"Tensoring a sequence with a direct sum of $\mathbb{Z}'s$ produces just a direct sum of copie sof the given sequence"
This is a quote from page 265 of hatcher. This statement is made in order to hlep show that $Tor(A,B) = 0$ if $B$ is free. Can somebody show me the details in proving this statement? Also any insight in general about tensoring with $\mathbb{Z}$ is greatly appreciated. Thanks!
There is a natural isomorphism $\mathbb Z \otimes_{\mathbb Z} A \cong A$ given by $n \otimes a \mapsto na$ with the inverse $a \mapsto 1 \otimes a$. Prove that each mapping is well-defined using the universal property of the tensor product, and then verify that they are inverses of each other. This should be straightforward.
Once done, construct a similar isomorphism $A \otimes_{\mathbb Z} (B \oplus C) \cong (A \otimes_{\mathbb Z} B) \oplus (A \otimes_{\mathbb Z} C)$.
Note that these isomorphisms aren't specific to $\mathbb Z$. Any commutative ring with unity would do.