Torsion modules, lattices and p-points

Answering a long-standing question in the theory of torsion modules, we show that weakly productively bounded domains are necessarily productively bounded. Moreover, we prove a twin result for the ideal lattice L of a domain equating weak and strong global intersection conditions for families (X_i)_{i in I} of subsets of L with the property that bigcap_{i in I} A_i not= 0 whenever A_i in X_i. Finally, we show that, for domains with Krull dimension (and countably generated extensions thereof), these lattice-theoretic conditions are equivalent to productive boundedness.