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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.

MSC:

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)
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References:

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