SOME RESULTS ON SUBRINGS OF $C(X)$ AND $C(X, \mathbb{C})$

Document Type : Original Manuscript

Authors

Department of Mathematics, North Eastern Hill University, P.O. Box 793022, Meghalaya, India.

10.22044/jas.2024.14897.1875

Abstract

Let $X$ be a Tychonoff space and $\mathcal{R}[X]$ be the collection of all subrings of $C(X)$ that separate points and contain the identity element 1. In this paper, we establish a correspondence between ideals in $A(X)$ $\in$ $\mathcal{R}[X]$ and $z^{\gamma}_A$-filters on the completion of $X$ with respect to a uniform structure arising from the functions in $A(X)$. We also explore some properties of $z$-ideals, $z_A$-ideals and maximal ideals in these types of subrings of $C(X)$. For each subset $A(X)$ of $C(X)$, let $[A(X)]_c$ $=$ $\{f + ig: f, g \in A(X)\}$. We demonstrate that $[A(X)]_c$ is a $c$-type subring of $C(X, \mathbb{C})$ when $A(X)$ $\in$ $\mathcal{R}[X]$ is a $c$-type subring of $C(X)$. Finally, for an intermediate subring $A(X)$ of $C(X)$, we show that the completion of $X$ with respect to a suitable uniform structure derived from $[A(X)]_c$ is equal to $\upsilon_AX$, the $A$-compactification of $X$.

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References

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Journal of Algebraic Systems
Volume 14, Issue 3
June 2026
Pages 455-463

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