A characterization of \(c\)-type subrings of \(C(X)\) of some kind. (English) Zbl 1464.54015

Summary: A commutative ring with unity is of a \(c\)-type if it is (ring) isomorphic with \(C(X)\) for some space \(X\). In this paper, we have described a structural representation of \(c\)-type subrings of \(C(X)\) that separates points and contains the constant function 1, but not necessarily containing \(C^*(X)\). It would be complete if all \(c\)-type subring of \(C(X)\) separate points and contains 1. But, we have produced an example of \(c\)-type ring which does not separate points. We have also produced an example of a subring of \(C(\mathbb{R})\) not containing \(C^*(\mathbb{R})\) which is of \(c\)-type, \(\mathbb{R}\) real line.


54C40 Algebraic properties of function spaces in general topology
54H10 Topological representations of algebraic systems
46E25 Rings and algebras of continuous, differentiable or analytic functions
54E15 Uniform structures and generalizations
06A15 Galois correspondences, closure operators (in relation to ordered sets)
Full Text: DOI


[1] Acharyya, SK; De, D., Characterization of function rings between \(C^*(X)\) and \(C(X)\), Kyungpook Math. J., 46, 503-507 (2006) · Zbl 1120.54014
[2] Chandler, RE, Hausdorff Compactification (1976), New York: Marcel Dekker, New York
[3] Davey, BA; Priestley, HA, Introduction to Lattices and Order (2002), Cambridge: Cambridge University Press, Cambridge
[4] Dominguez, JM; Gomez, J.; Mulero, MA, Intermediate algebras between \(C^*(X)\) and \(C(X)\) as rings of fractions of \(C^*(X)\), Topol. Appl., 77, 115-130 (1997) · Zbl 0870.54017
[5] Engelking, R., General Topology (1989), Berlin: Heldermann Verlag, Berlin
[6] Gillman, L.; Jerison, M., Rings of Continuous Functions. University Series in Higher Mathematics (1960), Princeton: Van Nostrand, Princeton · Zbl 0093.30001
[7] Henriksen, M.; Johnson, DG, On the structure of a class of archimedean lattice-ordered algebras, Fund. Math., 50, 73 (1961) · Zbl 0099.10101
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