I am breaking off a discussion with Mariano from here
I have a map $\mathbb{Z} \stackrel{\left(\begin{array}{c}2\\2\end{array}\right)}{\to} \mathbb{Z} \oplus \mathbb{Z} $
This is the map $(a) \mapsto (2a,2a)$. I thought that the image of this map was just $2 \mathbb{Z} \oplus 2\mathbb{Z}$. But to quote Mariano:
"the image of a map $\mathbb{Z}\to \mathbb{Z}\oplus \mathbb{Z}$ is either zero or a subgroup of rank 1, and $2\mathbb{Z}\oplus 2\mathbb{Z}$ has rank two."
I am trying to understand this further (I guess both Mariano's comment, and what the image would actually be)