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

$p(x)=x^n+p_1{x}^{n-1}+p_2{x}^{n-2}+\dots +p_n$

such that $p(\varphi )=0$ as an endomorphism of $M$ and such that each $p_j\in I^j$.