Co-local subgroups of abelian groups II

In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25–33] the notion of a co-local subgroup of an abelian group was introduced. A subgroup KK of AA is called co-local if the natural map Hom(A,A)→Hom(A,A/K) is an isomorphism. At the center of attention in [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25–33] were co-local subgroups of torsion-free abelian groups. In the present paper we shift our attention to co-local subgroups KK of mixed, non-splitting abelian groups AA with torsion subgroup t(A)t(A). We will show that any co-local subgroup KK is a pure, cotorsion-free subgroup and if D/t(A)D/t(A) is the divisible part of A/t(A)=D/t(A)⊕H/t(A)A/t(A)=D/t(A)⊕H/t(A), then K∩D=0K∩D=0, and one may assume that K⊆HK⊆H. We will construct examples to show that KK need not be a co-local subgroup of HH. Moreover, we will investigate connections between co-local subgroups of AA and A/t(A)A/t(A).