TITLE: Example of an ideal $I$ that is maximal in $R$ but not maximal in $R[x].$
QUESTION [5 upvotes]: I was reading Dummit and Foote and the following remark showed up:

Note that it is not true that if $I$ is a maximal ideal of $R$ then $I[x]$ is a maximal ideal of $R[x]$. 

If $I$ is maximal in $R$ then $R/I$ is a field and this means that $R[x]/I[x]$ is a field since $R[x]/I[x]\cong (R/I)[x].$ Beyond this, I am not sure how I can come up with an example of this claim.

REPLY [6 votes]: Take $\mathbb{R}$, $0$ is the maximal ideal of since $\mathbb{R}$ is a field. But $0$ is not maximal ideal in $\mathbb{R}[X]$.

REPLY [6 votes]: Since $R[x]/I[x]\cong (R/I)[x]$ and polynomial rings are never fields, $I[x]$ is never maximal in $R[x]$.