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.

String Numbers of Abelian Groups

“…In particular, if (A, +) is a cotorsion-free abelian group (i.e., (A, +) does not contain any copy of Q, Z(p), or J p for every prime p), then there exist cotorsion-free abelian groups G of arbitrarily large order such that End(G) ∼ = A (just take A = Z \ {0} in [6, Theorem (6.3)]). Finally, using arguments similar to those in the proof of [17,Theorem 2.11], one can show that for such G all φ ∈ End(G) are injective.…”

The Hopfian exponent of an abelian group

“…To deduce the remaining part we appeal to realization theory: it is well known that such a ring R can be realized as the endomorphism ring of a torsion-free group G of arbitrary large cardinality-see Corner and Göbel [5]. Moreover, the fact that R is a domain ensures that every nonzero endomorphism of G is monic-see the proof of [10,Theorem 2.11]. In particular, if r ∈ R is regular and rg = 0, then certainly g = 0 so that G is torsion-free as a left R-module.…”

On Transitivity-like Properties for Torsion-Free Abelian Groups

“…Mimicking [11], we state the following important concept: Definition 1.3. A group G is called weakly transitive if, for any pair of elements x, y ∈ G and endomorphisms φ, ψ of G such that φ(x) = y, ψ(y) = x, there exists an automorphism η of G with η(x) = y.…”