PROBLEM: Let $A$ be a Noetherian domain that is an integral extension of $\mathbb{Z}[x]$ and $J$ be a non-zero ideal of $A$. Show that if $I=J\cap \mathbb{Z}[x]$ has height 2, then $J$ has a single primary decomposition.
Attempt: Since $A$ is an integer extent of $\mathbb{Z}[x]$, I got $$\dim (A) = \dim (\mathbb{Z}[x]) = 2 = ht(I).$$ I think the biggest problem is not knowing for sure what I have to prove to ensure that $J$ admits of a single primary decomposition. At first I tried to prove that all components of the decomposition were single primes. I don't know if that fixes it. If anyone can help me, I would be very grateful. Thank you very much in advance!