### ${\aleph}_{k}$-free separable groups with prescribed endomorphism ring

We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of ${\aleph}_{k}$-free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is ${\aleph}_{k}$-free if every subset of size $<{\aleph}_{k}$ is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a module G is...