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It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with j3X. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C'{X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible / £ A(X), we define a z-filter Z>.A(/) on X which, in a sense, provides a measure of where / is 'locally invertible'. We show that the map ZA generates a correspondence between ideals of A(X) and z-filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of fiX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions. INTRODUCTIONIf X is a completely regular space then the collection C(X) of continuous realvalued functions on X is a ring under pointwise operations. The ring C(X) and its subring C*(X), consisting of the bounded elements of C{X), have been studied extensively (see [4] and [1]). The resulting theory of 'rings of continuous functions' is beautifully presented in the now classic text of Gillman and Jerison by that title. One of the main achievements in this theory is the characterisation of the maximal ideals in the rings C(X) and 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. This

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