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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?

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a) What are the (additive) generators? And what does "does this ring have identity" mean? b) The way I learned it, ring homomorphisms must map the additive and multiplicative identities to themselves (e.g. $1 \times 1$ to $1$). This might influence the possible ring homomorphisms -- I suspicion that there are no ring homomorphisms. – Michael Chen Apr 11 '11 at 6:03
@Michael Chen: Whether ring homomorphisms between rings with identity must map 1 to 1 or not is a matter of convention; I would not assume it either way and request clarification when not explicitly indicated. – Arturo Magidin May 11 '11 at 15:39

Firstly you are correct, the multiplicative identity of the ring is $(1,1)$. The question is to describe the multiplicative and additive structure of the ring.

As for $(b)$, once you've described the multiplicative and additive structure what is $(0,1)\cdot(1,0)$? What is $(0,1) + (1,0)$? Knowing this and the fact stated be Michael Chan, that a homomorphism sends $(1,1)$ in $\mathbb{Z} \times \mathbb{Z}$ to $1$ in $\mathbb{Z}$, you can say what it does on every element of the ring.

Note that $\mathbb{Z}$ is not a field. What does it mean for a ring to be an integral domain? If you take a product of two integers $n$ and $m$, such that $nm=0$, what can you say about them?

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