א_n-free modules with trivial duals

In the first part of this paper we introduce a simplified version of a new Black Box from Shelah [11] which can be used to construct complicated א_n-free abelian groups for any natural number n ∈ ℕ. In the second part we apply this prediction principle to derive for many commutative rings R the existence of א_n-free R-modules M with trivial dual M* = 0, where M* = Hom(M,R). The minimal size of the א_n-free abelian groups constructed below is {square superset}_n, and this lower bound is also necessary as can be seen immediately if we apply GCH.