Show $\Bbb R$ and $\Bbb R[X]$ not isomorphic

Question

Why are the rings $\mathbb{R}$ and $\mathbb{R}[ x ]$ not isomorphic to eachother ?

Think it might have to do with multiplicative inverses but I'm not sure.

Answer 1

You are right: the element "$x$" has no multiplicative inverse, that is there is no polynomial $p(x)$ such that $x\cdot p(x)=1$.

Answer 2

Well $\Bbb R$ is a field while $\Bbb R[X]$ is not.

Question: if a ring is ring-isomorphic to a field, is it necessary a field?