PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 93, Number 1, January 1985 ON THE INTRINSIC TOPOLOGY AND SOME RELATED IDEALS OF C(X) O. A. S. KARAMZADEH AND M. ROSTAMI ABSTRACT. 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 idempo- tent (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 uni- form 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 fol- lowing 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. Keywords and phrases. Isolated maximal ideals, intrinsic topology, P-space, minimal ideal, socle, real pseudo-finite space, P-ideal, z-ideal. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 179 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use