Consider the collection $C$ of all ideals $I$ of $R$ such that $\sqrt{I}$ is not a finite intersection of prime ideals. Now using Zorn's lemma $C$ has a maximal element and that won't be prime. Continuing it seems we can get that radical of any ideal $I$ is a finite intersection of prime ideals. Where is my mistake?
I have seen this result stated with Noetherian hypothesis, but the above indicated proof does not use that hypothesis.
Yes, it is.
Let $R=K^{\mathbb N}$ be the ring of sequences with elements in a field $K$. Then $R$ is reduced, and every prime ideal of $R$ is maximal. Now suppose that $(0)=\bigcap_{i=1}^nP_i$. Then, by CRT we get that $R\simeq\prod_{i=1}^nR/P_i$, that is, $R$ is isomorphic to a finite direct product of fields, so $R$ is noetherian, a contradiction.