Lemma 10.35.2. Let $\varphi : R \to S$ be a ring map. Let $y \in S$. If there exists a finite $R$-submodule $M$ of $S$ such that $1 \in M$ and $yM \subset M$, then $y$ is integral over $R$.

**Proof.**
Let $x_1 = 1 \in M$ and $x_ i \in M$, $i = 2, \ldots , n$ be a finite set of elements generating $M$ as an $R$-module. Write $yx_ i = \sum \varphi (a_{ij}) x_ j$ for some $a_{ij} \in R$. Let $P(T) \in R[T]$ be the characteristic polynomial of the $n \times n$ matrix $A = (a_{ij})$. By Lemma 10.15.1 we see $P(A) = 0$. By construction the map $\pi : R^ n \to M$, $(a_1, \ldots , a_ n) \mapsto \sum \varphi (a_ i) x_ i$ commutes with $A : R^ n \to R^ n$ and multiplication by $y$. In a formula $\pi (Av) = y\pi (v)$. Thus $P(y) = P(y) \cdot 1 = P(y) \cdot x_1 = P(y) \cdot \pi ((1, 0, \ldots , 0)) = \pi (P(A)(1, 0, \ldots , 0)) = 0$.
$\square$

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