The following is a part of proposition $5.1$ in An Introduction to Commutative Algebra by Atiyah&Mcdonald. First let $A\subset B$ be commutative rings with identity and $b\in B$.
If there is a faithful $A[b]$-module $M$ such that $M$ is finitely generated as an $A$-module, then $b$ is integral over $A$.
I cannot see how the faithfulness of $M$ is used, so could anyone please give a counterexample when $M$ is not faithful? The proof in the book uses Cayley-Hamilton with $\phi$ to be the right multiplication by $b$. Thank you.