Localization of a polynomial ring at a maximal ideal

Question

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.

Answer 1

$$\frac{R[X]}{mR[X]+(X)}\simeq\frac{R[X]/(X)}{mR[X]+(X)/(X)}\simeq R/m$$

$R[X]_N$ 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.)

Answer 2

For the first question, it is easy to see that $N$ is indeed an ideal. The elements consists of sums of products of the form $p(x)m+rx$ where $p(x)$ is a polynomial in $R[x]$, $m$ is an element of $M$, and $r$ is an element of $R$.

Now I claim that the quotient $R[x]/N$ is isomorphic to $R/M$, which is a field (since $M$ is maximal). Here's a hint: note that $x \in N$, so that $x=0$ in the quotient ring.

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For the other part, you probably want to localize at $N$ and not $M$ (otherwise there should be plently of maximal ideals).