# Characterizations of Ideals in IntermediateC-Rings A(X) via the A-Compactifications of X

Authors: Joshua Sack; Saleem Watson
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

LetX 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 and Watson (1987) and in Panman et al. (2012), correspondences 𝒵A and ℨA are defined between ideals in A(X) and z-filters on X, and it is shown that these extend the well-known correspondences studied separately for C∗(X) and C(X), respectively, to any intermediate ring. Moreover, the inverse map 𝒵A←
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 function 𝔎A that, in the case that A(X) is a C-ring, describes ℨA in terms of extensions of functions to realcompactifications of X. For such rings, we show that ℨA← maps z-filters to ideals. We also give a characterization of the maximal ideals in A(X) that generalize the Gelfand-Kolmogorov theorem from C(X) to A(X).


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## Body

## 1. Introduction

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


## 2. Preliminaries

For convenience we state some of the definitions and results needed in this paper.Following the notation in [1], we set
(1)Z(X)=Z[C(X)]={Z(f)∣f∈C(X)}
to be the collection of the zero sets Z(f)=f-1({0}) of all functions f∈C(X). In this paper, we generally work with functions f on a fixed set X, as well as some extensions g of f to larger domains. As expected, Z(g) then denotes the zero set of g on the larger domain.Az-filter (z-ultrafilter, resp.) on X is the intersection of Z(X) with a filter (ultrafilter, resp.) on X. The kernelk of a set 𝒮 of z-ultrafilters is defined by
(2)k(𝒮)=⋂‍{𝒰∣𝒰∈𝒮}.One can verify that the kernel of a set ofz-ultrafilters is a z-filter. The hullh of a z-filter ℱ is defined by
(3)h(ℱ)={𝒰∣𝒰isaz-ultrafilteronX,𝒰⊇ℱ}.
Given a set E⊆X, let 〈E〉 denote the set of all zero sets F⊆X, such that E⊆F.Given any two functionsf and g on a set X and H⊆X, we write f≡g on H if f(x)=g(x) for all x∈H. For each intermediate ring A(X) of continuous functions, each noninvertible f∈A(X), and each H⊆X, we say that f is H-regular inA(X) if there exists g∈A(X) such that fg≡1 on H. We just say f is H-regular if A(X) is understood by context. For any set E⊆X, let Ec be the complement of E in X.For eachf∈A(X), set
(4)𝒵A(f)={E∈Z(X)∣fisEc-regular},ℨA(f)={E∈Z(X)∣fisH-regularforeachzero-setH⊆Ec}.
For an ideal I⊂A(X), we set 𝒵A[I]=⋃‍{𝒵A(f)∣f∈I} and ℨA[I]=⋃‍{ℨA(f)∣f∈I}. Several properties of 𝒵A and ℨA are proved in [3, 5, 6]. In particular, we have the following lemma, which we state here for convenience.Lemma 1.
LetA(X) be an intermediate ring. Then the following hold.(a)
For any noninvertiblef∈A(X), both 𝒵A(f) and ℨA(f) are z-filters on X. If f is invertible, then 𝒵A(f)=ℨA(f)=Z[X], the set of all zero sets in X.(b)
For an idealI⊂A(X), both 𝒵A[I] and ℨA[I] are z-filters on X.(c)
For anyf∈A(X), we have ℨA(f)=kh𝒵A(f).(d)
Ifℱ is a z-filter on X, then 𝒵A(f)⊆ℱ if and only if limℱfh=0 for all h∈A(X).Item (a) is from [2, 6], Item (b) is from [3, 6], Item (c) is from [6], and Item (d) is from [2].TheStone-Čech compactification of X, denoted βX, is any topological space homeomorphic to the space of z-ultrafilters on X topologized with the hull-kernel closure operator as follows: the closure of any set U of z-ultrafilters is hk(U). Throughout this paper, we will in particular take βX to consist of a superset of X, whose points, which we denote p,q,…, can be viewed as indices of z-ultrafilters on X. Let 𝒰 be the map which associates every p∈βX with a z-ultrafilter 𝒰p:
(5)p⟼𝒰𝒰p,
such that for each p∈X, 𝒰p=〈{p}〉 is the fixed z-ultrafilter containing p, and for each p∈βX∖X,𝒰p is a unique free z-ultrafilter, such that 𝒰 is a one-to-one correspondence between βX and the set of z-ultrafilters on X. The topology on βX 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 X homeomorphically onto the subspace of fixed z-ultrafilters, and hence X is a subspace of βX.Az-filter ℱ on X is called A-stable if for every f∈A(X) there exists a set in ℱ on which f is bounded (see [7, 8]). Following [7], for each A(X) we define the A-compactificationυAX of X as the subspace of βX where
(6)υAX={p∈βX∣𝒰pisA-stable}.
From [5, Theorem 4.6], it holds that υAX is a realcompactification of X. Note that if A(X)=C*(X), then υAX=βX. If A(X)=C(X), then υAX=υX is the Hewitt realcompactification.To refine our understanding of the topology onυAX, we define the A-stable hullhA of a z-filter ℱ by
(7)hA(ℱ)={𝒰∣𝒰isanA-stablez-ultrafilteronX,𝒰⊇ℱ}.
It is immediate from the definition of the subspace topology that υAX is homeomorphic (via 𝒰∣υAX) to the hull-kernel topology restricted to A-stable z-ultrafilters, that is, the topology with the following closure operator: the closure of any set U is hAk(U). It follows that
(8)p∈clυAXEiffE∈𝒰p.From [7], we have that the space υAX consists of the points of βX to which every function f∈A(X) can be continuously extended. We denote the extension of f to υAX by fυA. From [7, Theorem 9], we have the value of fυA at a point p∈υAX is given by
(9)fυA(p)=lim𝒰pf.In [7], a ring A(X) of continuous functions is called a C-ring if there is a completely regular space Y such that A(X) is isomorphic to C(Y). Clearly C(X) and C*(X) are C-rings (with C*(X) isomorphic to C(βX)). We use the following result from [7, Theorem 7].Lemma 2.
LetA(X) be an intermediate ring. Then the following hold. (a)
A(X) is a C-ring if and only if A(X) is isomorphic to C(υAX).(b)
IfA(X) is a C-ring, then f∈A(X) is invertible in A(X) if and only if Z(fυA)=∅.In addition, it is shown in [5, Theorem 4.7] that there is a bijective correspondence between the realcompactifications of X and the C-rings on X.


## 3. Characterizations Using Realcompactifications

In this section, we utilize the realcompactifications ofX to provide a new description of the function ℨA and of maximal ideals of A(X), when A(X) is an intermediate C-ring. The new description of the maximal ideals of A(X) generalizes the Gelfand-Kolmogorov theorem [1, page 102] for C(X) to all intermediate C-rings.


### 3.1. A New Description ofℨA for C-Rings

While the definition ofℨA is essentially algebraic (using the property of local invertibility), we now define a function 𝔎A that provides a “topological description" (using realcompactifications of X) of a mapping from ideals of an intermediate C-ring A(X) to z-filters on X, and we will show (Theorem 6) that 𝔎A coincides with ℨA when A(X) is an intermediate C-ring.Definition 3.
LetA(X) be an intermediate ring and f∈A(X). We set
(10)𝔎A(f)={E∈Z(X)∣Z(fυA)⊆clυAXE}.
For an ideal I⊂A(X), we set 𝔎A[I]=⋃‍{𝔎A(f)∣f∈I}.That𝔎A is indeed a mapping from ideals of an intermediate C-ring A(X) to z-filters on X will follow from our main result of the section that establishes that 𝔎A=ℨA when A(X) is an intermediate C-ring. We also show that 𝔎A does not necessarily map ideals in A(X) to z-filters on X when A(X) is not an intermediate C-ring (Example 7).First, to illustrate some connections that motivated the development of𝔎A, let us now observe a similarity between 𝔎A and a simple function Z^ that we define in terms of Z. We define Z^X on C(X) by
(11)Z^X:f⟼〈Z(f)〉={E∈Z(X)∣Z(f)⊆E}.
We drop the subscript X when it is understood by context. It is easy to see that given an ideal I,Z^[I]=Z[I].We now turn our attention toC*(X) and we note that C*(X) is an intermediate C-ring, as C*(X) is isomorphic to C(βX). In light of this, we have for f∈C(X) that
(12)Z^βX:fβ⟼〈Z(fβ)〉={Eβ∈Z(βX)∣Z(fβ)⊆Eβ}.
Ultimately, we would like to find a map defined on C*(X) itself rather than C(βX) and that maps to z-filters on X rather than βX (it is possible that a z-filter on βX may contain a set that does not meet X; for example, βℕ∖ℕ is the zero set of the function fβℕ, where f(n)=1/n). Observe that
(13)⋂‍{Eβ∈Z(βX)∣Z(fβ)⊆Eβ}=Z(fβ)=⋂‍{clβXE∣E∈Z(X),Z(fβ)⊆clβXE},
where the first equality is immediate from the fact that Z(fβ)∈Z(βX), and the second equality holds because the sets of the form clβXE are a base for the closed sets in βX (see [1, page 94]). This motivates the following definition that relates Z(fβ) to X:
(14)𝔎C*:f⟼{E∈Z(X)∣Z(fβ)⊆clβXE}.
We then generalize 𝔎C* to all intermediate rings A(X) to arrive at Definition 3.In order to prove our main theorem that𝔎A=ℨA whenever A is an intermediate C-ring, we need some lemmas. We first show that the zero set of fυA, viewed as a set of z-ultrafilters, is the A-stable hull of 𝒵A(f).Lemma 4.
IfA(X) is an intermediate ring of continuous functions,
(15)Z(fυA)≡hA(𝒵A(f)),
where the symbol ≡ indicates that one set is the homeomorphic image of the other. In particular, if f∈A(X), then p∈Z(fυA) if and only if 𝒰p∈hA(𝒵A(f)).Proof.
We observe that the following are equivalent:(i)
p∈Z(fυA),(ii)
𝒰p is A-stable and lim𝒰pf=0,(iii)
𝒰p is A-stable and lim𝒰pfh=0 for all h∈A(X),(iv)
𝒰p is A-stable and 𝒰p⊇𝒵A(f), that is, 𝒰p∈hA(𝒵A(f)).The equivalence (i)⇔(ii) follows from (6) and (9). The equivalence (ii)⇔(iii) follows from the fact that 𝒰p is A-stable, and hence h is bounded on some set in 𝒰p. The equivalence (iii)⇔(iv) follows from Lemma 1(d).We use the notationkhℱ to denote the kernel of the hull of the z-filter ℱ, that is, the intersection of the set of z-ultrafilters containing ℱ.Lemma 5.
LetA(X) be an intermediate C-ring and f∈A(X). Then
(16)kh𝒵A(f)=khA𝒵A(f).Proof.
Without loss of generality, we may assume thatf≥0 (because 𝒵A(f)=𝒵A(f2)). Of course kh𝒵A(f)⊆khA𝒵A(f). For the other containment, suppose E∈khA𝒵A(f), so E belongs to every A-stable z-ultrafilter containing 𝒵A(f). We first note that by Lemma 4, 𝒰p∈hA𝒵A(f) if and only if p∈Z(fυA). In other words, for any E∈Z[X], we have E∈khA𝒵A(f) if and only if E∈𝒰p for every p in Z(fυA) if and only if clυAXE⊇Z(fυA). Now suppose 𝒰q is any z-ultrafilter (A-stable or not) containing 𝒵A(f). We show that E belongs to 𝒰q. If not, then there exists F∈𝒰q such that
(17)E∩F=∅.
Since E and F are disjoint zero sets in X, they are completely separated (see [1, page 17]). So there is a continuous function h that takes the value 1 on E and 0 on F and 0≤h≤1, so h∈A(X). Consider the function k=f+h∈A(X). Note that k(x)=f(x) for x∈F. Also, since k≥1 on E and since clυAXE⊇Z(fυA), it follows that Z(kυA)=∅. By Lemma 2(b), k is invertible, and as k≡f on F, it follows that k-1f≡1 on F. Since F∈𝒰q, it also follows that lim𝒰qk-1f≠0, but this contradicts the fact that 𝒰q⊇𝒵A(f) because of Lemma 1(d). This contradiction stems from the assumption that E does not belong to 𝒰q. So E∈𝒰q for every 𝒰q⊇𝒵A(f), that is, E∈kh𝒵A(f).Recall that for any intermediate ringA(X), Lemma 1(c) gives the relationship ℨA(f)=kh𝒵A(f). For intermediate C-rings, we can characterize ℨA topologically as 𝔎A.Theorem 6.
LetA(X) be an intermediate C-ring and f∈A(X). Then ℨA(f)=𝔎A(f).Proof.
SupposeE∈ℨA(f). By Lemma 1(c), E∈kh𝒵A(f). We show that in this case p∈Z(fυA) implies p∈clυAXE. Now if p∈Z(fυA), then by Lemma 4, 𝒰p∈hA(𝒵A(f)). Since E∈kh𝒵A(f), it follows that E∈𝒰p, and hence by (8), we have that p∈clυAXE.
For the other containment, supposeE∈𝔎A(f), that is, Z(fυA)⊆clυAXE. We show that E belongs to every A-stable z-ultrafilter containing 𝒵A(f). First, let 𝒰p be an A-stable z-ultrafilter containing 𝒵A(f). Then by Lemma 4, p∈Z(fυA), so by hypothesis, p∈clυAXE, and hence E∈𝒰p. Thus E belongs to every A-stable z-ultrafilter containing 𝒵A(f); in other words, E∈khA𝒵A(f). Since A(X) is an intermediate C-ring, we can apply Lemma 5 so that E∈kh𝒵A(f), and hence by Lemma 1(c), E∈ℨA(f).The theorem does not hold if the assumption thatA(X) being an intermediate C-ring is removed. In fact, 𝔎A need not map ideals in A(X) to z-filters on X when A(X) is not a C-ring, as the following example shows.Example 7.
LetX=[1,∞). Let A(X) be the smallest ring of continuous functions containing both C*(X) and f(x)=x. Then each function h∈A(X) has the form h(x)=∑k=0nfk(x)xk for fk∈C*(X). Note that any function h∈A(X) is therefore bounded by some function of the form Mxn for M∈ℝ and n∈ℕ, where M is n+1 times a common bound for all of fk. We now observe that ex∉A(X), since ex cannot be bounded by any function of the form Mxn for M∈ℝ and n∈ℕ. Let g=e-x. Then g is in A(X) but is not invertible in A(X). Thus A(X) and C(X) are not isomorphic.
Since every set of a freez-ultrafilter on X=[1,∞) must be unbounded, the identity function f is not bounded on any such set either. Hence any free z-ultrafilter on X is not A-stable, and υA(X)=X. Then by Lemma 2(a), A(X) is not a C-ring, since C(υA(X))=C(X) and C(X)≠A(X).
Finally, sincegυA=g and Z(g)=∅, the set 𝔎A(g)={E∈Z(X)∣Z(gυA)⊆clυAXE} consists of all zero sets of X (hence is not a z-filter), while ℨA(g) is the z-filter consisting of all zero sets in X whose complement in X has an upper bound. Thus 𝔎A(g)≠ℨA(g), which is in contrast to the conclusion of Theorem 6.We leave open the question as to precisely what ringsA(X) are such that 𝔎A=ℨA. We also leave open the question as to whether there exists a ring A(X) such that 𝔎A does map ideals in A(X) to z-filters on X, but 𝔎A≠ℨA.


### 3.2. Characterizing Maximal Ideals inC-Rings

The following characterization for maximal ideals inA(X) is proved in [6]. Every maximal ideal in A(X) is of the form
(18)MAp={f∈A(X)∣ℨA(f)⊆𝒰p},
for p∈βX. By Lemma 1(c), ℨA(f)⊆𝒰p whenever 𝒵A(f)⊆𝒰p, and hence by (9) and Lemma 1(d), we have
(19)MC*p={f∈C*(X)∣fβ(p)=0}.SinceℨC(f)=〈Z(f)〉, we have by (8) that
(20)MCp={f∈C(X)∣〈Z(f)〉⊆𝒰p}={f∈C(X)∣p∈clβXZ(f)}.The characterizations in (19) and (20) that we obtained from (18) agree with those given in [1, pages 101-102]; the latter characterization of MCp is called the Gelfand-Kolmogorov theorem [1, page 102]. The following theorem provides a characterization of MAp that we see extends both (20) and (19) to all intermediate C-rings A(X).Theorem 8.
LetA(X) be an intermediate C-ring. Then each maximal ideal in A(X) is of the form
(21)MAp={f∈A(X):p∈clβXZ(fυA)},
where p∈βX.Proof.
f
∈
M
A
p if and only if 𝒵A(f)⊆𝒰p if and only if 𝒰p∈h(𝒵A(f)). By identifying a set in βX with its image under 𝒰, it follows from the definition of closure in βX, Lemmas 4 and 5, that
(22)clβXZ(fυA)=hkZ(fυA)=hkhA𝒵A(f)=hkh𝒵A(f)=h𝒵A(f).
The result now follows from the fact that h(𝒵A(f))=clβXZ(fυA).We now verify that this theorem generalizes the Gelfand-Kolmogorov Theorem [1, page 102] to intermediate C-rings. If A(X)=C(X), then υAX=υX, the Hewitt realcompactification of X. Now using the fact that clυXZ(f)=Z(fυ) [1, page 118] and Theorem 8, we have
(23)MCp={f∈C(X)∣p∈clβXZ(fυ)}={f∈C(X)∣p∈clβXclυXZ(f)}={f∈C(X)∣p∈clβXZ(f)}.For the case whereA(X)=C*(X), we have
(24)MC*p={f∈C*(X)∣p∈clβXZ(fβ)}={f∈C*(X)∣fβ(p)=0}.
So Theorem 8 simultaneously generalizes the results of (20) for C(X) and (19) for C*(X) to all intermediate C-rings.


## 3.1. A New Description ofℨA for C-Rings

While the definition ofℨA is essentially algebraic (using the property of local invertibility), we now define a function 𝔎A that provides a “topological description" (using realcompactifications of X) of a mapping from ideals of an intermediate C-ring A(X) to z-filters on X, and we will show (Theorem 6) that 𝔎A coincides with ℨA when A(X) is an intermediate C-ring.Definition 3.
LetA(X) be an intermediate ring and f∈A(X). We set
(10)𝔎A(f)={E∈Z(X)∣Z(fυA)⊆clυAXE}.
For an ideal I⊂A(X), we set 𝔎A[I]=⋃‍{𝔎A(f)∣f∈I}.That𝔎A is indeed a mapping from ideals of an intermediate C-ring A(X) to z-filters on X will follow from our main result of the section that establishes that 𝔎A=ℨA when A(X) is an intermediate C-ring. We also show that 𝔎A does not necessarily map ideals in A(X) to z-filters on X when A(X) is not an intermediate C-ring (Example 7).First, to illustrate some connections that motivated the development of𝔎A, let us now observe a similarity between 𝔎A and a simple function Z^ that we define in terms of Z. We define Z^X on C(X) by
(11)Z^X:f⟼〈Z(f)〉={E∈Z(X)∣Z(f)⊆E}.
We drop the subscript X when it is understood by context. It is easy to see that given an ideal I,Z^[I]=Z[I].We now turn our attention toC*(X) and we note that C*(X) is an intermediate C-ring, as C*(X) is isomorphic to C(βX). In light of this, we have for f∈C(X) that
(12)Z^βX:fβ⟼〈Z(fβ)〉={Eβ∈Z(βX)∣Z(fβ)⊆Eβ}.
Ultimately, we would like to find a map defined on C*(X) itself rather than C(βX) and that maps to z-filters on X rather than βX (it is possible that a z-filter on βX may contain a set that does not meet X; for example, βℕ∖ℕ is the zero set of the function fβℕ, where f(n)=1/n). Observe that
(13)⋂‍{Eβ∈Z(βX)∣Z(fβ)⊆Eβ}=Z(fβ)=⋂‍{clβXE∣E∈Z(X),Z(fβ)⊆clβXE},
where the first equality is immediate from the fact that Z(fβ)∈Z(βX), and the second equality holds because the sets of the form clβXE are a base for the closed sets in βX (see [1, page 94]). This motivates the following definition that relates Z(fβ) to X:
(14)𝔎C*:f⟼{E∈Z(X)∣Z(fβ)⊆clβXE}.
We then generalize 𝔎C* to all intermediate rings A(X) to arrive at Definition 3.In order to prove our main theorem that𝔎A=ℨA whenever A is an intermediate C-ring, we need some lemmas. We first show that the zero set of fυA, viewed as a set of z-ultrafilters, is the A-stable hull of 𝒵A(f).Lemma 4.
IfA(X) is an intermediate ring of continuous functions,
(15)Z(fυA)≡hA(𝒵A(f)),
where the symbol ≡ indicates that one set is the homeomorphic image of the other. In particular, if f∈A(X), then p∈Z(fυA) if and only if 𝒰p∈hA(𝒵A(f)).Proof.
We observe that the following are equivalent:(i)
p∈Z(fυA),(ii)
𝒰p is A-stable and lim𝒰pf=0,(iii)
𝒰p is A-stable and lim𝒰pfh=0 for all h∈A(X),(iv)
𝒰p is A-stable and 𝒰p⊇𝒵A(f), that is, 𝒰p∈hA(𝒵A(f)).The equivalence (i)⇔(ii) follows from (6) and (9). The equivalence (ii)⇔(iii) follows from the fact that 𝒰p is A-stable, and hence h is bounded on some set in 𝒰p. The equivalence (iii)⇔(iv) follows from Lemma 1(d).We use the notationkhℱ to denote the kernel of the hull of the z-filter ℱ, that is, the intersection of the set of z-ultrafilters containing ℱ.Lemma 5.
LetA(X) be an intermediate C-ring and f∈A(X). Then
(16)kh𝒵A(f)=khA𝒵A(f).Proof.
Without loss of generality, we may assume thatf≥0 (because 𝒵A(f)=𝒵A(f2)). Of course kh𝒵A(f)⊆khA𝒵A(f). For the other containment, suppose E∈khA𝒵A(f), so E belongs to every A-stable z-ultrafilter containing 𝒵A(f). We first note that by Lemma 4, 𝒰p∈hA𝒵A(f) if and only if p∈Z(fυA). In other words, for any E∈Z[X], we have E∈khA𝒵A(f) if and only if E∈𝒰p for every p in Z(fυA) if and only if clυAXE⊇Z(fυA). Now suppose 𝒰q is any z-ultrafilter (A-stable or not) containing 𝒵A(f). We show that E belongs to 𝒰q. If not, then there exists F∈𝒰q such that
(17)E∩F=∅.
Since E and F are disjoint zero sets in X, they are completely separated (see [1, page 17]). So there is a continuous function h that takes the value 1 on E and 0 on F and 0≤h≤1, so h∈A(X). Consider the function k=f+h∈A(X). Note that k(x)=f(x) for x∈F. Also, since k≥1 on E and since clυAXE⊇Z(fυA), it follows that Z(kυA)=∅. By Lemma 2(b), k is invertible, and as k≡f on F, it follows that k-1f≡1 on F. Since F∈𝒰q, it also follows that lim𝒰qk-1f≠0, but this contradicts the fact that 𝒰q⊇𝒵A(f) because of Lemma 1(d). This contradiction stems from the assumption that E does not belong to 𝒰q. So E∈𝒰q for every 𝒰q⊇𝒵A(f), that is, E∈kh𝒵A(f).Recall that for any intermediate ringA(X), Lemma 1(c) gives the relationship ℨA(f)=kh𝒵A(f). For intermediate C-rings, we can characterize ℨA topologically as 𝔎A.Theorem 6.
LetA(X) be an intermediate C-ring and f∈A(X). Then ℨA(f)=𝔎A(f).Proof.
SupposeE∈ℨA(f). By Lemma 1(c), E∈kh𝒵A(f). We show that in this case p∈Z(fυA) implies p∈clυAXE. Now if p∈Z(fυA), then by Lemma 4, 𝒰p∈hA(𝒵A(f)). Since E∈kh𝒵A(f), it follows that E∈𝒰p, and hence by (8), we have that p∈clυAXE.
For the other containment, supposeE∈𝔎A(f), that is, Z(fυA)⊆clυAXE. We show that E belongs to every A-stable z-ultrafilter containing 𝒵A(f). First, let 𝒰p be an A-stable z-ultrafilter containing 𝒵A(f). Then by Lemma 4, p∈Z(fυA), so by hypothesis, p∈clυAXE, and hence E∈𝒰p. Thus E belongs to every A-stable z-ultrafilter containing 𝒵A(f); in other words, E∈khA𝒵A(f). Since A(X) is an intermediate C-ring, we can apply Lemma 5 so that E∈kh𝒵A(f), and hence by Lemma 1(c), E∈ℨA(f).The theorem does not hold if the assumption thatA(X) being an intermediate C-ring is removed. In fact, 𝔎A need not map ideals in A(X) to z-filters on X when A(X) is not a C-ring, as the following example shows.Example 7.
LetX=[1,∞). Let A(X) be the smallest ring of continuous functions containing both C*(X) and f(x)=x. Then each function h∈A(X) has the form h(x)=∑k=0nfk(x)xk for fk∈C*(X). Note that any function h∈A(X) is therefore bounded by some function of the form Mxn for M∈ℝ and n∈ℕ, where M is n+1 times a common bound for all of fk. We now observe that ex∉A(X), since ex cannot be bounded by any function of the form Mxn for M∈ℝ and n∈ℕ. Let g=e-x. Then g is in A(X) but is not invertible in A(X). Thus A(X) and C(X) are not isomorphic.
Since every set of a freez-ultrafilter on X=[1,∞) must be unbounded, the identity function f is not bounded on any such set either. Hence any free z-ultrafilter on X is not A-stable, and υA(X)=X. Then by Lemma 2(a), A(X) is not a C-ring, since C(υA(X))=C(X) and C(X)≠A(X).
Finally, sincegυA=g and Z(g)=∅, the set 𝔎A(g)={E∈Z(X)∣Z(gυA)⊆clυAXE} consists of all zero sets of X (hence is not a z-filter), while ℨA(g) is the z-filter consisting of all zero sets in X whose complement in X has an upper bound. Thus 𝔎A(g)≠ℨA(g), which is in contrast to the conclusion of Theorem 6.We leave open the question as to precisely what ringsA(X) are such that 𝔎A=ℨA. We also leave open the question as to whether there exists a ring A(X) such that 𝔎A does map ideals in A(X) to z-filters on X, but 𝔎A≠ℨA.


## 3.2. Characterizing Maximal Ideals inC-Rings

The following characterization for maximal ideals inA(X) is proved in [6]. Every maximal ideal in A(X) is of the form
(18)MAp={f∈A(X)∣ℨA(f)⊆𝒰p},
for p∈βX. By Lemma 1(c), ℨA(f)⊆𝒰p whenever 𝒵A(f)⊆𝒰p, and hence by (9) and Lemma 1(d), we have
(19)MC*p={f∈C*(X)∣fβ(p)=0}.SinceℨC(f)=〈Z(f)〉, we have by (8) that
(20)MCp={f∈C(X)∣〈Z(f)〉⊆𝒰p}={f∈C(X)∣p∈clβXZ(f)}.The characterizations in (19) and (20) that we obtained from (18) agree with those given in [1, pages 101-102]; the latter characterization of MCp is called the Gelfand-Kolmogorov theorem [1, page 102]. The following theorem provides a characterization of MAp that we see extends both (20) and (19) to all intermediate C-rings A(X).Theorem 8.
LetA(X) be an intermediate C-ring. Then each maximal ideal in A(X) is of the form
(21)MAp={f∈A(X):p∈clβXZ(fυA)},
where p∈βX.Proof.
f
∈
M
A
p if and only if 𝒵A(f)⊆𝒰p if and only if 𝒰p∈h(𝒵A(f)). By identifying a set in βX with its image under 𝒰, it follows from the definition of closure in βX, Lemmas 4 and 5, that
(22)clβXZ(fυA)=hkZ(fυA)=hkhA𝒵A(f)=hkh𝒵A(f)=h𝒵A(f).
The result now follows from the fact that h(𝒵A(f))=clβXZ(fυA).We now verify that this theorem generalizes the Gelfand-Kolmogorov Theorem [1, page 102] to intermediate C-rings. If A(X)=C(X), then υAX=υX, the Hewitt realcompactification of X. Now using the fact that clυXZ(f)=Z(fυ) [1, page 118] and Theorem 8, we have
(23)MCp={f∈C(X)∣p∈clβXZ(fυ)}={f∈C(X)∣p∈clβXclυXZ(f)}={f∈C(X)∣p∈clβXZ(f)}.For the case whereA(X)=C*(X), we have
(24)MC*p={f∈C*(X)∣p∈clβXZ(fβ)}={f∈C*(X)∣fβ(p)=0}.
So Theorem 8 simultaneously generalizes the results of (20) for C(X) and (19) for C*(X) to all intermediate C-rings.


## 4. The MapℨA← and Ideals in A(X)

Recall thatℨA maps ideals to z-filters (Lemma 1(a)). Here we show that for an intermediate C-ring A(X), the inverse map ℨA←, defined by
(25)ℨA←[ℱ]={f∈A(X):ℨA(f)⊆ℱ}
maps z-filters on X to ideals in A(X) (Theorem 14). The corresponding result for the maps Z (for C(X)) and 𝒵A (for any intermediate ring A(X)) is proved, respectively, in [1] and [3]. Our proof for ℨA makes use of Theorem 6.We need some lemmas concerning meets and joins on the lattice ofz-filters. Recall that ℱ∨𝒢 is the smallest z-filter containing both the z-filters ℱ and 𝒢. Similarly, ℱ∧𝒢 is the largest z-filter contained in both the z-filters ℱ and 𝒢.The following lemma is from [6].Lemma 9.
LetA(X) be an intermediate ring and f,g∈A(X).(a)
𝒵A(fg)=𝒵A(f)∧𝒵A(g).(b)
𝒵A(f+g)⊆𝒵A(f)∨𝒵A(g).(c)
Iff,g≥0, then 𝒵A(f+g)=𝒵A(f)∨𝒵A(g).The following is a special case of [6, Lemma 4.2(a)].Lemma 10.
For allf,g∈A(X), kh(𝒵A(f)∧𝒵A(g))=kh(𝒵A(f))∧kh(𝒵A(g)).To obtain the analog of Lemma10 for joins, we need the following lemma.Lemma 11.
IfE is a zero set in X and if Z(fυA)∩Z(gυA)⊆clυAXE then there exist zero sets Z1 and Z2 in X such that Z(fυA)⊆clυAXZ1, Z(gυA)⊆clυAXZ2, and Z1∩Z2=E.Proof.
LetH=clυAXE and let Y=υAX∖H. Now, the sets Z(fυA)∩Y and Z(gυA)∩Y are disjoint zero sets in Y, so are contained in disjoint zero set neighborhoods W1 and W2 in Y. Moreover, since W1 is a neighborhood of Z(fυA)∩Y, it follows that
(26)Z(fυA)∩Y⊆clυAX(W1)=clυAX(W1∩X).
Similarly Z(gυA)∩Y⊆clυAX(W2∩X). Let
(27)Z1=(W1∩X)∪E,Z2=(W2∩X)∪E.
Since W1∩X is a zero set in X∖E, it follows that Z1 is a zero set in X by the fact that the cozero set of a cozero set is a cozero set [9, Proposition 1.1]. Similarly, Z2 is a zero set in X. Also, Z(fυA)⊆clυAXZ1, because
(28)Z(fυA)=(Z(fυA)∩Y)∪(Z(fυA)∩H)⊆clυAX(W1∩X)∪clυAXE=clυAX((W1∩X)∪E)=clυAXZ1.
Similarly, Z(gυA)⊆clυAXZ2. Finally, Z1∩Z2=E because W1∩X and W2∩X are disjoint.The next lemma shows that the kernel-hull operation distributes over the join operation on the lattice ofz-filters.Theorem 12.
IfA(X) is an intermediate C-ring, then
(29)kh(𝒵A(f)∨𝒵A(g))=kh(𝒵A(f))∨kh(𝒵A(g)).Proof.
The containmentkh(𝒵A(f)∨𝒵A(g))⊇kh(𝒵A(f))∨kh(𝒵A(g)) is a special case of [6, Lemma 4.2(b)].
We show that the other containment is equivalent to Lemma11 and hence must hold. First, we show the equivalence of the premises by showing that the following are equivalent. Recall that 𝒵A(f)=𝒵A(f2), and hence we can assume without loss of generality that f,g≥0.(i)
E∈kh(𝒵A(f)∨𝒵A(g)).(ii)
E∈kh(𝒵A(f+g)).(iii)
Z((f+g)υA)⊆clυAXE.(iv)
Z(fυA)∩Z(gυA)⊆clυAXE.
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 f,g≥0. This establishes the equivalence between the premises of left-to-right containment of this lemma and Lemma 11.
For the equivalence of the conclusions, note thatE∈kh(𝒵A(f))∨kh(𝒵A(g)) if and only if E is an intersection of a set in kh(𝒵A(f)) with a set in kh(𝒵A(g)). In other words, there must exist Z1∈kh(𝒵A(f)) and Z2∈kh(𝒵A(g)) such that Z1∩Z2=E. By Lemmas 4 and 5, this is equivalent to the statement that there exist zero sets Z1, Z2 such that Z(fυA)⊆clυAXZ1, Z(gυA)⊆clυAXZ2, and Z1∩Z2=E, which is the conclusion of Lemma 11.Corollary 13.
LetA(X) be an intermediate ring of continuous functions and f,g∈A(X). Then(a)
ℨA(fg)=ℨA(f)∧ℨA(g),(b)
ℨA(f+g)⊆ℨA(f)∨ℨA(g).Proof.
Item (a) immediately follows from Lemmas10 and 9(a).
Item (b) immediately follows from Theorem12 and Lemma 9(b).We are now ready to prove the main result of this section.Theorem 14.
LetA(X) be an intermediate C-ring. If ℱ is a z-filter on X, then ℨA←[ℱ] is an ideal in A(X).Proof.
Letℱ be a z-filter and I=ℨA←[ℱ]. If f∈I and g∈A(X), then ℨA(f)⊆ℱ. By Corollary 13(a), ℨA(fg)=ℨA(f)∧ℨA(g)⊆ℱ, so fg∈I. If f,g∈I, then both ℨA(f)⊆ℱ and ℨA(g)⊆ℱ. By Corollary 13(b), ℨA(f+g)⊆ℨA(f)∨ℨA(g)⊆ℱ, so f+g∈I. Thus I is an ideal.



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