Combinatorial Principles and ﬡ_k-free Modules
For a natural number k > 1 and a ring A with free additive structure, we realize two different constructions of arbitrarily large ﬡ_k-free A-modules which are separable as abelian groups. The main tool to construct these modules is a new variant of Shelah's Black Box principle, which is provable in ZFC. In the first case, we take A = Z and construct an ﬡ_k-free abelian group G with no epimorphisms onto a free abelian group of countably infinite rank. In the second case, we take A to be a separably realizable ring and construct an ﬡ_k-free A-module G with the prescribed endomorphism ring End G = A \oplus Fin G, where Fin G is the ideal of all endomorphisms of G whose images have finite rank.