Pullbacks of arithmetical rings

We give necessary and sufficient conditions that the pullback of a conductor square he a chain ring (i.e., a ring whose ideals are totally ordered by inclusion). We also give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (i.e., a ring which is locally a chain ring at every maximal ideal). For any integral domain D with field of fractions K, we characterize all Prufer domains R between D[X] and K[X] such that the conductor C of K[X] into R is nonzero. As an application, we show that for n >= 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.