Maximal Ideals in Subalgebras of C(X)

Abstract

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 ßX. 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 ßX.

Essential ideals in subrings of C(X) that contain C*(X)

“…In other words, Z A ( f ) consists of those zero sets such that f is locally invertible in A(X) on their complements. In [24], Redlin and Waston defined Z…”

“…It is well known that any maximal ideal of A(X) is of the form M p A , where p ∈ βX and M p A = { f ∈ A(X) : p is cluster point of Z A ( f ) in βX}, see [24,Theorem 5]. We also note that a maximal ideal in A(X) is free if and only if it is of the form M p A for some p ∈ βX \ X. H.L.…”

“…Also, they have been investigated as intermediate algebras by H.L. Byun, L. Redlin and S. Watson in [6], [7] and [24], and also [22] and [26]. In 1997, J.M.…”

“…We show that in A(X) the intersection of all essential minimal prime ideals coincides with Soc(C(X)). The map Z A was introduced in [24], and its properties were further investigated in [6], [7] and [26]. In [22], another correspondence Z A between ideals of A(X) and z-filters on X is introduced.…”

Local invertibility in subrings of C*(X)

“…Although the problems of characterising the maximal ideals in C(X) and C*(X) are quite different, they have the same solution-namely, the set of maximal ideals in each is in one-to-one correspondence with /3X, the Stone-Cech compactification of X. In [6], a method was developed to characterise the maximal ideals in any ring A(X) of continuous functions on X with A(X) D C*(X), and in [2] the method was used to study deeper properties of these rings. The technique of 'local invertibility' used in these papers has the desirable effect of putting the problems of characterising the maximal ideals in C{X) and C*(X) into a common setting.…”