  In the class of reduced Abelian torsion-free groups $G$ of finite rank, we
describe TI-groups, this means that every associative ring on $G$ is filial. If
every associative multiplication on $G$ is the zero multiplication, then $G$ is
called a $nil_a$-group. It is proved that a reduced Abelian torsion-free group
$G$ of finite rank is a $TI$-group if and only if $G$ is a homogeneous Murley
group or $G$ is a $nil_a$-group. We also study the interrelations between the
class of homogeneous Murley groups and the class of $nil_a$-groups. For any
type $t\ne (\infty,\infty,\ldots)$ and every integer $n>1$, there exist
$2^{\aleph_0}$ pairwise non-quasi-isomorphic homogeneous Murley groups of type
$t$ and rank $n$ which are $nil_a$-groups. We describe types $t$ such that
there exists a homogeneous Murley group of type $t$ which is not a
$nil_a$-group. This paper will be published in Beitr\"{a}ge zur Algebra und
Geometrie / Contributions to Algebra and Geometry.
