeh, Z. Keshtkar, and A.R. Olfati, On maximal ideals of $C_c(X)$ and the uniformity of its localizations, Rocky Mountain J. Math. 48 (2018), 345-384. DOI: https://doi.org/10.1216/RMJ-2018-48-2-345
F. Azarpanah and M. Parsinia, On the sum of $z$-ideals in subrings of $C(X)$, J. Commut. Algebra, 12(2019), 459-466. DOI: https://doi.org/10.1216/jca.2020.12.459
J. G. Brookshear, On projective prime ideals in $C(X)$, Proc. Amer. Math. Soc. 69 (1978), 203-204. DOI: https://doi.org/10.1090/S0002-9939-1978-0470929-5
G. De Marco, Projectivity of pure ideals, Rend. Sem. Mat. Univ. Padova 68 (1983), 289-304.
B. Diamond, Algebraic subrings and perfect compactifications, Topology and its Applications, 39 (1991), 217-228. DOI: https://doi.org/10.1016/0166-8641(91)90115-3
J.M. Dom'inguez, J. G'omez, and M.A. Mulero, Intermediate algebras between $C^*(X)$ and $C(X)$ as rings of fractions of $C^*(X)$,
Topology and its Applications, 77 (1997), 115-130. DOI: https://doi.org/10.1016/S0166-8641(96)00136-8
M. Ghadermazi, O.A.S. Karamzadeh, and M. Namdari, $C(X)$ versus its functionally countable subalgebra, Bull. Iran Math. Soc. 45 (2019), 173-187. DOI: https://doi.org/10.1007/s41980-018-0124-8
M. Ghadermazi, O.A.S. Karamzadeh, and M. Namdari, On the functionally countable subalgebra of $C(X)$, Rend. Semin. Mat. Univ. Padova 129 (2013), 47-69. DOI: https://doi.org/10.4171/rsmup/129-4
L. Gillman and M. Henriksen, Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc, 82 (1956), 366-391. DOI: https://doi.org/10.1090/S0002-9947-1956-0078980-4
L. Gillman and M. Jerison, Rings of Continuous Functions, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. DOI: https://doi.org/10.1007/978-1-4615-7819-2
M. Henriksen, S. Larson, J. Martinez, and R.G. Woods, Lattice-ordered algebras that are subdirect products of valuation domains, Trans. Amer. Math. Soc. 345 (1994), 195-221. DOI: https://doi.org/10.1090/S0002-9947-1994-1239640-0
H. Joris, Une $C^{infty}$-application non-immersive qui possede la propri'et'e universelle des immersions, Arch. Math. 39 (1982), 269-277. DOI: https://doi.org/10.1007/BF01899535
M.L. Knox, R. Levy, W.Wm. McGovern, and J. Shapiro, Generalizations of complemented rings with applications to rings of functions, J. Alg. Appl. 8 (2009) 17-40. DOI: https://doi.org/10.1142/S0219498809003138
T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189. Springer-Verlag, New York, 1999. DOI: https://doi.org/10.1007/978-1-4612-0525-8
W. Murray, J. Sack, and S. Watson, $P$-spaces and intermediate rings of continuous functions, Rocky Mountain J. Math. 47 (2017), 2757-2775. DOI: https://doi.org/10.1216/RMJ-2017-47-8-2757
L.D. Nel and D. Riordan, Note on a subalgebra of $C(X)$, Canadian Mathematical Bulletin 15 (1972), 607-608. DOI: https://doi.org/10.4153/CMB-1972-108-4
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Studia Universitatis Babes-Bolyai Matematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Transfer of copyright agreement: When the article is accepted for publication, the authors and the representative of the coauthors, hereby agree to transfer to Studia Universitatis Babeș-Bolyai Mathematica all rights, including those pertaining to electronic forms and transmissions, under existing copyright laws, except for the following, which the authors specifically retain: the authors can use the material however they want as long as it fits the NC ND terms of the license. The authors have all rights for reuse according to the license.
