Absolute nil-ideals of Abelian groups

It is known that in an Abelian group $G$ that contains no nonzero divisible torsion-free subgroups the intersection of upper nil-radicals of all the rings on $G$ is $\bigcap_ppT(G)$, where $T(G)$ is the torsion part of $G$. In this work, we define a pure fully invariant subgroup $G^*\supseteq T(G)$ of an arbitrary Abelian mixed group $G$ and prove that if $G$ contains no nonzero torsion-free subgroups, then the subgroup $\bigcap_ppG^*$ is a nil-ideal in any ring on $G$, and the first Ulm subgroup $G^1$ is its nilpotent ideal.