integral over proof

Question

Let $S$ be a sub ring of a commutative ring $R$ let $y$ be in $R$. Prove that if $y$ is integral over $S$, then $S[y]$ is integral over $S$. I'm puzzled as how to start this one.

Answer 1

You should use the following Lemma, which can be proved using the Cayley-Hamilton Theorem:

An element $x \in R$ is integral over $S$, if and only if there is a faithful $S[x]$-module which is finitely generated as $S$-module.

(We say that an $A$-module $M$ is faithful, if for every $0 \neq a \in A$ there is an $z \in M$ such that $a \cdot z \neq 0$)