Cayley-Hamilton theorem

Statement

Let $R$ be a commutative unital ring, and $I\subset R$ be an ideal. Let $M$ be a $R$-module that can be generated by $n$ elements.

if $\phi$ is an endomorphism of $M$ such that $\phi (M)=IM$, then there exists a monic polynomial:

$p(x)=x^{n}+p_{1}x^{n-1}+p_{2}x^{n-2}+\ldots +p_{n}$

such that $p(\phi )=0$ as an endomorphism of $M$ and such that each $p_{j}\in I^{j}$.