Countably Generated Ideals in Rings of Continuous Functions

1. Results. Scattered results about countably2 generated ideals in C(X) are established in [2] and [4] : Op is countably generated if and only if pEX and p has a countable base of neighborhoods; Op is both prime and countably generated if and only if Mp is principal, and if and only if pEX and p is isolated; no lower prime ideal is countably generated. We generalize these as follows: if Mp is countably generated, then pEX and p is isolated (5.4) ; no free prime ideal is countably generated (4.5); a fixed, nonmaximal prime ideal can be countably generated (6.2), but not if it is a z-ideal (6.1);3 the intersection of a countable chain of lower prime ideals, or of a strictly decreasing sequence of arbitrary prime ideals, cannot be countably generated (6.3, 6.4). Some of the above are corollaries of much more general theorems: any free, countably generated ideal is contained in 2C hyper-real maximal ideals (4.4) ; if the fixed ideal of all functions vanishing on a compact set F is countably generated, then F is an open set (5.2). These theorems, in turn, can be formulated more generally. In some of the results, the hypothesis that an ideal is countably generated can be replaced by the weaker hypothesis that its z-filter be countably generated.