TORSION-FREE WEAKLY TRANSITIVE ABELIAN GROUPS

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.

Key Words:

• Full transitivity
• Transitivity

Mathematical Subject Classification:

• 20K01
• 20K10
• 20K30

ACKNOWLEDGMENTS

Supported by the Deutsche Forschungs Gesellschaft. The authors would like to thank Dani for her beautiful typing of this manuscript. We would like to thank Chris Meehan for suggesting the examples 3.25, 3.26, and 3.27.

Notes

^#Communicated by K. Rangaswamy.