Show that the rings Z[x] and Z are not isomorphic

[Mr Davis 97]

Homework Statement

Show that the rings Z[x] and Z are not isomorphic

Homework Equations

The Attempt at a Solution

I want to show that these are not isomorphic. The thing is that I already know that $\mathbb{Z}/(x) \cong \mathbb{Z}$, but for some reason I can't find specific structural properties of $\mathbb{Z}[x]$ that are different than $\mathbb{Z}$

 

[fresh_42]

$\mathbb{Z}/(x) \cong \mathbb{Z}$?

 

[Mr Davis 97]

$\mathbb{Z}/(x) \cong \mathbb{Z}$?

I meant to write $\mathbb{Z}[x]/(x) \cong \mathbb{Z}$

 

[fresh_42]

What if you assumed an isomorphsm $\varphi\, : \,\mathbb{Z}[x] \longrightarrow \mathbb{Z}$. Then $\varphi((x))$ is an ideal in $\mathbb{Z}$, say $\varphi((x)) = n\mathbb{Z}$ with $\varphi(x)=n$. What do you get if you factor this ideal on both sides?

 

[fresh_42]

Homework Statement

Show that the rings Z[x] and Z are not isomorphic

Homework Equations

The Attempt at a Solution

I want to show that these are not isomorphic. The thing is that I already know that $\mathbb{Z}/(x) \cong \mathbb{Z}$, but for some reason I can't find specific structural properties of $\mathbb{Z}[x]$ that are different than $\mathbb{Z}$

Another idea is to use the fact that $\mathbb{Z}[x]$ is no PID, e.g. $I:=\langle 2, 1+x \rangle$.