Relations between Cayley-Hamilton theorem and the classification theorem for finitely generated modules over a PID

Question

There is a sentence in the book "Algebra Chapter 0" that the Cayley-Hamilton theorem would be evident if we connect it with the classification theorem for finitely generated modules over a PID. I don't think I fully understand this sentence. Given an integral domain $R$ and its free module $F$, let $\alpha\in End_{R}(F)$, I guess $\{f\in R[t]|f(\alpha)=0\}$ would somehow be related to annihilator ideal of a finite generated module, and $P_{\alpha}(t)$ is related to the characteristic ideal, and since the characteristic ideal is contained in the annihilator, we have $P_{\alpha}(t)\in \{f\in R[t]|f(\alpha)=0\}$. Unfortunately, I don't have a good idea about that. Can someone help me with that?

Thank you!

Answer 1

I find that actually section VI.7 of this book "Algebra Chapter 0" has somehow answered this question.