Research Article | Open Access
Joshua Sack, Saleem Watson, "Characterizations of Ideals in Intermediate -Rings via the -Compactifications of ", International Journal of Mathematics and Mathematical Sciences, vol. 2013, Article ID 635361, 6 pages, 2013. https://doi.org/10.1155/2013/635361
Characterizations of Ideals in Intermediate -Rings via the -Compactifications of
Let be a completely regular topological space. An intermediate ring is a ring of continuous functions satisfying . In Redlin and Watson (1987) and in Panman et al. (2012), correspondences and are defined between ideals in and -filters on , and it is shown that these extend the well-known correspondences studied separately for and , respectively, to any intermediate ring. Moreover, the inverse map sets up a one-one correspondence between the maximal ideals of and the -ultrafilters on . In this paper, we define a function that, in the case that is a -ring, describes in terms of extensions of functions to realcompactifications of . For such rings, we show that maps -filters to ideals. We also give a characterization of the maximal ideals in that generalize the Gelfand-Kolmogorov theorem from to .
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 between ideals of and -filters on X as described in [1, pages 26-27]. Such a correspondence also exists for [1, Problem 2L]. In , a correspondence between the ideals of any and the -filters on was introduced, and its properties were further investigated in [3–5]. In , another correspondence between ideals of any and -filters on is introduced. It is shown in  that the correspondences and extend the correspondences and 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 for intermediate -rings in terms of the -compactifications of introduced in . In this setting, we show (Theorem 14) that the inverse map of the set map 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].
For convenience we state some of the definitions and results needed in this paper.
Following the notation in , we set to be the collection of the zero sets 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, then denotes the zero set of on the larger domain.
A -filter (-ultrafilter, resp.) on is the intersection of with a filter (ultrafilter, resp.) on . The kernel of a set of -ultrafilters is defined by
One can verify that the kernel of a set of -ultrafilters is a -filter. The hull of a -filter 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 on . We just say is -regular if is understood by context. For any set , let be the complement of in .
Lemma 1. Let be an intermediate ring. Then the following hold.(a)For any noninvertible , both and are -filters on . If is invertible, then , the set of all zero sets in . (b)For an ideal , both and are -filters on . (c)For any , we have . (d)If is a -filter on , then if and only if for all .
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 . 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 be the map which associates every with a -ultrafilter : such that for each , is the fixed -ultrafilter containing , and for each is a unique free -ultrafilter, such that 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 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 maps homeomorphically onto the subspace of fixed -ultrafilters, and hence is a subspace of .
A -filter on is called -stable if for every there exists a set in on which is bounded (see [7, 8]). Following , 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 . If , then is the Hewitt realcompactification.
To refine our understanding of the topology on , we define the -stable hull of a -filter by It is immediate from the definition of the subspace topology that is homeomorphic (via ) to the hull-kernel topology restricted to -stable -ultrafilters, that is, the topology with the following closure operator: the closure of any set is . It follows that
From , 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 , 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 be an intermediate ring. Then the following hold. (a) is a -ring if and only if is isomorphic to .(b)If is a -ring, then is invertible in if and only if .
In addition, it is shown in [5, Theorem 4.7] that there is a bijective correspondence between the realcompactifications of and the -rings on .
3. Characterizations Using Realcompactifications
In this section, we utilize the realcompactifications of to provide a new description of the function 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.
3.1. A New Description of for -Rings
While the definition of is essentially algebraic (using the property of local invertibility), we now define a function 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 coincides with when is an intermediate -ring.
Definition 3. Let be an intermediate ring and . We set For an ideal , we set .
That 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 when is an intermediate -ring. We also show that 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 , let us now observe a similarity between and a simple function that we define in terms of . We define on by We drop the subscript when it is understood by context. It is easy to see that given an ideal .
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, is the zero set of the function , where ). Observe that where the first equality is immediate from the fact that , and the second equality holds because the sets of the form are a base for the closed sets in (see [1, page 94]). This motivates the following definition that relates to : We then generalize to all intermediate rings to arrive at Definition 3.
In order to prove our main theorem that 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 .
Lemma 4. If 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 if and only if .
Proof. We observe that the following are equivalent:(i), (ii) is -stable and , (iii) is -stable and for all , (iv) is -stable and , that is, . The equivalence (i)(ii) follows from (6) and (9). The equivalence (ii)(iii) follows from the fact that is -stable, and hence is bounded on some set in . The equivalence (iii)(iv) follows from Lemma 1(d).
We use the notation to denote the kernel of the hull of the -filter , that is, the intersection of the set of -ultrafilters containing .
Lemma 5. Let be an intermediate -ring and . Then
Proof. Without loss of generality, we may assume that (because ). Of course . For the other containment, suppose , so belongs to every -stable -ultrafilter containing . We first note that by Lemma 4, if and only if . In other words, for any , we have if and only if for every in if and only if . Now suppose is any -ultrafilter (-stable or not) containing . We show that belongs to . If not, then there exists such that Since and are disjoint zero sets in , they are completely separated (see [1, page 17]). So there is a continuous function that takes the value on and on and , so . Consider the function . Note that for . Also, since on and since , it follows that . By Lemma 2(b), is invertible, and as on , it follows that on . Since , it also follows that , but this contradicts the fact that because of Lemma 1(d). This contradiction stems from the assumption that does not belong to . So for every , that is, .
Recall that for any intermediate ring , Lemma 1(c) gives the relationship . For intermediate -rings, we can characterize topologically as .
Theorem 6. Let be an intermediate -ring and . Then .
Proof. Suppose . By Lemma 1(c), . We show that in this case implies . Now if , then by Lemma 4, . Since , it follows that , and hence by (8), we have that .
For the other containment, suppose , that is, . We show that belongs to every -stable -ultrafilter containing . First, let be an -stable -ultrafilter containing . Then by Lemma 4, , so by hypothesis, , and hence . Thus belongs to every -stable -ultrafilter containing ; in other words, . Since is an intermediate -ring, we can apply Lemma 5 so that , and hence by Lemma 1(c), .
The theorem does not hold if the assumption that being an intermediate -ring is removed. In fact, need not map ideals in to -filters on when is not a -ring, as the following example shows.
Example 7. Let . Let be the smallest ring of continuous functions containing both and . Then each function has the form for . Note that any function is therefore bounded by some function of the form for and , where is times a common bound for all of . We now observe that , since cannot be bounded by any function of the form for and . Let . Then is in but is not invertible in . Thus and are not isomorphic.
Since every set of a free -ultrafilter on 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 , the set consists of all zero sets of (hence is not a -filter), while is the -filter consisting of all zero sets in whose complement in has an upper bound. Thus , which is in contrast to the conclusion of Theorem 6.
We leave open the question as to precisely what rings are such that . We also leave open the question as to whether there exists a ring such that does map ideals in to -filters on , but .
3.2. Characterizing Maximal Ideals in -Rings
Since , we have by (8) that
The characterizations in (19) and (20) that we obtained from (18) agree with those given in [1, pages 101-102]; the latter characterization of is called the Gelfand-Kolmogorov theorem [1, page 102]. The following theorem provides a characterization of that we see extends both (20) and (19) to all intermediate -rings .
Theorem 8. Let be an intermediate -ring. Then each maximal ideal in is of the form where .
We now verify that this theorem generalizes the Gelfand-Kolmogorov Theorem [1, page 102] to intermediate -rings. If , then , the Hewitt realcompactification of . Now using the fact that [1, page 118] and Theorem 8, we have
4. The Map and Ideals in
Recall that maps ideals to -filters (Lemma 1(a)). Here we show that for an intermediate -ring , the inverse map , defined by maps -filters on to ideals in (Theorem 14). The corresponding result for the maps (for ) and (for any intermediate ring ) is proved, respectively, in  and . Our proof for makes use of Theorem 6.
We need some lemmas concerning meets and joins on the lattice of -filters. Recall that is the smallest -filter containing both the -filters and . Similarly, is the largest -filter contained in both the -filters and .
The following lemma is from .
Lemma 9. Let be an intermediate ring and .(a). (b). (c)If , then .
The following is a special case of [6, Lemma 4.2(a)].
Lemma 10. For all , .
To obtain the analog of Lemma 10 for joins, we need the following lemma.
Lemma 11. If is a zero set in and if then there exist zero sets and in such that , , and .
Proof. Let and let . Now, the sets and are disjoint zero sets in , so are contained in disjoint zero set neighborhoods and in . Moreover, since is a neighborhood of , it follows that Similarly . Let Since is a zero set in , it follows that is a zero set in by the fact that the cozero set of a cozero set is a cozero set [9, Proposition 1.1]. Similarly, is a zero set in . Also, , because Similarly, . Finally, because and are disjoint.
The next lemma shows that the kernel-hull operation distributes over the join operation on the lattice of -filters.
Theorem 12. If is an intermediate -ring, then
Proof. The containment is a special case of [6, Lemma 4.2(b)].
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 , and hence we can assume without loss of generality that .(i). (ii). (iii). (iv).
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 . 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 that if and only if is an intersection of a set in with a set in . In other words, there must exist and such that . By Lemmas 4 and 5, this is equivalent to the statement that there exist zero sets , such that , , and , which is the conclusion of Lemma 11.
Corollary 13. Let be an intermediate ring of continuous functions and . Then(a), (b).
We are now ready to prove the main result of this section.
Theorem 14. Let be an intermediate -ring. If is a -filter on , then is an ideal in .
Joshua Sack was partially supported by The Netherlands Organisation for Scientific Research VIDI Project 639.072.904.
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Copyright © 2013 Joshua Sack and Saleem Watson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.