a) Please describe the ring structure on $\mathbb Z \times \mathbb Z$. Does this ring have identity?
b) Describe all ring homomorphisms of $\mathbb Z \times \mathbb Z \to \mathbb Z$.
Here's what I tried:
a) The identity is $1 \times 1$, and when they say ring structure, do they mean talk about ring addition and multiplication? What else could I talk about or explain.
b) The only ring homomorphism I am able to get is to try out $\{1\times 1, 0\times 1, 0\times 0\}$, but I am unsure what these mean and why is Integers field an integral domain?