Absolute ideals of algebraically compact Abelian groups

An absolute ideal of an Abelian group $G$ is a subgroup that is an ideal in every ring whose additive group coincides with $G$. We describe reduced algebraically compact Abelian groups $G$ that admit at least one ring structure $R$ such that every ideal of $R$ is an absolute ideal of $G$ (Problem 93 in L. Fuchs' book “Infinite Abelian Groups”). Reduced, algebraically compact, Abelian groups that have only fully invariant subgroups as absolute ideal are characterized.