The above topology is defined and studied on C(X), the ring of real-valued continuous functions on a completely regular HausdorfT space X. The minimal ideals and the socle of C(X) are characterized via their corresponding z-filters. We observe that these ideals are z-ideals and X is discrete if and only if the socle of C(X) is a free ideal. It is also shown that for a class of topological spaces, containing all P-spaces, the family Ck{X) of functions with compact support is identical with the intersection of the free maximal ideals of C(X). Introduction. Clearly every maximal ideal in any commutative ring with unity either intersects every nonzero ideal nontrivially or else is generated by an idempotent (in this case we call it isolated; see [10-13] for more details). The existence of isolated maximal ideals of C(X) is equivalent to the existence of isolated points in X, which plays a fundamental role in the study of some important topological spaces (see [7, 22, 14]). In this paper we study the intrinsic topology on C(X) and relate the density of the set of isolated points in X to some algebraic properties of C(X). The minimal ideals and the socle of C(X), which is the sum of all the minimal ideals (see [1]), are characterized via their corresponding 2-filters. The equality of Gfc(X), the set of functions with compact support, with the intersection of the free maximal ideals was first proved for discrete spaces by Kaplansky [9], who asked if the equality holds in general. It is well known that the equality may fail in general and Kohls [15], Gillman-Jerison [4], and Robinson [20] have proved the result for P-spaces, realcompact spaces, and spaces admitting a complete uniform structure, respectively. We add a new class of topological spaces, larger than P-spaces, to these. The realcompactness of a discrete space with cardinality less than or equal to c (the cardinality of the reals) is proved in a very simple way. In this paper we denote by X a completely regular HausdorfT space. If / G G = C(X), then Z(f) denotes the set of zeros of /. If / is an ideal of C, then Z[I] — {Z(f): f G 1} is a 2-filter. The Stone-Cech compactification of X is denoted by ßX, and by |X| we mean the cardinality of X. The reader is referred to [4] for undefined terms and notations. 1. Isolated points and isolated maximal ideals. We begin with the following well-known result. PROPOSITION 1.1. A point x G X is isolated if and only if Mx = {f G C: f(x) = 0} is an isolated maximal ideal of C. Received by the editors March 29, 1983 and, in revised form, February 29, 1984. 1980 Mathematics Subject Classification. Primary 54C40.

https://www.ams.org/proc/1985-093-01/S0002-9939-1985-0766552-9/S0002-9939-1985-0766552-9.pdf

On the intrinsic topology and some related ideals of $C(X)$

It is shown thatX is finite if and only ifC(X) has a finite Goldie dimension. More generally we observe that the Goldie dimension ofC(X) is equal to the Souslin number ofX. Essential ideals inC(X) are characterized via their corresponding z-filters and a topological criterion is given for recognizing essential ideals inC(X). It is proved that the Frechet z-filter (cofinite z-filter) is the intersection of essential z-filters. The intersection of idealsOx wherex runs through nonisolated points inX is the socle ofC(X) if and only if every open set containing all nonisolated points is cofinite. Finally it is shown that if every essential ideal inC(X) is a z-ideal thenX is a P-space.

Essential ideals inC(X)

Let X be a completely regular space, and let A(X) be a subal- gebra of C(X) containing C*{X). We study the maximal ideals in A(X) by associating a filter Z(f) to each / 6 A(X). This association extends to a one- to-one correspondence between M(A) (the set of maximal ideals of A(X)) and sX. We use the filters Z(f) to characterize the maximal ideals and to describe the intersection of the free maximal ideals in A(X). Finally, we outline some of the applications of our results to compactifications between vX and sX. 1. Introduction. The algebra C(X) of continuous real-valued functions on a completely regular space X and its subalgebra C*(X) of bounded functions have been studied extensively (see Gillman and Jerison (3), and Aull (1)). One of the interesting problems considered in (3) is that of characterizing the maximal ideals in these two algebras. It is a remarkable fact that the distinct problems of identifying the maximal ideals in C(X) and C*(X) have a common solution—the maximal ideals are in one-to-one correspondence with the points of sX in a natural way. The methods of achieving this correspondence, however, are quite different in the two cases. In this paper we consider this problem for subalgebras A(X) of C(X) that contain C*(X). We show that for such algebras the maximal ideals are in one-to-one correspondence with sX. The correspondence we construct reduces to that in (3) for the cases of C(X) and C*(X). Thus our result puts in a common setting these apparently distinct problems. A function is invertible in C(X) if it is never zero, and in C*(X) if it is bounded away from zero. In an arbitrary A(X), of course, there is no such description of invertibility which is independent of the structure of the algebra. Thus in §2 we associate to each noninvertible / E A(X) a z-filter Z(f) that is a measure of where / is "locally" invertible in A(X). This correspondence extends to one between maximal ideals of A(X) and z-ultrafilters on X. In §3 we use the filters Z(f) to describe the intersection of the free maximal ideals in any algebra A(X). Finally, our main result allows us to introduce the notion of A(X)-compactness of which compactness and realcompactness are special cases. In §4 we show how the Banach-Stone theorem extends to A(X)-compact spaces. 2. The structure space. Throughout this paper X will denote a completely regular Hausdorff space and A(X) a subalgebra of C(X) containing C*(X). In this section we construct the correspondence mentioned in the introduction. A zero set in X is a set of the form Z(f) = {x E X: f(x) = 0} for some / E C(X). The complement of a zero set is a cozero set. Z(X) will denote the

/pdf/maximal-ideals-in-subalgebras-of-c-x-22mel6vsyw.pdf

Maximal ideals in subalgebras of c(x)

Let X be a completely regular Hausdorff space, C(X) the ring of real-valued continuous functions on X, CK the ideal of functions with compact support, I the intersection of the free maximal ideals of C(X), and Cψ the ideal of functions with pseudocompact support. For any space, CK ⊆ I ⊆ Cψ. When CK = I, or I = Cψ, or CK = Cψ , it is said that X is μ-compact, η-compact, or ψ-compact, respectively. These ideals and spaces are characterized in terms of the ideal structure of C(X) and the topological structure of βX. Also, μ-, η-, and ψ-compactifications are constructed. Examples and counterexamples are given.

Functions with pseudocompact support

Prime and maximal ideals in subrings of C(X)

Gillman and Jerison have shown that when A" is a realcompact space, every function in C(X) that belongs to all the free maximal ideals has compact support. A space with the latter property will be called fi-compact. In this paper we give several characterizations of /?-compact spaces and also introduce and study a related class of spaces, the ^-compact spaces ; these are spaces X with the property that every function in C(X) with pseudocompact support has compact support. It is shown that every realcompact space is ^-compact and every i/i-compact space is /?-compact. A family & of subsets of a space X is said to be stable if every function in C(X) is bounded on some member of #". We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.

/pdf/supports-of-continuous-functions-1yx6g7cb5o.pdf

Mark Mandelker

Papers

Supports of continuous functions