Prime and maximal ideals in subrings of C(X) (Q808465)

C(X) (respectively \(C^*(X))\) denotes the ring of all continuous (respectively all bounded continuous) real-valued functions on a Tychonov space X. This paper studies prime and maximal ideals in subrings A(X) of C(X) which contain \(C^*(X)\). Many known results about C(X) and \(C^*(X)\) follow as special cases. Central to the theme is the concept of a \({\mathfrak z}\)-ideal which plays a role analogous to that of a z-ideal in C(X). Other sections of the paper deal with the relationship between maximal ideals of A(X) and z-ultrafilters on X (Section 3), analogues of the ideals \(O^ p\) (Section 4) and in Section 5 the intersection of all free ideals and the intersection of all free maximal ideals of A(X) are characterized.