Abelian SACR-groups

A homomorphism $\mu: G \otimes G \to G$ is called a multiplication on an abelian group $G$. An abelian group $G$ with a multiplication on it is called a ring on $G$. The study of abelian groups supporting only a certain ring is one of the trends in the additive group theory. An abelian group on which every ring is associative and commutative is called an SACR-group (this abbreviation comes from: “strongly associative and commutative ring”). In this paper, we study SACR-groups in the following classes of abelian groups: homogeneous completely decomposable quotient divisible groups and indecomposable torsion-free groups of rank $2$. Together with associative and commutative rings, we are also interested in additive groups of filial rings. An associative ring in which all meta-ideals of finite index are ideals is called filial. Certainly, an associative ring $R$ is called filial if the relation of being an ideal in $R$ is transitive. An abelian group on which every associative ring is filial is called a TI-group.
In Section 1, homogeneous completely decomposable quotient divisible abelian SACR-groups are described (Theorem 7). The proof of this theorem is based on Theorem 4: every quotient divisible group of rank $1$ is an SACR-group. Further, in Section 3, it is shown that every indecomposable torsion-free group of rank 2 is an SACR-group. In particular, TI-groups are described in the class of indecomposable torsion-free abelian groups of rank $2$. It is shown that the concepts of a TI-group and a nil-group in the class of rank $2$ torsion-free indecomposable groups are equivalent. Until now, all known torsion-free TI-groups are SACR-groups. However, the converse is not true; an example is given in Section 3.