The following claim is made in Massey's Singular Homology Theory: If $G$ is any abelian group, and $F$ is a free abelian group with basis $\{T_\alpha\}$, then every element of $G \otimes F$ can be written uniquely as $a_1 \otimes T_1 + \cdots + a_l \otimes T_k$, for $T_1,\ldots,T_k$ basis elements and $a_1,\ldots,a_k \in G$.
I can see why every element of $G \otimes F$ can be written in this form, but I can't prove uniqueness. Even in the easiest case $F = \mathbb{Z}$ this is unclear: Why does $x \otimes 1 = 0$ imply $x = 0$? Handling this case will also cover the more general situation since tensor products distribute over direct sums.