Is $\mathbb Z_3[x]$ isomorphic with $\mathbb Z$?

Question

Is $\mathbb Z_3[x]$ isomorphic with $\mathbb Z$ ?

(This question arose in trying to determine whether there is a commutative ring $R$ with unity such that $R[x]\cong\mathbb Z$ . It is easy to see that if such a ring exists then $R$ must be a field with two units i.e. $R \cong \mathbb Z_3$ )

Please help . Thanks in advance

Answer 1

It's not isomorphic to $\mathbb Z$ because it has characteristic 3.

Answer 2

$\Bbb Z_3[x]$ has elements $x$ such that $x+x+x=0$ (in particular, $x=1_3$), but this is not true for $\Bbb Z$. An isomorphism would have to send $1_3 \mapsto 1$, and then we'd have to have $3=0$ in $\Bbb Z$.

Answer 3

$\mathbb{Z}$ is not isomorphic to any ring of the form $R[x]$. In particular, note that every nonzero element in $\mathbb{Z}$ can be written as either:

$$ 1+1+...+1$$ Or $$(-1)+(-1)+...+(-1)$$

But in a polynomial ring, the element $p(x)=x$ cannot be written in either form.