JamirKharkamniDutta

Abstract.

Let \(X\) be a Tychonoff space and \(C{(X)}\), \(C{(X,{\mathbb{C}})}\) be the rings of all real-valued and complex-valued continuous functions defined on \(X\) respectively. For each intermediate subring \(A{(X)}\) of \(C{(X)}\), Acharyya and De have introduced the notion of \(z_{A}^{\beta}\)-ideals in \(A{(X)}\) and \(z_{A}^{\beta}\)-filters on \(\beta X\). For each \(A{(X)}\), we extend this notion to \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals and \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filters for an intermediate subring \({\lbrack{A{(X)}}\rbrack}_{c}\) of \(C{(X,{\mathbb{C}})}\), where \({\lbrack{A{(X)}}\rbrack}_{c} = {\{{f + {ig}}:{{f,g} \in {A{(X)}}}\}}\). We establish a correspondence between the collection of \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals in the subrings \({\lbrack{A{(X)}}\rbrack}_{c}\) of \(C{(X,{\mathbb{C}})}\) and the collection of \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filters on \(\beta X\). We study the properties of the \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals. We also deduce that the structure space of the subrings \({\lbrack{A{(X)}}\rbrack}_{c}\) is homeomorphic to \(\beta X\).

keywords:

ring of complex-valued continuous functions; \(z -\)ideals; absolutely convex ideals; \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals; hull-kernel topology; structure space.

MSC:

54C40; 46E25.

1. Introduction

Throughout this paper, we take \(X\) to be a Tychonoff space. Let \(C{(X,{\mathbb{C}})}\), \(C^{\ast}{(X,{\mathbb{C}})}\) denote the rings of all complex-valued and bounded complex-valued continuous functions defined on \(X\) respectively, while \(C{(X)}\), \(C^{\ast}{(X)}\) denote the rings of all real-valued and bounded real-valued continuous functions defined on \(X\) respectively. An intermediate ring of complex-valued continuous functions is a subring of \(C{(X,{\mathbb{C}})}\) which contains \(C^{\ast}{(X,{\mathbb{C}})}\) and we denote the collection of all these subrings by \(\sum{(X,{\mathbb{C}})}\). Similarly, an intermediate ring of real-valued continuous functions is a subring of \(C{(X)}\) which contains \(C^{\ast}{(X)}\) and we denote the collection of all these subrings by \(\sum{(X)}\). Let \(\beta X\) denote the Stone-Čech compactificaton of \(X\). In this paper, we consider an ideal to be a proper ideal.
An elegant bridge between a topological space \(X\) and the algebraic properties of its associated ring \(C{(X)}\) is the correspondence between the ideals in \(C{(X)}\) and the \(z\)-filters on \(X\) which is well documented by Gillman and Jerison in [5]. Mason [6] introduced the concept of \(z\)-ideals to study the ideal structure of the ring \(C{(X)}\). There is a bijective correspondence between the \(z\)-ideals in \(C{(X)}\) and \(z\)-filters on \(X\). Redlin and Watson [11] extended the correspondence between ideals and \(z\)-filters to intermediate rings \(A{(X)}\) of real-valued continuous functions by defining \(\mathcal{Z}_{A}{(f)}\) = \(\{ E \in Z{(X)}:\exists g \in A{(X)}\) such that \({(fg)}|_{E} = 1\}\) where \(f \in {A{(X)}}\). For any ideal \(I\) in \(A{(X)}\), \(\mathcal{Z}_{A}{(I)}\) = \(\{{\mathcal{Z}_{A}{(f)}}:{f \in I}\}\) forms a \(z\)-filter on \(X\). We refer the reader to [4], [7] and [9] for more on this correspondence. Acharyya et al. [3], introduced a duality between special types of ideals called \(z_{A}^{\beta}\)-ideals in intermediate rings \(A{(X)}\) of real-valued continuous functions and special types of \(z\)-filters on \(\beta X\) called \(z_{A}^{\beta}\)-filters.

The structure space of a commutative ring \(R\) with identity is the set \(\mathcal{M}{(R)}\) of all maximal ideals in \(R\) equipped with the hull-kernel topology. It is known that the structure space of \(C{(X)}\) is homeomorphic to \(\beta X\) [5]. Plank [10] and Redlin et al. [11] independently proved that the structure space of an intermediate ring of real-valued continuous functions is homeomorphic to the space \(\beta X\). For any subring \(A{(X)}\) of \(C{(X)}\), let \({\lbrack{A{(X)}}\rbrack}_{c}\) = \(\{{f' + {if^{\operatorname{\prime\prime}}}}:{{f',f^{\operatorname{\prime\prime}}} \in {A{(X)}}}\}\). Clearly, \({\lbrack{C{(X)}}\rbrack}_{c}\) = \(C{(X,{\mathbb{C}})}\) and \({\lbrack{C^{\ast}{(X)}}\rbrack}_{c}\) = \(C^{\ast}{(X,{\mathbb{C}})}\). If \(A{(X)}\) \(\in\) \(\sum{(X)}\), then \({\lbrack{A{(X)}}\rbrack}_{c}\) \(\in\) \(\sum{(X,{\mathbb{C}})}\), in fact, \({\lbrack{A{(X)}}\rbrack}_{c}\) is the smallest such ring containing \(A{(X)}\) and the constant function \(i\). Acharyya et al. [1] showed that the structure space of each \({\lbrack{A{(X)}}\rbrack}_{c}\) is homeomorphic to \(\beta X\) .
For the ring \({\lbrack{A{(X)}}\rbrack}_{c}\), where \({A{(X)}} \in {\sum{(X)}}\), we establish a correspondence between a special collection of ideals in \({\lbrack{A{(X)}}\rbrack}_{c}\) called \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals and a special collection of \(z -\)filters on \(\beta X\) called \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filters in Section 2 of the paper. In Section 3, we examine the properties of these \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals in \({\lbrack{A{(X)}}\rbrack}_{c}\), where \({A{(X)}} \in {\sum{(X)}}\). Section 4 is devoted to the study of the structure spaces of intermediate rings \({\lbrack{A{(X)}}\rbrack}_{c}\), where \({A{(X)}} \in {\sum{(X)}}\), and we show that they are homeomorphic to \(\beta X\). This is achieved by extending the known correspondence between maximal ideals in \(A{(X)}\) and \(z\)-ultrafilters on \(X\).

2. Ideals of \({\lbrack{A{(X)}}\rbrack}_{c}\) and \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filters on \(\beta X\)

For each \(f\) \(\in\) \(C{(X)}\), there exists a unique continuous extension \(f^{\ast}\) from \(\beta X\) to \({\mathbb{R}}^{\ast}\), the one-point compactification of \(\mathbb{R}\) [5]. For any subring \(A{(X)}\) of \(C{(X)}\) and \(p\) \(\in\) \(\beta X\), Plank [10], defined \(M_{A}^{p}\) = \(\{ f \in A{(X)}:{(fg)}^{\ast}{(p)} = 0\) for all \(g \in A{(X)}\}\). For \(p\) \(\in\) \(\beta X\), \(M_{A}^{p}\) is shown to be a prime ideal and in particular, when \(A{(X)}\) is an \(LBI\)-subalgebra of \(C{(X)}\), \(M_{A}^{p}\) is maximal in \(A{(X)}\) (\(A{(X)}\) is called closed under local bounded inversion, briefly, \(LBI\)-subalgebra, if whenever \(f\) \(\in\) \(A{(X)}\) is bounded away from zero on some cozero-set \(E\), then there exists \(g\) \(\in\) \(A{(X)}\) such that \({{fg}|}_{E}\) = 1). In fact, for these subalgebras, the collection of maximal ideals in \(A{(X)}\) is precisely \(\mathcal{M}{(A)}\) = \(\{ M_{A}^{p}:{p \in {\beta X}}\}\) [8]. For each \(f \in {A{(X)}}\) of \(C{(X)}\), Plank [10] defined the set \({S_{A}{(f)}} = {\{{p \in {\beta X}}:{{{({fg})}^{\ast}{(p)}} = {0{\forall g}} \in {A{(X)}}}\}}\). From the definition of \(M_{A}^{p}\), we can clearly see that \(S_{A}{(f)}\) = \(\{{p \in {\beta X}}:{f \in M_{A}^{p}}\}\).
From now on, \(A{(X)}\) or \(B{(X)}\) will mean intermediate rings of real-valued continuous functions. We set \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(\{{p \in {\beta X}}:{f \in {\lbrack M_{A}^{p}\rbrack}_{c}}\}\). In view of [1, Theorem 2.6] and using the fact that \(M_{A}^{p}\) is absolutely convex (in fact it is maximal), we see that \({\lbrack M_{A}^{p}\rbrack}_{c}\) = \(\{{h \in {\lbrack{A{(X)}}\rbrack}_{c}}:{{|h|} \in M_{A}^{p}}\}\). Therefore, if \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\), then \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \({\{{p \in {\beta X}}:{{|f|} \in M_{A}^{p}}\}} = {S_{A}{({|f|})}}\). Since \(M_{A}^{p}\) is absolutely convex, it is easy to see that \({S_{A}{(f)}} = {S_{A}{({|f|})}}\) for all \(f \in {A{(X)}}\).
For \(f\), \(g\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\), \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) \(\cup\) \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(g)}\) = \(S_{A}{({|f|})}\) \(\cup\) \(S_{A}{({|g|})}\) = \(S_{A}{({{|f|}{|g|}})}\) = \(S_{A}{({|{fg}|})}\) = \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{({fg})}\). Also, \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) \(\cap\) \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(g)}\) = \(S_{A}{({|f|})}\) \(\cap\) \(S_{A}{({|g|})}\) = \(S_{A}{({{|f|}^{2} + {|g|}^{2}})}\). If we let \(h\) = \(({{|f|}^{2} + {|g|}^{2}})\) + \(i0\), then \(h\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\) and \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) \(\cap\) \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(g)}\) = \(S_{A}{({|h|})}\) = \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(h)}\).
We know that the collection of all maximal ideals in \(A{(X)}\) is \(\mathcal{M}{({A{(X)}})}\) = \(\{ M_{A}^{p}:{p \in {\beta X}}\}\). Therefore, by [1, Theorem 2.13], the collection of all maximal ideals in \({\lbrack{A{(X)}}\rbrack}_{c}\) is \(\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}\) = \(\{{\lbrack M_{A}^{p}\rbrack}_{c}:{p \in {\beta X}}\}\). Thus, we have the following Lemma.

Lemma 2.1.

Let \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\). Then \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(\phi\) if and only if \(f\) is invertible in \({\lbrack{A{(X)}}\rbrack}_{c}\).

Proof 2.2.

Suppose \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(\phi\). Then by definition of \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\), we have \(f\) \(\notin\) \({\lbrack M_{A}^{p}\rbrack}_{c}\) for all \(p\) \(\in\) \(\beta X\). So \(f\) does not belong to any maximal ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\). Therefore, \(f\) is invertible in \({\lbrack{A{(X)}}\rbrack}_{c}\). Conversely, suppose \(f\) is invertible in \({\lbrack{A{(X)}}\rbrack}_{c}\). Then for each \(p\) \(\in\) \(\beta X\), \(f\) \(\notin\) \({\lbrack M_{A}^{p}\rbrack}_{c}\). Hence, \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(\phi\).

For any subset \(I\) of \({\lbrack{A{(X)}}\rbrack}_{c}\), we shall denote the set \(\{{S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}}:{f \in I}\}\) by \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I\rbrack}\) and we denote the set \(\{{S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}}:{f \in {\lbrack{A{(X)}}\rbrack}_{c}}\}\) by \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack X\rbrack}\).

Definition 2.3.

Let \(\mathcal{F}\) be a non-empty subset of \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack X\rbrack}\). We say that \(\mathcal{F}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filter on \(\beta X\) if the following conditions hold.

(1)

\(\phi\) \(\notin\) \(\mathcal{F}\),
(2)

if \(S_{1}\) and \(S_{2}\) \(\in\) \(\mathcal{F}\), then \(S_{1}\) \(\cap\) \(S_{2}\) \(\in\) \(\mathcal{F}\),
(3)

if \(S_{1}\) \(\in\) \(\mathcal{F}\) and \(S_{2}\) \(\in\) \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack X\rbrack}\) such that \(S_{1}\) \(\subseteq\) \(S_{2}\), then \(S_{2}\) \(\in\) \(\mathcal{F}\).

Because of condition (c), condition (b) is equivalent to “if \(S_{1}\) and \(S_{2}\) \(\in\) \(\mathcal{F}\), then \(S_{1}\) \(\cap\) \(S_{2}\) contains a member of \(\mathcal{F}\)”.

Theorem 2.4.

The following statements easily follow from Lemma 2.1.

(1)

Let \(I\) be an ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\). Then \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I\rbrack}\) = \(\{{S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}}:{f \in I}\}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filter on \(\beta X\).
(2)

Let \(\mathcal{F}\) be a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filter on \(\beta X\). Then \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta\leftarrow}{\lbrack\mathcal{F}\rbrack}\) = \(\{{f \in {\lbrack{A{(X)}}\rbrack}_{c}}:{{S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}} \in \mathcal{F}}\}\) is an ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\).

Definition 2.5.

A \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filter on \(\beta X\) is said to be a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilter on \(\beta X\) if it is not contained in any other \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filter on \(\beta X\).

We have the following theorem which is a complex analogue of [3, Theorem 2.5].

Theorem 2.6.

For any \({\lbrack{A{(X)}}\rbrack}_{c}\), the following are equivalent.

(1)

Every \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filter on \(\beta X\) can be extended to a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilter on \(\beta X\).
(2)

Every subfamily of \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack X\rbrack}\) with finite intersection property can be extended to a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilter on \(\beta X\) and therefore a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilter on \(\beta X\) is a subfamily of \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack X\rbrack}\) which is maximal with respect to having the finite intersection property. Conversely, a subfamily \(\mathcal{F}\) of \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack X\rbrack}\) having the finite intersection property and which is maximal with respect to this property is necessarily a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilter on \(\beta X\).
(3)

A \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-filter \(\mathcal{F}\) on \(\beta X\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilter on \(\beta X\) if and only if for any \(S\) \(\in\) \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack X\rbrack}\) such that \(S\) \(\cap\) \(S' \neq \phi\) for any \(S'\) \(\in\) \(\mathcal{F}\), then \(S\) \(\in\) \(\mathcal{F}\).

We already have the duality between maximal ideals in \(A{(X)}\) and \(z_{A}^{\beta}\)-ultrafilters on \(\beta X\) as given in [3, Theorem 2.6]. The analogous duality between maximal ideals in \({\lbrack{A{(X)}}\rbrack}_{c}\) and \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilters on \(\beta X\) is stated in the following theorem.

Theorem 2.7.

For any \({\lbrack{A{(X)}}\rbrack}_{c}\), we have

(1)

if \(M\) is a maximal ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\), then \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack M\rbrack}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilter on \(\beta X\) and
(2)

if \(\mathcal{F}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ultrafilter on \(\beta X\), then \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta\leftarrow}{\lbrack\mathcal{F}\rbrack}\) is a maximal ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\).

Next we state the following theorem which is a complex analogue of [3, Theorem 2.7].

Theorem 2.8.

Let \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\). If \(M\) is a maximal ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\) and \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) meets every member of \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack M\rbrack}\), then \(f\) \(\in\) \(M\).

3. \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals and \(z_{A}^{\beta}\)-ideals

For any ideal \(I\) in \({\lbrack{A{(X)}}\rbrack}_{c}\), it is easy to see that \(I\) \(\subseteq\) \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta\leftarrow}{\lbrack{Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I\rbrack}}\rbrack}\). An ideal \(I\) in \({\lbrack{A{(X)}}\rbrack}_{c}\) is called a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideal if \(I\) = \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta\leftarrow}{\lbrack{Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I\rbrack}}\rbrack}\). There are examples of ideals in \({\lbrack{A{(X)}}\rbrack}_{c}\) which are not \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals. For instance, if we take \(I\) to be the ideal in \(C{({\mathbb{R}})}\) generated by the identity function \(\mathbf{i}:{{\mathbb{R}}\rightarrow{\mathbb{R}}}\), then the ideal \(I_{c}\) in \(\lbrack C{({\mathbb{R}}\rbrack}_{c}\) is not a \(z_{\lbrack C{({\mathbb{R}}\rbrack}_{c}}^{\beta}\)-ideal as \(S_{{\lbrack{C{({\mathbb{R}})}}\rbrack}_{c}}{(\mathbf{i}^{\mathbf{1}/\mathbf{3}})}\) = \(S_{C{({\mathbb{R}})}}{({|\mathbf{i}^{\mathbf{1}/\mathbf{3}}|})}\) = \(S_{C{({\mathbb{R}})}}{(\mathbf{i}^{\mathbf{1}/\mathbf{3}})}\) since \(\mathbf{i}^{\mathbf{1}/\mathbf{3}}\) \(\in\) \(C{({\mathbb{R}})}\). But \(S_{C{({\mathbb{R}})}}{(\mathbf{i}^{\mathbf{1}/\mathbf{3}})}\) = \(cl_{\beta{\mathbb{R}}}Z{(\mathbf{i}^{\mathbf{1}/\mathbf{3}})}\). Also, clearly \(cl_{\beta{\mathbb{R}}}Z{(\mathbf{i}^{\mathbf{1}/\mathbf{3}})}\) = \(cl_{\beta{\mathbb{R}}}Z{(\mathbf{i})}\). Thus, \(S_{{\lbrack{C{({\mathbb{R}})}}\rbrack}_{c}}{(\mathbf{i}^{\mathbf{1}/\mathbf{3}})}\) = \(cl_{\beta{\mathbb{R}}}Z{(\mathbf{i})}\) = \(S_{C{({\mathbb{R}})}}{(\mathbf{i})}\). Therefore, \(S_{{\lbrack{C{({\mathbb{R}})}}\rbrack}_{c}}{(\mathbf{i}^{\mathbf{1}/\mathbf{3}})}\) \(\in\) \(Z_{{\lbrack{C{({\mathbb{R}})}}\rbrack}_{c}}^{\beta\leftarrow}{\lbrack{Z_{{\lbrack{C{({\mathbb{R}})}}\rbrack}_{c}}^{\beta}{\lbrack I\rbrack}}\rbrack}\), but clearly \(\mathbf{i}^{\mathbf{1}/\mathbf{3}}\) \(\notin\) \(I_{c}\).

Remark 3.1.

In [3, Theorem 3.8], the authors have proved that in any \(A{(X)}\), the \(z_{A}^{\beta}\)-ideals are precisely the same as \(z\)-ideals. Also by [6, Lemma 1.0], for any commutative ring \(R\) with identity, if \(I\) is a \(z\)-ideal, then \(I\) is the intersection of the minimal prime ideals in \(R\) containing \(I\).

Lemma 3.2.

For any ideal \(I\) in \(A{(X)}\), we have that \(Z_{A}^{\beta}{\lbrack I\rbrack}\) \(\subseteq\) \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I_{c}\rbrack}\).

Proof 3.3.

Let \(S_{A}{(g)}\) \(\in\) \(Z_{A}^{\beta}{\lbrack I\rbrack}\). Then \(S_{A}{(g)}\) = \(S_{A}{(h)}\) for some \(h\) \(\in\) \(I\). Since \(h\) \(\in\) \(I\) and \(I\) \(\subseteq\) \(A{(X)}\), \(h\) \(\in\) \(I_{c}\). Also, \(S_{A}{(h)}\) = \(S_{A}{({|h|})}\) = \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(h)}\) and \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(h)}\) \(\in\) \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I_{c}\rbrack}\). Thus, \(S_{A}{(g)}\) is also a member of \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I_{c}\rbrack}\).

Theorem 3.4.

Let \(I\) be an ideal in \(A{(X)}\). Then \(I\) is a \(z_{A}^{\beta}\)-ideal in \(A{(X)}\) if and only if \(I_{c}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\).

Proof 3.5.

Suppose \(I\) is a \(z_{A}^{\beta}\)-ideal in \(A{(X)}\). Let \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\) such that \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) \(\in\) \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I_{c}\rbrack}\). Then \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(g)}\), for some \(g\) \(\in\) \(I_{c}\). By [1, Theorem 2.6], \(|g|\) \(\in\) \(I\). Therefore, \(S_{A}{({|f|})}\) = \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(g)}\) = \(S_{A}{({|g|})}\) \(\in\) \(Z_{A}^{\beta}{\lbrack I\rbrack}\). Since \(I\) is a \(z_{A}^{\beta}\)-ideal in \(A{(X)}\), \(|f|\) \(\in\) \(I\). From Remark 3.1, it follows that \(I\) is absolutely convex, hence by [1, Theorem 2.6], \(f\) \(\in\) \(I_{c}\), showing that \(I_{c}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\).
Conversely, suppose \(I_{c}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\). Let \(f\) \(\in\) \(A{(X)}\) such that \(S_{A}{(f)}\) \(\in\) \(Z_{A}^{\beta}{\lbrack I\rbrack}\). By Lemma 3.2 and using the fact that \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(S_{A}{({|f|})}\) = \(S_{A}{(f)}\), we have \(S_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) \(\in\) \(Z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}{\lbrack I_{c}\rbrack}\). Since \(I_{c}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\), we get \(f\) \(\in\) \(I_{c}\). Therefore, \(f\) = \(f'\) + \(if^{\operatorname{\prime\prime}}\), where \(f'\) and \(f^{\operatorname{\prime\prime}}\) \(\in\) \(I\). But since \(f\) \(\in\) \(A{(X)}\), we must have \(f^{\operatorname{\prime\prime}}\) = 0. Hence, \(f\) = \(f'\) \(\in\) \(I\), showing that \(I\) is a \(z_{A}^{\beta}\)-ideal in \(A{(X)}\).

Theorem 3.6.

Let \(I\) be an ideal in \(A{(X)}\). Then \(I_{c}\) is a \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\) if and only if given any \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\), there exists \(g\) \(\in\) \(I_{c}\) such that whenever \(f\) belongs to every maximal ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\) containing \(g\), then \(f\) \(\in\) \(I_{c}\).

Thus the \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals in the ring \({\lbrack{A{(X)}}\rbrack}_{c}\) which are of the form \(I_{c}\) for some ideal \(I\) in \(A{(X)}\) are essentially the same as \(z\)-ideals in \({\lbrack{A{(X)}}\rbrack}_{c}\). Consequently, since maximal and minimal prime ideals in a commutative ring are \(z\)-ideals, by [1, Remark 2.12, Theorem 2.13] we see that maximal and minimal prime ideals in \({\lbrack{A{(X)}}\rbrack}_{c}\) are \(z_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\beta}\)-ideals. Moreover, if \(I\) is a \(z_{A}^{\beta}\)-ideal in \(A{(X)}\), then together with [6, Lemma1.0], \(I_{c}\) is equal to the intersection of all the minimal prime ideals in \({\lbrack{A{(X)}}\rbrack}_{c}\) containing it.

4. On the mapping \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}\) and the structure space of \({\lbrack{A{(X)}}\rbrack}_{c}\)

In this section, we show that \(\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}\) equipped with the hull-kernel topology is homeomorphic to the Stone-\(\mathcal{C}\)ech compactification \(\beta X\). As in [11], we associate each element \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\) with a \(z\)-filter \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) on \(X\) and we extend this correspondence to one between \(\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}\) and \(z\)-ultrafilters on \(X\).

Definition 4.1 ([11]).

For any cozero-set \(E\) in \(X\), \(f\) \(\in\) \(A{(X)}\) is said to be \(E\)-regular if there exists \(g\) \(\in\) \(A{(X)}\) such that \({{fg}|}_{E} = 1\).

Let \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\). We define \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(\{ E \in Z{\lbrack X\rbrack}:|f|\) is \(E^{c}\)-regular\(\}\). It is clear that \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(\mathcal{Z}_{A}{({|f|})}\).

Lemma 4.2.

A function \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\) is invertible in \({\lbrack{A{(X)}}\rbrack}_{c}\) if and only if \(|f|\) is invertible in \(A{(X)}\).

Proof 4.3.

By [1, Theorem 2.3], we have \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\) if and only if \(|f|\) \(\in\) \(A{(X)}\). Also, for any \(p\) in \(\beta X\), \(f\) \(\in\) \({\lbrack M_{A}^{p}\rbrack}_{c}\) if and only if \(|f|\) \(\in\) \(M_{A}^{p}\). Thus, the result follows from the fact that \(\mathcal{M}{({A{(X)}})}\) = \(\{ M_{A}^{p}:{p \in {\beta X}}\}\) and \(\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}\) = \(\{{\lbrack M_{A}^{p}\rbrack}_{c}:{p \in {\beta X}}\}\).

Theorem 4.4.

Suppose \(f\) is not invertible in \({\lbrack{A{(X)}}\rbrack}_{c}\), then \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) is a \(z\)-filter on \(X\) and the converse also holds.

Proof 4.5.

Let \(f\) be not invertible in \({\lbrack{A{(X)}}\rbrack}_{c}\). Then

(1)

\(\phi\) \(\notin\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\). For if \(\phi\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(\mathcal{Z}_{A}{({|f|})}\), then \(|f|\) would be invertible in \(A{(X)}\), which is a contradiction by Lemma 4.2.
(2)

Let \(E\), \(F\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\). Then by [11, Lemma 1(b)], \(E \cap F\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\).
(3)

Let \(E\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\) and \(F\) be a zero-set in \(X\) containing \(E\). Then, by [11, Lemma 1(a)], \(F\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\).

Thus, \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) is a \(z\)-filter on \(X\). The converse is trivial.

Theorem 4.6.

Let \(I\) be a prime ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\). Then \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\) = \(\bigcup_{f \in I}{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}}\) is a \(z\)-filter on \(X\).

Proof 4.7.

Since \(I\) is a prime ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\), then \(I\) = \(Q_{c}\) for some prime ideal \(Q\) in \({\lbrack{A{(X)}}\rbrack}_{c}\) \(\cap\) \(C{(X)}\) = \(A{(X)}\).

(1)

Since \(I\) is a proper ideal, clearly \(\phi\) \(\notin\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\).
(2)

Let \(E\), \(F\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\). So, \(E\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) and \(F\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(g)}\), for some \(f\) and \(g\) in \(I\). Then by [11, Lemma 1(e)], \({|f|}^{2} + {|g|}^{2}\) is \({({E \cap F})}^{c}\)-regular. Also, since \(I\) is absolutely convex, by [1, Theorem 2.6], we have \({|f|}^{2} + {|g|}^{2}\) \(\in\) \(Q\) \(\subseteq\) \(Q_{c}\) = \(I\). Therefore, \(E \cap F\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\).
(3)

Let \(E\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\) and \(F\) be a zero-set in \(X\) containing \(E\). Then \(E\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) for some \(f\) \(\in\) \(I\). Hence, by Theorem 4.4, \(F\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\). Thus, \(F\) \(\in\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\).

A topological description on \(\mathcal{Z}_{A}\) given by Parsinia in [9] for any \(f\) \(\in\) \(A{(X)}\) is \(\mathcal{Z}_{A}{(f)}\) = \(\{{E \in {Z{\lbrack X\rbrack}}}:{{S_{A}{(f)}} \subseteq {int_{\beta X}cl_{\beta X}E}}\}\). So, for any \(f\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\), we have \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\) = \(\mathcal{Z}_{A}{({|f|})}\) = \(\{{E \in {Z{\lbrack X\rbrack}}}:{{S_{A}{({|f|})}} \subseteq {int_{\beta X}cl_{\beta X}E}}\}\).

Lemma 4.8.

If \(f\), \(g\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\) be such that \(|f|\) \(\leq\) \(|g|\), then \(S_{A}{({|g|})}\) \(\subseteq\) \(S_{A}{({|f|})}\). Hence, if \(|f|\) \(\leq\) \(|g|\), then \(\mathcal{Z}_{A}{({|f|})}\) \(\subseteq\) \(\mathcal{Z}_{A}{({|g|})}\).

Proof 4.9.

Let \(p\) \(\in\) \(S_{A}{({|g|})}\). Then \(|g|\) \(\in\) \(M_{A}^{p}\). Again since \(|f|\) \(\leq\) \(|g|\), \(|f|\) \(\in\) \(M_{A}^{p}\). Thus, \(p\) \(\in\) \(S_{A}{({|f|})}\) and hence \(S_{A}{({|g|})}\) \(\subseteq\) \(S_{A}{({|f|})}\). So, it follows that if \(|f|\) \(\leq\) \(|g|\), then \(\mathcal{Z}_{A}{({|f|})}\) \(\subseteq\) \(\mathcal{Z}_{A}{({|g|})}\).

Theorem 4.10.

For any \(z\)-filter \(\mathcal{F}\) on \(X\), \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}\) = \(\{{f \in {\lbrack{A{(X)}}\rbrack}_{c}}:{{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}} \subseteq \mathcal{F}}\}\) is an ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\).

Proof 4.11.

Let \(I\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}\). Let \(f\) \(\in\) \(I\) and \(g\) \(\in\) \({\lbrack{A{(X)}}\rbrack}_{c}\). Since \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{({fg})}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}\), we have \(fg\) \(\in\) \(I\).
If \(f\), \(g\) \(\in\) \(I\), then \(|f|\), \(|g|\) \(\in\) \(A{(X)}\) and using [11, Lemma 2], we have \(lim_{\mathcal{Z}_{A}{({|f|})}}{|f|}h\) = \(lim_{\mathcal{Z}_{A}{({|g|})}}{|g|}h\) = 0 for every \(h\) \(\in\) \(A{(X)}\). But since \(\mathcal{Z}_{A}{({|f|})}\) \(\subseteq\) \(\mathcal{F}\) and \(\mathcal{Z}_{A}{({|g|})}\) \(\subseteq\) \(\mathcal{F}\), we have \(lim_{\mathcal{F}}{|f|}h\) = \(lim_{\mathcal{F}}{|g|}h\) = 0 for every \(h\) \(\in\) \(A{(X)}\). Therefore, \(lim_{\mathcal{F}}{|f|}h\) + \(lim_{\mathcal{F}}{|g|}h\) = \(lim_{\mathcal{F}}{({{|f|} + {|g|}})}h\) = 0 for every \(h\) \(\in\) \(A{(X)}\). Thus, by [11, Lemma 3], \(\mathcal{Z}_{A}{({{|f|} + {|g|}})}\) \(\subseteq\) \(\mathcal{F}\). Again, since \(|{f + g}|\) \(\leq\) \({|f|} + {|g|}\), by Lemma 4.8 we have \(\mathcal{Z}_{A}{({|{f + g}|})}\) \(\subseteq\) \(\mathcal{Z}_{A}{({{|f|} + {|g|}})}\) and hence \(f + g\) \(\in\) \(I\). It is easy to see that \(I\) contains no invertible element of \({\lbrack{A{(X)}}\rbrack}_{c}\). Therefore, \(I\) is an ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\).

Lemma 4.12.

Let \(I\) be an ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\) and \(\mathcal{F}\) be a \(z\)-filter on \(X\). Then

(1)

\(I\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}}\rbrack}\) and the equality holds when \(I\) is maximal,
(2)

\(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}}\rbrack}}\rbrack}\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\),
(3)

\(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}}\rbrack}\) \(\subseteq\) \(\mathcal{F}\),
(4)

\(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}}\rbrack}}\rbrack}\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}\).

Proof 4.13.

The proof clearly follows from the definition of \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}\) and \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}\).

Lemma 4.14.

Let \(B{(X)}\) \(\subseteq\) \(A{(X)}\). Then for any absolutely convex ideal \(I\) of \({\lbrack{A{(X)}}\rbrack}_{c}\), \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}\) = \(\mathcal{Z}_{{\lbrack{B{(X)}}\rbrack}_{c}}{\lbrack{I \cap {\lbrack{B{(X)}}\rbrack}_{c}}\rbrack}\).

Proof 4.15.

It is sufficient to show that \({\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}} \subseteq {\mathcal{Z}_{{\lbrack{C^{\ast}{(X)}}\rbrack}_{c}}{\lbrack{I \cap {\lbrack{C^{\ast}{(X)}}\rbrack}_{c}}\rbrack}}\). Let \(E \in {\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack I\rbrack}}\). Then there exists \(f \in I\) and \(g \in {\lbrack{A{(X)}}\rbrack}_{c}\) such that \({(|f|.g)}|_{({X\backslash E})} = 1\). Let \(h = \frac{{2{|f|}}.g}{1 + |f.g|}\). Then \(h \in {I \cap {\lbrack{C^{\ast}{(X)}}\rbrack}_{c}}\) and \({h|}_{({X\backslash E})} = 1\). This shows that \(E \in {\mathcal{Z}_{{\lbrack{C^{\ast}{(X)}}\rbrack}_{c}}{\lbrack{I \cap {\lbrack{C^{\ast}{(X)}}\rbrack}_{c}}\rbrack}}\).

Theorem 4.16.

For any maximal ideal \(M\) in \(A{(X)}\), \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}\) is contained in a unique \(z\)-ultrafilter on \(X\).

Proof 4.17.

Given that \(M\) is a maximal ideal in \(A{(X)}\). Let \(\mathcal{F}\) be a \(z\)-ultrafilter on \(X\) containing \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}\). Since the mapping \(Z\) defined by \(Z{\lbrack I\rbrack}\) = \(\{{Z{(f)}}:{f \in I}\}\), where \(I\) is an ideal in \(C{(X)}\), is a bijection from the set of all maximal ideals in \(C{(X)}\) and \(z\)-ultrafilters on \(X\), there is a maximal ideal \(J\) in \(C{(X)}\) with \(\mathcal{F}\) = \(Z{\lbrack J\rbrack}\). So, we get \(M\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}}\rbrack}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{Z{\lbrack J\rbrack}}\rbrack}\). Since \(M\) is maximal so \(M\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{Z{\lbrack J\rbrack}}\rbrack}\). Also, \(J \cap {\lbrack{A{(X)}}\rbrack}_{c}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack{J \cap {\lbrack{A{(X)}}\rbrack}_{c}}\rbrack}}\rbrack}\) and by Lemma 4.14, we have
\(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack{J \cap {\lbrack{A{(X)}}\rbrack}_{c}}\rbrack}}\rbrack}\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J\rbrack}}\rbrack}\). Again, \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J\rbrack}}\rbrack}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{Z{\lbrack J\rbrack}}\rbrack}\) = \(M\). Thus, \(\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J\rbrack}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}\) \(\subseteq\) \(Z{\lbrack J\rbrack}\).
If there exists another \(z\)-ultrafilter \(Z{\lbrack J^{'}\rbrack}\), where \(J^{'}\) is a maximal ideal in \(C{(X)}\), with \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}\) \(\subseteq\) \(Z{\lbrack J^{'}\rbrack}\), then similarly as above we get \(\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J^{'}\rbrack}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}\) \(\subseteq\) \(Z{\lbrack J^{'}\rbrack}\). So, it follows that any zero-set \(F\) in \(Z{\lbrack J^{'}\rbrack}\) intersects every member of \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}\) and thus \(F\) intersects with every member of \(\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J\rbrack}\). Therefore, \(F\) intersects with every member of \(Z{\lbrack J^{'}\rbrack}\) giving that \(F\) \(\in\) \(Z{\lbrack J\rbrack}\). Hence, \(Z{\lbrack J\rbrack}\) = \(Z{\lbrack J^{'}\rbrack}\).

Theorem 4.18.

Let \(\mathcal{F}\) be a \(z\)-ultrafilter on \(X\). Then \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}\) is a maximal ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\).

Proof 4.19.

By Theorem 4.10, \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}\) is an ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\). Since \(\mathcal{F}\) is a \(z\)-ultrafilter, so \(\mathcal{F}\) = \(Z{\lbrack J\rbrack}\) for some maximal ideal \(J\) in \(C{(X)}\). Now, \(J \cap {\lbrack{A{(X)}}\rbrack}_{c}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack{J \cap {\lbrack{A{(X)}}\rbrack}_{c}}\rbrack}}\rbrack}\) and by Lemma 4.14, we have \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack{J \cap {\lbrack{A{(X)}}\rbrack}_{c}}\rbrack}}\rbrack}\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J\rbrack}}\rbrack}\). Again, \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J\rbrack}}\rbrack}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{Z{\lbrack J\rbrack}}\rbrack}\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}\). If \(J^{'}\) is a maximal ideal in \({\lbrack{A{(X)}}\rbrack}_{c}\) with \(J\) \(\cap\) \({\lbrack{A{(X)}}\rbrack}_{c}\) \(\subseteq\) \(J^{'}\), then \(\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J\rbrack}\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack{J \cap {\lbrack{A{(X)}}\rbrack}_{c}}\rbrack}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack J^{'}\rbrack}\). If we take \(J^{^{\operatorname{\prime\prime}}}\) to be a maximal ideal in \(C{(X)}\) such that \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack J^{'}\rbrack}\) \(\subseteq\) \(Z{\lbrack J^{^{\operatorname{\prime\prime}}}\rbrack}\), then by same argument as above, we get \(\mathcal{Z}_{{\lbrack{C{(X)}}\rbrack}_{c}}{\lbrack J^{^{\operatorname{\prime\prime}}}\rbrack}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack J^{'}\rbrack}\) and \(Z{\lbrack J^{^{\operatorname{\prime\prime}}}\rbrack}\) = \(Z{\lbrack J\rbrack}\). Hence, \(J^{'}\) \(\subseteq\) \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{Z{\lbrack J\rbrack}}\rbrack}\). But \(J^{'}\) is maximal, therefore \(J^{'}\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{Z{\lbrack J\rbrack}}\rbrack}\) = \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{F}\rbrack}\).

Therefore, by Theorems 4.16 and 4.18, we have the following theorem as in [4].

Theorem 4.20.

For each \(A{(X)}\), the structure space of \({\lbrack{A{(X)}}\rbrack}_{c}\) is homeomorphic to \(\beta X\).

Proof 4.21.

Let \(\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}\) be the structure space of \({\lbrack{A{(X)}}\rbrack}_{c}\). Since for each \(M \in {\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}}\), \(\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}\) is contained in a unique \(z\)-ultrafilter \(\mathcal{U}_{M}\) on \(X\), this defines a map \(\psi:{{\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}}\rightarrow{\beta X}}\) given by \({\psi{(M)}} = \mathcal{U}_{M}\) and note that \({\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{U}_{M}\rbrack}} \supseteq {\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{\lbrack M\rbrack}}\rbrack}} = M\) (because of the maximality of \(M\) in \({\lbrack{A{(X)}}\rbrack}_{c}\)) implies that \({\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}^{\leftarrow}{\lbrack\mathcal{U}_{M}\rbrack}} = M\). Thus, \(\psi\) is a bijection onto \(\beta X\). For any \(f \in {\lbrack{A{(X)}}\rbrack}_{c}\), \({\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}} \subseteq \mathcal{U}_{M}\) if and only \(f \in M\). Now, a typical basic closed set in \(\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}\) is \({\mathcal{M}{(f)}} = {\{{M \in {\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}}}:{f \in M}\}}\) and therefore, \({\psi{({\mathcal{M}{(f)}})}} = {\{\mathcal{U}_{M{(f)}}:{{\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}} \subseteq \mathcal{U}_{M{(f)}}}\}} = {\bigcap\limits_{Z \in {\mathcal{Z}_{{\lbrack{A{(X)}}\rbrack}_{c}}{(f)}}}{\{{\mathcal{U} \in {\beta X}}:{Z \in \mathcal{U}}\}}}\), a basic closed set in \(\beta X\). Thus, \(\psi:{{\mathcal{M}{({\lbrack{A{(X)}}\rbrack}_{c})}}\rightarrow{\beta X}}\) becomes a closed bijection on the domain space onto the range space. Since both the domain and range spaces are compact Hausdorff space, \(\psi\) turns out to be a homeomorphism.

Acknowledgements.

The authors are deeply grateful to the referee(s) for the valuable suggestions for improving this paper.

Funding.

The second author would also like to thank the Council of Scientific and Industrial Research \((\)CSIR\()\) for financial support under CSIR-JRF scheme.

Author Contributions.

Conceptualization, resources, funding acquisition and writing—review and editing, visualization, Y. J., L. K. and S. D.; software, Y. J. and L. K.; investigation, L. K. and S. D.; validation, formal analysis, Y. J. and S. D.; methodology and writing—original draft, L. K.; data curation, supervision, project administration, S. D. All authors have read and agreed to the published version of the manuscript.

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