Checking the maximality of a principal ideal in $R[x]$

Question

Let $R = \mathbb{Z}_{(2)}$ be the localization of $\mathbb{Z}$ at the prime ideal generated by $2$ in $\mathbb{Z}$. Then prove that the ideal generated by $(2x-1)$ is maximal in $R[x]$.

Answer 1

Hint. Prove that $R[x]/(2x-1)\simeq R[\frac12]$ and $R[\frac12]=\mathbb Q$.

Answer 2

The above hint is correct. Sending $x$ to $1/2$ we get a surjective map from $R[x]$ to $\mathbb Q$ and then look kernel of this homomorphism that will be exactly $(2x-1)$. So we are done.