Let $R$ be an integral domain such that every non zero prime ideal is maximal in $R[x]$. Choose the correct statement.
a) $R$ is a Field.
b) $R$ contains $\mathbb Z$ as a subring
c) $R$ contains $F_p$ as a subring for some prime $p$
Now I think c) is false clearly because taking $R$ as $\mathbb Q$ then c is out. But do not know further how to look for others?
Indeed a is the correct choice. If $R$ is not a field, then it contains a nonzero proper ideal $I$. $I+(x)$ is then a proper ideal of $R[x]$, while $(x)$ is a prime ideal properly contained in $I+(x)$.