prime ideals of polynomial rings over a local ring

Question

Let $(R,\ m)$ be a local commutative ring with identity, and $R[x]$ be the polynomial ring over $R$. Is the following true?

The prime ideals of $R[x]$ above $m$ are of the following form:
1. $mR[x]$ and
2. $\langle m, q\rangle R[x]$ for each polynomial $q\in R[x]$ which is irreducible modulo $m$.

If it is true, is there any proof?

Answer 1

The prime ideals of $R[x]$ above $m$ correspond to the prime ideals of $S=R[x]/mR[x]$. The ring $S\cong k[x]$ where $k=R/m$. The prime ideals of a polynomial ring over a field are the zero ideal, and the ideals generated by irreducibles.

So the prime ideals of $R[x]$ above $m$ are $mR[x]$ and $mR[x]+\left <f\right>$ where $f$ is irreducible modulo $m$.