Multiplications on mixed abelian groups

A multiplication on an abelian group $G$ is a homomorphism $\mu: G\otimes G\rightarrow G$. An mixed abelian group $G$ is called an $MT$-group if every multiplication on the torsion part of the group $G$ can be extended uniquely to a multiplication on $G$. $MT$-groups have been studied in many articles on the theory of additive groups of rings, but their complete description has not yet been obtained. In this paper, a pure fully invariant subgroup $G^*_\Lambda$ is considered for an abelian $MT$-group $G$. One of the main properties of this subgroup is that $\bigcap\limits_{p \in \Lambda (G)}pG^*_\Lambda$ is a nil-ideal in every ring with the additive group $G$ (here $\Lambda (G)$ is the set of all primes $p$, for which the $p$-primary component of $G$ is non-zero). It is shown that for every $MT$-group $G$ either $G=G^*_\Lambda$ or the quotient group $G/G^*_\Lambda$ is uncountable.