# Sh:1045

• Shelah, S. Quite free Abelian groups with prescribed endomorphism ring. Preprint.
• Abstract:
In [Sh:1028] we like to build Abelian groups (or R-modules) which on the one hand are quite free, say \aleph_{\omega +1}-free, and on the other hand, are complicated in suitable sense. We choose as our test problem having no non-trivial homomorphism to \mathbb Z (known classically for \aleph_1-free, recently for \aleph_n-free). We get even \aleph_{\omega_1 \cdot n}-free. Other applications were delayed to the present work.

The construction (there and here) requires building n-dimensional black boxes, which are quite free. Here we continue [Sh:1028] (in some ways). In particular, we consider building quite free Abelian groups with a pre-assigned ring of endomorphism.

In §2 we try to elaborate [Sh:1028, §(2B)], on “minimal" {\rm Hom}(G,{}_R R), e.g. for separable p-groups. In §(3C), §(3D) are some old continuation of [Sh:1028, §3], so of unclear status. In §(4A) we control {\rm End}(G,R,+), e.g. =R. In §(4B), done when we have a 1-witness; seems nearly done. In §(4C) we intend to fill the 4-witness case.

Presently (2014.1.13) it seems that §(4A), §(4B) do something, construct from a 1-witness, which §(4C) has to be completed. Also §(2A) do something, not clear how final. In SEPT 2017 lecture on it in Simonfest; have added in the end of sec 1, analysis of the simple case- R is co-torsion free

@article{Sh:1045,
}