Zip rings (resp. czip rings) and some applications

A commutative ring with identity is called a zip ring if each of its faithful ideals contains a finitely generated faithful ideal. A natural generalization of this notion is to require every faithful ideal to contain a countably generated faithful ideal. This is what we do in this paper. If a ring has this property, we call it a czip ring. Clearly, every zip ring is a czip ring. We give several characterizations of czip rings, including some in terms of subspaces of the space of minimal prime ideals with the Zariski topology. In rings of continuous functions we characterize the Tychonoff spaces X for which C(X) is a czip ring. This enables us to show that the class of czip rings properly contains that of zip rings.