Strongly homogeneous torsion-free Abelian groups (Q791668)

All groups in this paper are abelian. A torsion-free group G is called strongly homogeneous if for any rank 1 pure subgroups A and B of G there exists \(\alpha \in Aut G\) such that \(\alpha A=B\). An associative ring R with identity is called strongly homogeneous if every element of R is an integral multiple of a unit. Dave Arnold completely classified finite rank strongly homogeneous groups [\textit{D. M. Arnold}, Proc. Am. Math. Soc. 56, 67-72 (1976; Zbl 0358.20060)]. This paper considers strongly homogeneous groups of arbitrary rank. Let G be a strongly homogeneous torsion free group. Then the center C of the endomorphism ring E(G) is a strongly homogeneous ring, and \(G\cong F\otimes_ ZA,\) where F is a C-module all whose countable rank submodules are free; A is a rank 1 torsion-free group of the same type t(G) as G. A countable torsion-free group G is strongly homogeneous if and only if \(G\cong F\otimes_ ZA,\) where F is a finitely or countably generated free module over some strongly homogeneous torsion-free E-ring T, A is a rank 1 torsion-free group, such that \(pA=A\) implies \(pF=F\) for any prime p. The center C of E(G) is isomorphic to T.