For some finitely many nonzero prime ideals, the contraction and extension of their product is zero

Question

I was reading P.M. Eakin's thesis paper, The converse to a well known theorem on Noetherian Rings. The following is taken from Theorem 2, page 281 of that paper, and that's where I'm stuck.

Let $R$ be a ring and $S$ a finite integral overring of $R$. Suppose that $S$ is Noetherian and not an integral domain. If every prime ideal of $S$ contracts to a nonzero prime ideal of R, then for some nonzero prime ideals $P_{1},P_{2}, ... , P_{n},$ of $R$, $(P_{1}...P_{n})^{ec}=0$, where $e$ and $c$ are extension and contraction of ideals, respectively.

I have no idea how to prove it. Any suggestions?

Answer 1

Hint. In a noetherian ring every ideal contains a product of prime ideals.