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

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 ۱. 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.

کلیدواژه ها:

c-type subrings ، Stone-\mathcal{C}ech compactification ، uniform structure ، z^{\gamma}_A-filter ، z^{\gamma}_A-ideal