Bézout domains and lattice-valued modules

Let B be a commutative Bézout domain and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class Mod-B of all (right) B-modules. We analyze the definable sets in terms, on the one hand, of the definable sets in the classes Mod-BM, where BM ranges over the localizations of B at M, M ∈ MSpec(B), and on the other hand, of the constructible subsets of MSpec(B). This allows us to derive decidability results for the class Mod-B, in particular when B is the ring Z of algebraic integers or one of the rings Z ∩ R, Z ∩ Qp.