Fully inert subgroups of divisible Abelian groups

A subgroup H of an Abelian group G is said to be fully inert if the quotient (H + phi(H)/H is finite for every endomorphism phi of G. Clearly, this is a common generalization of the notions of fully invariant, finite and finite-index subgroups. We investigate the fully inert subgroups of divisible Abelian groups, and in particular, those Abelian groups that are fully inert in their divisible hull, called inert groups. We prove that the inert torsion-free groups coincide with the completely decomposable homogeneous groups of finite rank and we give a complete description of the inert groups in the general case. This yields a characterization of the fully inert subgroups of divisible Abelian groups.