Quite free Abelian groups with prescribed endomorphism ring

by Shelah. [Sh:1045]

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 Z (known classically for aleph_1-free, recently for aleph_n-free). We get even aleph_{omega_1 . 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 section 2 we try to elaborate [Sh:1028, section(2B)], on ``minimal'' Hom (G,{}_R R), e.g. for separable p-groups. In section(3C), section(3D) are some old continuation of [Sh:1028, section 3], so of unclear status. In section(4A) we control End (G,R,+), e.g. =R . In section(4B), done when we have a 1-witness; seems nearly done. In section(4C) we intend to fill the 4-witness case. Presently (2014.1.13) it seems that section(4A), section(4B) do something, construct from a 1-witness, which section(4C) has to be completed. Also section(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

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