Small almost free modules with prescribed topological endomorphism rings. (Q2569823)

Summary: We will realize certain topological rings as endomorphism rings of \(\aleph_1\)-free Abelian groups of cardinality \(\aleph_1\) where the isomorphism is also a homeomorphism relating the topology on the given ring to the finite topology on the endomorphism ring. This way we also find \(\aleph_1\)-free Abelian groups of cardinality \(\aleph_1\) such that any non-trivial summand is a proper direct sum of an infinite number of summands. This answers a problem raised by the authors [in Proc. Lond. Math. Soc., III. Ser. 50, 447-479 (1985; Zbl 0562.20030), p. 447, (1)], (saying that \(\aleph_1\in\text{vat}(R)\) in the cotorsion-free case). The fact that the size of the continuum could and in particular universes of set theory will be much larger then \(\aleph_1\) causes difficulties in constructing pathological Abelian groups \(G\) of size \(\aleph_1\) in ``ordinary'' set theory using just ZFC: There are less possibilities to prevent potential, unwanted endomorphisms of \(G\) not to become members of \(\text{End\,}G\). Thus additional combinatorial arguments are needed. They come from [\textit{R. Göbel, S. Shelah}, Can. J. Math. 50, No. 4, 719-738 (1998; Zbl 0959.20049)], were improved for this paper, and are now ready for applications for other algebraic aspects. The results are formulated for modules over a large class of commutative rings.