On a property of Abelian groups related to direct sums and products

Let $T$ be an infinite set of prime numbers, $\mathcal M$ be a set of groups $\{\mathbb Z(p)\mid p \in T\}$. An Abelian group $A$ is said to be $\mathcal M$-large if
$$ \mathrm{Hom}\Bigl(A,\bigoplus_{p\in T}\mathbb Z(p)\Bigr)=\mathrm{Hom}\Bigl(A,\prod_{p\in T}\mathbb Z(p)\Bigr). $$
This paper presents a characterization of $\mathcal M$-large torsion-free and mixed groups.