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In Communications in Algebra, Volume 33, Issue 4, 2004, Pages 1177 – 1191 DOI: 10.1081/AGB-200053836 http://www.informaworld.com/smpp/content~db=all~content=a725341666?words=goldsmith&hash=3234069153

Abstract

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ϕ, ψ ∈ End(G) such that xϕ = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.

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