Nk-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 N-k-free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is N-k-free if every subset of size < N-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 almost free and admits many decompositions, we are able to control the endomorphism ring End G of its additive structure in a strong way: we are able to find arbitrarily large G with End G = A circle plus Fin G (so End G = Fin G = A, where Fin G is the ideal of End G of all endomorphisms of finite rank) and a special choice of A permits interesting separable N-k-free abelian groups G. This result includes as a special case the existence of non-free separable N-k-free abelian groups G (e.g. with End G = Z circle plus Fin G), known until recently only for k = 1.