Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X

Let X be a completely regular topological space. An intermediate ring is a ring A(X) of continuous functions satisfying C∗(X) ⊆ A(X) ⊆ C(X). In Redlin andWatson (1987) and in Panman et al. (2012), correspondencesZ A andZ A are defined between ideals in A(X) and z-filters onX, and it is shown that these extend the well-known correspondences studied separately for C∗(X) andC(X), respectively, to any intermediate ring. Moreover, the inverse map Z← A sets up a one-one correspondence between the maximal ideals of A(X) and the z-ultrafilters on X. In this paper, we define a functionK A that, in the case that A(X) is a C-ring, describes


Introduction
Let  be a completely regular space and () an intermediate ring of continuous real-valued functions; that is,  * () ⊆ () ⊆ ().It is well known that there is a natural correspondence Z between ideals of () and -filters on X as described in [1, pages 26-27].Such a correspondence E also exists for  * () [1,Problem 2L].In [2], a correspondence Z  between the ideals of any () and the -filters on  was introduced, and its properties were further investigated in [3][4][5].In [6], another correspondence Z  between ideals of any () and -filters on  is introduced.It is shown in [6] that the correspondences Z  and Z  extend the correspondences E and Z from  * () and (), respectively, to all intermediate rings, and an explicit formula is stated that relates the two correspondences.In this paper, we give a characterization (Definition 3 and Theorem 6) of the correspondence Z  for intermediate -rings () in terms of the -compactifications of  introduced in [7].In this setting, we show (Theorem 14) that the inverse map Z ←  of the set map Z  maps ideals in () to -filters on .We also give a characterization of the maximal ideals in ().This characterization generalizes from () to () the Gelfand-Kolmogorov theorem (Theorem 8).We follow the notation in [1,6].

Preliminaries
For convenience we state some of the definitions and results needed in this paper.
Following the notation in [1], we set to be the collection of the zero sets Z() =  −1 ({0}) of all functions  ∈ ().In this paper, we generally work with functions  on a fixed set , as well as some extensions  of  to larger domains.As expected, Z() then denotes the zero set of  on the larger domain.
A -filter (-ultrafilter, resp.) on  is the intersection of Z() with a filter (ultrafilter, resp.) on .The kernel  of a set S of -ultrafilters is defined by One can verify that the kernel of a set of -ultrafilters is a -filter.The hull ℎ of a -filter F is defined by Given a set  ⊆ , let ⟨⟩ denote the set of all zero sets  ⊆ , such that  ⊆ .
Given any two functions  and  on a set  and  ⊆ , we write  ≡  on  if () = () for all  ∈ .For each intermediate ring () of continuous functions, each noninvertible  ∈ (), and each  ⊆ , we say that  is -regular in () if there exists  ∈ () such that  ≡ 1 on .We just say  is -regular if () is understood by context.For any set  ⊆ , let   be the complement of  in .
For each  ∈ (), set For an ideal  ⊂ (), we set Several properties of Z  and Z  are proved in [3,5,6].In particular, we have the following lemma, which we state here for convenience.Item (a) is from [2,6], Item (b) is from [3,6], Item (c) is from [6], and Item (d) is from [2].
The Stone-Čech compactification of , denoted , is any topological space homeomorphic to the space of -ultrafilters on  topologized with the hull-kernel closure operator as follows: the closure of any set  of -ultrafilters is ℎ().Throughout this paper, we will in particular take  to consist of a superset of , whose points, which we denote , , . .., can be viewed as indices of -ultrafilters on .Let U be the map which associates every  ∈  with a -ultrafilter U  : such that for each  ∈ , U  = ⟨{}⟩ is the fixed -ultrafilter containing , and for each  ∈  \ , U  is a unique free -ultrafilter, such that U is a one-to-one correspondence between  and the set of -ultrafilters on .The topology on  is defined in such a way that the map U is a homeomorphism.Making use of the fact that the zero sets form a base for the collection of closed sets [1, page 38], one can check that U maps  homeomorphically onto the subspace of fixed -ultrafilters, and hence  is a subspace of .
A -filter F on  is called -stable if for every  ∈ () there exists a set in F on which  is bounded (see [7,8]).
Following [7], for each () we define the -compactification    of  as the subspace of  where From [5,Theorem 4.6], it holds that    is a realcompactification of .Note that if () =  * (), then    = .
To refine our understanding of the topology on   , we define the -stable hull ℎ  of a -filter F by It is immediate from the definition of the subspace topology that    is homeomorphic (via U|    ) to the hull-kernel topology restricted to -stable -ultrafilters, that is, the topology with the following closure operator: the closure of any set  is ℎ  ().It follows that From [7], we have that the space    consists of the points of  to which every function  ∈ () can be continuously extended.We denote the extension of  to    by    .From [7, Theorem 9], we have the value of    at a point  ∈    is given by In [7], a ring () of continuous functions is called a ring if there is a completely regular space  such that () is isomorphic to ().Clearly () and  * () are -rings (with  * () isomorphic to ()).We use the following result from [7,Theorem 7].

Lemma 2. Let 𝐴(𝑋) be an intermediate ring. Then the follow
In addition, it is shown in [5,Theorem 4.7] that there is a bijective correspondence between the realcompactifications of  and the -rings on .

Characterizations Using Realcompactifications
In We set For an ideal  ⊂ (), we set That K  is indeed a mapping from ideals of an intermediate -ring () to -filters on  will follow from our main result of the section that establishes that K  = Z  when () is an intermediate -ring.We also show that K  does not necessarily map ideals in () to -filters on  when () is not an intermediate -ring (Example 7).
First, to illustrate some connections that motivated the development of K  , let us now observe a similarity between K  and a simple function Ẑ that we define in terms of Z.We define Ẑ on () by We drop the subscript  when it is understood by context.It is easy to see that given an ideal , Ẑ We now turn our attention to  * () and we note that  * () is an intermediate -ring, as  * () is isomorphic to ().In light of this, we have for  ∈ () that Ultimately, we would like to find a map defined on  * () itself rather than () and that maps to -filters on  rather than  (it is possible that a -filter on  may contain a set that does not meet ; for example, N \ N is the zero set of the function  N , where () = 1/).Observe that where the first equality is immediate from the fact that Z(  ) ∈ Z(), and the second equality holds because the sets of the form cl   are a base for the closed sets in  (see [1, page 94]).This motivates the following definition that relates Z(  ) to : We then generalize K  * to all intermediate rings () to arrive at Definition 3.
In order to prove our main theorem that K  = Z  whenever  is an intermediate -ring, we need some lemmas.We first show that the zero set of    , viewed as a set of ultrafilters, is the -stable hull of Z  ().

Lemma 4. If 𝐴(𝑋) is an intermediate ring of continuous functions,
where the symbol ≡ indicates that one set is the homeomorphic image of the other.In particular, if  ∈ (), then  ∈ Z(   ) if and only if U  ∈ ℎ  (Z  ()).
Proof.We observe that the following are equivalent: (ii) U  is -stable and lim U   = 0, (iii) U  is -stable and lim U  ℎ = 0 for all ℎ ∈ (), The equivalence (i)⇔(ii) follows from ( 6) and ( 9).The equivalence (ii)⇔(iii) follows from the fact that U  is -stable, and hence ℎ is bounded on some set in U  .The equivalence (iii)⇔(iv) follows from Lemma 1(d).
We use the notation ℎF to denote the kernel of the hull of the -filter F, that is, the intersection of the set of ultrafilters containing F. (16) Proof.Without loss of generality, we may assume that  ≥ 0 (because Z  () = Z  ( Since  and  are disjoint zero sets in , they are completely separated (see [1, page 17]).So there is a continuous function ℎ that takes the value 1 on  and 0 on  and 0 ≤ ℎ ≤ 1, so ℎ ∈ ().Consider the function  =  + ℎ ∈ ().Note that () = () for  ∈ .Also, since  ≥ 1 on  and since cl     ⊇ Z(   ), it follows that Z(   ) = 0.By Lemma 2(b),  is invertible, and as  ≡  on , it follows that  −1  ≡ 1 on .Since  ∈ U  , it also follows that lim U   −1  ̸ = 0, but this contradicts the fact that U  ⊇ Z  () because of Lemma 1(d).This contradiction stems from the assumption that  does not belong to U  .So  ∈ U  for every U  ⊇ Z  (), that is,  ∈ ℎZ  ().
The theorem does not hold if the assumption that () being an intermediate -ring is removed.In fact, K  need not map ideals in () to -filters on  when () is not a -ring, as the following example shows.
Example 7. Let  = [1, ∞).Let () be the smallest ring of continuous functions containing both  * () and () = .Then each function ℎ ∈ () has the form ℎ() = ∑  =0   ()  for   ∈  * ().Note that any function ℎ ∈ () is therefore bounded by some function of the form   for  ∈ R and  ∈ N, where  is  + 1 times a common bound for all of   .We now observe that   ∉ (), since   cannot be bounded by any function of the form   for  ∈ R and  ∈ N. Let  =  − .Then  is in () but is not invertible in ().Thus () and () are not isomorphic.
Since every set of a free -ultrafilter on  = [1, ∞) must be unbounded, the identity function  is not bounded on any such set either.Hence any free -ultrafilter on  is not stable, and   () = .Then by Lemma 2(a), () is not a -ring, since (  ()) = () and () ̸ = ().Finally, since    =  and Z() = 0, the set K  () = { ∈ Z() | Z(   ) ⊆ cl    } consists of all zero sets of  (hence is not a -filter), while Z  () is the -filter consisting of all zero sets in  whose complement in  has an upper bound.Thus K  () ̸ = Z  (), which is in contrast to the conclusion of Theorem 6.
We leave open the question as to precisely what rings () are such that K  = Z  .We also leave open the question as to whether there exists a ring () such that K  does map ideals in () to -filters on , but K  ̸ = Z  .

Characterizing Maximal Ideals in 𝐶-Rings.
The following characterization for maximal ideals in () is proved in [6].Every maximal ideal in () is of the form for  ∈ .By Lemma 1(c), Z  () ⊆ U  whenever Z  () ⊆ U  , and hence by ( 9) and Lemma 1(d), we have Since Z  () = ⟨Z()⟩, we have by (8) that The characterizations in ( 19) and (20) that we obtained from (18) agree with those given in [ where  ∈ .

Proof. 𝑓 ∈ 𝑀
if and only if Z  () ⊆ U  if and only if U  ∈ ℎ(Z  ()).By identifying a set in  with its image under U, it follows from the definition of closure in , Lemmas 4 and 5, that The result now follows from the fact that ℎ(Z  ()) = cl  Z(   ).
We now verify that this theorem generalizes the Gelfand-Kolmogorov Theorem [1, page 102]

The Map Z ←
and Ideals in () Recall that Z  maps ideals to -filters (Lemma 1(a)).Here we show that for an intermediate -ring (), the inverse map Z ←  , defined by maps -filters on  to ideals in () (Theorem 14).The corresponding result for the maps Z (for ()) and Z  (for any intermediate ring ()) is proved, respectively, in [1] and [3].Our proof for Z  makes use of Theorem 6.We need some lemmas concerning meets and joins on the lattice of -filters.Recall that F ∨ G is the smallest -filter containing both the -filters F and G. Similarly, F ∧ G is the largest -filter contained in both the -filters F and G.
The following lemma is from [6].
The following is a special case of [6, Lemma 4.2(a)].
To obtain the analog of Lemma 10 for joins, we need the following lemma. Similarly Since  1 ∩ is a zero set in \, it follows that  1 is a zero set in  by the fact that the cozero set of a cozero set is a cozero set Similarly, Z( We show that the other containment is equivalent to Lemma 11 and hence must hold.First, we show the equivalence of the premises by showing that the following are equivalent.Recall that Z  () = Z  ( 2 ), and hence we can assume without loss of generality that ,  ≥ 0.  The equivalence (i)⇔(ii) follows from Lemma 9(c).The equivalence (ii)⇔(iii) follows from Theorem 6.The equivalence (iii)⇔(iv) follows from the assumption that ,  ≥ 0. This establishes the equivalence between the premises of leftto-right containment of this lemma and Lemma 11.
For the equivalence of the conclusions, note that  ∈ ℎ(Z  ()) ∨ ℎ(Z  ()) if and only if  is an intersection of a set in ℎ(Z  ()) with a set in ℎ(Z  ()).In other words, there must exist  1 ∈ ℎ(Z  ()) and  2 ∈ ℎ(Z  ()) such that  1 ∩  2 = .By Lemmas 4 and 5, this is equivalent to the statement that there exist zero sets  1 ,  2 such that Z(   ) ⊆ cl     1 , Z(   ) ⊆ cl     2 , and  1 ∩  2 = , which is the conclusion of Lemma 11.We are now ready to prove the main result of this section.

Lemma 1 .
Let () be an intermediate ring.Then the following hold.(a) For any noninvertible  ∈ (), both Z  () and Z  () are -filters on .If  is invertible, then Z  () = Z  () = Z[], the set of all zero sets in .(b) For an ideal  ⊂ (), both Z  [] and Z  [] are filters on .(c) For any  ∈ (), we have Z  () = ℎZ  ().(d) If F is a -filter on , then Z  () ⊆ F if and only if lim F ℎ = 0 for all ℎ ∈ ().
for -Rings.While the definition of Z  is essentially algebraic (using the property of local invertibility), we now define a function K  that provides a "topological description" (using realcompactifications of ) of a mapping from ideals of an intermediate -ring () to filters on , and we will show (Theorem 6) that K  coincides with Z  when () is an intermediate -ring.
this section, we utilize the realcompactifications of  to provide a new description of the function Z  and of maximal ideals of (), when () is an intermediate -ring.The new description of the maximal ideals of () generalizes the Gelfand-Kolmogorov theorem [1, page 102] for () to all intermediate -rings.International Journal of Mathematics and Mathematical Sciences 3 3.1.A New Description of Z Definition 3. Let () be an intermediate ring and  ∈ ().
Let () be an intermediate -ring.Then each maximal ideal in () is of the form Lemma 11.If  is a zero set in  and if Z(   ) ∩ Z(   ) ⊆ cl     then there exist zero sets  1 and  2 in  such that Z(   ) ⊆ cl     1 , Z(   ) ⊆ cl     2 , and  1 ∩  2 = .Proof.Let  = cl     and let  =    \ .Now, the sets Z(   ) ∩  and Z(   ) ∩  are disjoint zero sets in , so are contained in disjoint zero set neighborhoods  1 and  2 in .Moreover, since  1 is a neighborhood of Z(   )∩, it follows that ) ⊆ cl     2 .Finally,  1 ∩  2 =  because  1 ∩  and  2 ∩  are disjoint.The next lemma shows that the kernel-hull operation distributes over the join operation on the lattice of -filters.If () is an intermediate -ring, then ℎ (Z  () ∨ Z  ()) = ℎ (Z  ()) ∨ ℎ (Z  ()) .