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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.

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    I should note that in Atiyah/McDonald, they use the word 'adjoint' in a way that we don't usually today. When they say 'adjoint,' they in fact mean the 'adjugate.' This makes their proof more meaningful (it confused me when I first read it)2012-04-17

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