ABSTRACT
Abstract Dualizing the notion of a localization of an abelian group, we call a subgroup K = {0} of the abelian group G a co-local subgroup if the natural map σ : Hom(G, G) → Hom(G, G/K ) is an isomorphism, i.e., Hom(G, K ) = 0 and each ϕ ∈ Hom(G, G/K ) is induced by some (unique) ϕ′ ∈ Hom(G, G). While purely indecomposable abelian groups and torsion groups have no colocal subgroups, many co-purely indecomposable groups do have completely decomposable colocal subgroups. If K is a co-local subgroup of a reduced, torsion-free abelian group A, then K is cotorsion-free and a pure subgroup of A. We show that each cotorsion-free group K is isomorphic to a co-local subgroup of some cotorsion-free group G.
