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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) |

### Keywords:

rings of continuous function; \(c\)-type rings; uniform structure; uniform completion; Galois correspondences
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\textit{B. Mitra} and \textit{D. Chowdhury}, Positivity 24, No. 5, 1181--1190 (2020; Zbl 1464.54015)

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### References:

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