I encountered the following very interesting proposition in Atiyah's and McDonald's Introduction to Commutative Algebra:
Let $A$ be commutative ring with identity, $M$ a finitely generated $A$-module and $\phi$ an endomorphism of $M$. Then there exists positive integer $n$ and $\alpha_0, \dots, \alpha_{n-1} \in A$ such that $\phi^n + \alpha_{n-1} \phi^{n-1} + \cdots + \alpha_1 \phi + \alpha_0=0$.
The proof that is given is short and based on an argument involving the determinant of a matrix. I don't find it though very insightful algebraically. Does anybody know any alternative proof? I tried induction on the number of generators, but i have difficulty completing the induction step.