TITLE: Localization of a polynomial ring at a maximal ideal
QUESTION [5 upvotes]: Let $R$ be a regular local Noetherian ring, with maximal ideal $M$. Show that $N=R[x]M+(x)$ is a maximal ideal in the polynomial ring $R[x]$, and that the localization $R[x]_N$ is again regular local.
I have no idea how to attempt this problem. Any hints would be highly appreciated.

REPLY [3 votes]: $$\frac{R[X]}{mR[X]+(X)}\simeq\frac{R[X]/(X)}{mR[X]+(X)/(X)}\simeq R/m$$
$R[X]_N$ is regular since $R[X]$ is regular and localizations of regular rings are regular. (Alternatively, notice that $X\notin (NR[X]_N)^2$, and $R[X]_N/XR[X]_N\simeq R_m=R$ is regular.)