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

Abstract

Suppose L is a complete lattice containing no copy of the power-set 2ω and no uncountable well-ordered chains. It is shown that for any family of nonempty subsets $$X_1 \subseteq L (i \in A)$$ , one can choose elements p i ∈X i so that ∨ A p 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.