Prime and maximal ideals in subrings of C( X)

Abstract

The structure of ideals in the ring C( X) of continuous functions on a completely regular space X and its subring C ∗(X) consisting of the bounded functions is well known. In this paper we study the prime and maximal ideals in subrings A( X) of C( X) that contain C ∗(X) . We show that many of the results known separately for C( X) and C ∗(X) , often by different methods, are true for any such A( X). Our results put the problems of C( X) and C ∗(X) in a common setting by exhibiting these as special instances of the subrings A( X). We characterize prime and maximal ideals in any A( X) in terms of their residue class rings and in terms of certain z-filters on X that correspond to these ideals. We also characterize the intersection of the free ideals and the free maximal ideals in any A( X).