homomorphisms over direct sum I am trying to compute some homology groups and for that I need to figure out,
what are all homomorphisms from $\mathbb Z \oplus \mathbb Z$ into $\mathbb Z$. 
I would really appreciate any effort.

$\mathbf Z \oplus \mathbf Z \cong \mathbf Z^2$ is a free abelian group. So you can choose the images of the standard base vectors $e_1=(1,0)$ and $e_2=(0,1)$ of $\mathbf Z^2$ under a homomorphism in any abelain group in an arbitrary way. So any homomorphism from $\mathbf Z^2$ to $\mathbf Z$ is a follows:
$$
\varphi( x_1 e_1+x_2 e_2)=x_1 k + x_2 m
$$
where $k,m $ are fixed integers, the images of $e_1$ and $e_2,$ respectively. 
