Let $R$ be a Noetherian ring, and let $\mathfrak p$ be a prime of $R$ of codimension $d$. Suppose that $P\subset R[X]$, $P$ prime, intersects $R$ in $\mathfrak p$. Prove that if $P\neq\mathfrak pR[X]$, then codim $P=d+1$ and that there are infinitely many $P$ like this.
For the part about codimension (= height) see Theorem 149 from Kaplansky, Commutative Rings, 1974.
The second part can be easily proved by reduction modulo $\mathfrak p$: reduce all by $\mathfrak p$, get an integral domain $R$ and try to prove that there are infinitely many primes in $R[X]$ lying over $(0)$. These are in a one-to-one correspondence with the prime ideals of $K[X]$, where $K$ is the field of fractions of $R$. Now try to prove that $K[X]$ has infinitely many primes, that is, there are infinitely many irreducible polynomials in $K[X]$.