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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.2011-04-11
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    @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.2011-05-11

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