# Ring homomorphism

1. Sep 20, 2011

### burak100

1. The problem statement, all variables and given/known data

$\mathbb{Z}[x]/(x^3-x) \rightarrow \mathbb{Z}$

Show that is ring homomorphism, and count the number of homomorphism..?

2. Relevant equations

3. The attempt at a solution

the map $f$ is homomorphism if,

$f(x+y)=f(x)+f(y)$
$f(xy)=f(x)f(y)$

I think, I must find a map for the question , but how should I choose the map, I don't know....

2. Sep 20, 2011

### Robert1986

I'm a little confused, here. I don't think that the matter is as simple as finding a homomorphism as I believe mapping everything from Z[x]/(x^3 + x) to 0 is a homomorphism, isn't it? I think that the main point of this is to count how many homomorphisms there are into Z. Now, one idea might be to use the first isomorphism theorem. Do you see how this might be useful?