Filial rings on direct sums and direct products of torsion-free abelian groups

A ring whose additive group coincides with an abelian group $G$ is called a ring on $G$. An abelian group $G$ is called a $TI$-group if every associative ring on $G$ is filial. If every (associative) ring on an abelian group $G$ is an $SI$-ring (a hamiltonian ring), then $G$ is called an $SI$-group (an $SI_H$-group). In this article, $TI$-groups, $SI_H$-groups and $SI$-groups are described in the following classes of abelian groups: almost completely decomposable groups, separable torsion-free groups and non-measurable vector groups. Moreover, a complete description of non-reduced $TI$-groups, $SI_H$-groups and $SI$-groups is given. This allows us to only consider reduced groups when studying $TI$-groups.