Is there a ring homomorphism between $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$? Is there a ring homomorphism between $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$?
Also for any rings $R_{1}$ and $R_{2}$ does there exist a ring homomorphism  $\phi$ : $R_{1} \rightarrow$ $R_{1} \times R_{2}$?
Note: I am allowing 1 in both rings. So any homomorphism must map the identity on one ring to the identity on the other ring.

For any ring $R$ there is a unique homomorphism $\phi:\mathbb{Z}\to R$ defined $\phi(1)=1_R$. This is unique because then $\phi(n)=n\cdot 1_R$ for all $n\in\mathbb{Z}$. The category theoretic term for this is that $\mathbb{Z}$ is the initial object in the category of rings.
