Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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
    
@user8917: please do NOT just tag a question as homework and nothing else. The homework tag is meant to be used in conjunction with other tags. –  Willie Wong Apr 11 '11 at 11:04
    
@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

1 Answer 1

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?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.