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

[burak100]

Homework Statement

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

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

Homework Equations

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...

 

[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?