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 \textsf{absolute 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.