Transversals of families in complete lattices,and torsion in product modules

Suppose L is a complete lattice containing no copy of the power-set 2 and no uncountable well-ordered chains. It is shown that for any family of nonempty subsets , one can choose elements p_iX_i so that _Ap_i majorizes all elements of all but finitely many of the X_i. Ring-theoretic consequences are deduced: for instance, the direct product of a family of torsion modules over a commutative Noetherian integral domain R is torsion if and only if some element of R annihilates all but finitely many of the modules.