Prove that $I[x]$ is not maximal in $R[x]$.

Question

I've asked a lot tonight, but I'm grinding through my HW assignment and it's kicking my butt. This is my final problem, and I've managed to only struggle on 3 of the 13 problems, which I think is nice, but more to the point!

$R$ is a commutative ring with unity, and $I$ is some ideal of $R$. I'm asked to show that $I[x]$ is not maximal in $R[x]$.

I'm honestly lost, and I think I need the condition that $I$ is maximal. Is this true?

Thanks!

Answer 1

Assume that $I[x]$ is maximal. Then it is easy to see that $I$ is maximal. But then $R/I$ is a field. On the other hand, $R[x]/I[x]$ is also a field. But since we have $R[x]/I[x] \cong (R/I)[x]$ and the polynomial ring over a field is never a field (for example, $x$ is non-zero and not invertible), we have a contradiction.