I have trouble solving the exercise 4.14 in Eisenbud's book "Commutative algebra with a view toward algebraic geometry". The statement is the following
Let $R$ be a domain and $I$ be an ideal of $R$. Prove that an element $s\in R$ is integral over $I$ if and only if there is a finitely generated $R$-module $N$ which is not annihilated by any non-zero element of $R$, and such that $sN\subset IN$.
I recall that an element $s\in R$ is said to be integral over $I$ if there is $n\geq 1$ and elements $r_j \in I^j$ for $j=1,\ldots ,n$ such that $$s^n + r_1s^{n-1}+\cdots+r_n=0$$
I was able to prove the sufficient condition using Cayley-Hamilton theorem as stated previously in the book. However, I failed to prove the necessary condition. Namely, given an integral element $s$, I can't find which module $N$ would work. Could anybody give me a hint in order to conclude this exercise ?