Shahrood University of Technology Journal of Algebraic Systems 2345-5128 14 3 2026 06 01 SOME RESULTS ON SUBRINGS OF $C(X)$ AND $C(X, \mathbb{C})$ 455 463 3822 10.22044/jas.2024.14897.1875 EN Longjaijai Kharkamni Department of Mathematics, North Eastern Hill University, P.O. Box 793022, Meghalaya, India. Sanghita Dutta Department of Mathematics, North Eastern Hill University, P.O. Box 793022, Meghalaya, India. Journal Article 2024 08 07 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 $X$. https://jas.shahroodut.ac.ir/article_3822_f570affa91ac4410ab048aa6af0782b1.pdf