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

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}##

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

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

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?

Mr Davis 97

Homework Statement

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

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##.

Mr Davis 97