Correctness of Abelian torsion-free groups and determinability of Abelian groups by their subgroups

An Abelian group $A$ is called correct if for any Abelian group $B$ isomorphisms $A\cong B'$ and $B\cong A'$, where $A'$ and $B'$ are subgroups of the groups $A$ and $B$, respectively, imply the isomorphism $A\cong B$. We say that a group $A$ is determined by its subgroups (its proper subgroups) if for any group $B$ the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups $A$ and $B$ such that corresponding subgroups are isomorphic implies $A\cong B$.
In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of divisible torsion-free groups and completely decomposable groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained.