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.

As for the second question:
There is no homomorphism $\mathbb Z_5\rightarrow \mathbb Z_5\times \mathbb Z$.
This is because the image must be isomorphic to a quotient of $\mathbb Z_5$, and is therefore finite. On the other hand it contains $n(1,1)$ for every $n$, this gives us an infinite number of elements.
