Prime and Maximal ideals in the polynomial rings

Question

If $P$ is a prime ideal of $R[x]$, then I can prove that $P\cap R$ is a prime ideal of $R$. Does the same result follow in case of maximal ideals? I did not find any such thing anywhere. I tried to produce a counterexample to show that this is not true in case of maximal ideals, but failed. Please help!

Answer 1

No. Let $R$ be the ring $\mathbb{Z}_{(2)}$ of rational numbers whose denominators in lowest terms are odd. In $R[x]$, the ideal $(2x - 1)$ is maximal because it is the kernel of a surjective ring homomorphism

$$\mathbb{Z}_{(2)}[x] \to \mathbb{Q}$$

to a field, namely the one given by the obvious inclusion on $\mathbb{Z}_{(2)}$ and sending $x$ to $\frac{1}{2}$. Its intersection with $R$ is $(0)$, which is prime but not maximal because $R$ is an integral domain but not a field.