Absolute Ideals of Almost Completely Decomposable Abelian Groups

We consider the class ${\mathcal{A}}_0$ of Abelian block-rigid $CRQ$-groups of ring type. A subgroup $A$ of an Abelian group $G$ is called an $\mathsf{\text{absolute}}\mathsf{\text{ideal}}$ of the group $G$ if $A$ is an ideal in any ring on $G$. We describe principal absolute ideals of groups in ${\mathcal{A}}_0$. This allows to prove that any group in ${\mathcal{A}}_0$ is an $afi$-group, i.e., a group $G$ such that any absolute ideal of $G$ is a fully invariant subgroup.