Find all homomorphism $\Phi : \mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z}$

Question

Knowing $\mathbb{Z}$ is cyclic can I infer that $\mathbb{Z} \oplus \mathbb{Z}$ is cyclic aswell?

I know I can assume this statement to be true for finite cyclic groups but is there a infinite case to this theorem?

How can I find all homomorphisms beside the trivial one: $\phi \equiv 0$?

Answer 1

Any such will be determined by $h(1,0)=m$ and $h(0,1)=n$, and vice-versa, since $\Bbb Z\oplus \Bbb Z=\langle (1,0),(0,1)\rangle $.

Thus $\operatorname {hom}(\Bbb Z\oplus\Bbb Z,\Bbb Z)\cong \Bbb Z\oplus \Bbb Z$.

See here.

Note: $\Bbb Z\oplus\Bbb Z$ isn't cyclic, since its homomorphic image $\Bbb Z_2\oplus\Bbb Z_2$ isn't.

Answer 2

On account of the universal property of the direct sum, you get $\operatorname{hom}(\mathbb Z\oplus\mathbb Z, \mathbb Z)\simeq \operatorname{hom}(\mathbb Z, \mathbb Z)\times \operatorname{hom}(\mathbb Z, \mathbb Z)\simeq \mathbb Z\times \mathbb Z$. Here the last isomorphism is obtained by sending a homomorphism $\phi\colon \mathbb Z\to\mathbb Z$ to $\phi(1)$.