Adelta;_n-Free modules over complete discrete valuation domains with almost trivial dual

A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the â? n-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large â? n-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.