Originally Posted by

**AlexP** My goal was to find all homomorphisms $\displaystyle \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$, and $\displaystyle \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}$, and $\displaystyle \mathbb{Q} \to \mathbb{Q}$. Here's what I have...

For $\displaystyle \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$ I believe they are $\displaystyle (a,b) \mapsto 0$, $\displaystyle (a,b) \mapsto (a,b)$, $\displaystyle (a,b) \mapsto (b,a)$, $\displaystyle (a,b) \mapsto (a,0)$, and $\displaystyle (a,b) \mapsto (0,b)$. (a,b) → (a,a) ? (a,b) → (b,b) ?

For $\displaystyle \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}$ I have the identity, the 0-map, $\displaystyle (a,b) \mapsto a$, and $\displaystyle (a,b) \mapsto b$.

And for $\displaystyle \mathbb{Q} \to \mathbb{Q}$ I just have the 0-map and the identity.

Did I get them all? And when there are more 'creative' ones like $\displaystyle (a,b) \mapsto (b,a)$, how do we know when we've found them all?