Abelian groups with finite primary quotients

An abelian group $A$ is called $\pi$-bounded for a set of prime numbers $\pi$, if all $p$-primary components $t_{p}(A/B)$ are finite for every subgroup $B\subset A$ and for every $p\in\pi$. E. V. Sokolov has introduced the class of $\pi$-bounded groups investigating $\mathcal{F}_{\pi}$-separable and $\pi^\prime$-isolated subgroups in the general group theory. The description of torsion $\pi$-bounded groups is trivial. E. V. Sokolov has proved that the description of mixed $\pi$-bounded groups can be reduced to the case of torsion free groups.
We consider the class of $\pi$-bounded torsion free groups in the present paper and we prove that this class of groups coincides with the class of $\pi$-local torsion free abelian groups of finite rank. We consider also abelian groups satisfying the condition $(\ast)$, that is such groups that their quotient groups don't contain subgroups of the form $\mathbb{Z}_{p^{\infty}}$ for all prime numbers $p\in\pi$, where $\pi$ is a fixed set of prime numbers. It is clear that all $\pi$-bounded groups satisfy the condition $(\ast)$. We prove that an abelian group $A$ satisfies the condition $(\ast)$ if and only if both groups $t(A)$ and $A/t(A)$ satisfy the condition $(\ast)$. We construct also an example of a non-splitting mixed group of rank $1$, satisfying the condition $(\ast)$, for every infinite set $\pi$ of prime numbers.