The Nullstellensatz says that any ideal $I \subset \mathbb{C}[x_1,\dots,x_n]$ has the property that
$\sqrt{I} = \bigcap_{\text{maximal } \mathfrak{m} \supset I} \mathfrak{m}$
Is it also true that we can write any ideal as above as
$I = \bigcap_{\text{maximal } \mathfrak{m} \supset I} \mathfrak{q}_{\mathfrak{m}}$
for certain $\mathfrak{m}$-primary ideals $\mathfrak{q}_{\mathfrak{m}}$? Note that I'm only allowing a single primary ideal for each maximal ideal $\mathfrak{m}$; if I allowed countably many then the problem would be trivial.
I suppose my question applies to any Jacobson ring instead of $\mathbb{C}[x_1,\dots,x_n]$ .