Fully inert subgroups of completely decomposable finite rank groups and their commensurability

A subgroup $H$ of an Abelian group $G$ is said to be fully inert in $G$ if the subgroup $H\cap\varphi H$ has a finite index in $\varphi H$ for any endomorphism $\varphi$ of the group $G$. Subgroups $H$ and $K$ of the group $G$ are said to be commensurable if the subgroup $K\cap H$ has a finite index in $H$ and in $K$. Some properties of fully inert and commensurable groups in the context of direct decompositions of the group and operations on subgroups are proved. For example, if a subgroup $H$ is commensurable with a subgroup $K$, then $H$ is commensurable with $H\cap K$ and with $H + K$; if a subgroup $H$ is commensurable with a subgroup $K$, then the subgroup $fH$ is commensurable with $fK$ for any homomorphism $f$. The main result of the paper is that every fully inert subgroup of a completely decomposable finite rank torsion-free group $G$ is commensurable with a fully invariant subgroup if and only if types of rank $1$ direct summands of the group $G$ are either equal or incomparable, and all rank $1$ direct summands of the group $G$ are not divisible by any prime number $p$.