Co-local subgroups of abelian groups II

Abstract

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 K of A 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 K of mixed, non-splitting abelian groups A with torsion subgroup t ( A ) . We will show that any co-local subgroup K is a pure, cotorsion-free subgroup and if D / t ( A ) is the divisible part of A / t ( A ) = D / t ( A ) ⊕ H / t ( A ) , then K ∩ D = 0 , and one may assume that K ⊆ H . We will construct examples to show that K need not be a co-local subgroup of H . Moreover, we will investigate connections between co-local subgroups of A and A / t ( A ) .