Lemma 115.5.1. Let $R$ be a ring and let $\varphi : R[x] \to S$ be a ring map. Let $t \in S$. If $t$ is integral over $R[x]$, then there exists an $\ell \geq 0$ such that for every $a \in R$ the element $\varphi (a)^\ell t$ is integral over $\varphi _ a : R[y] \to S$, defined by $y \mapsto \varphi (ax)$ and $r \mapsto \varphi (r)$ for $r\in R$.
Proof. Say $t^ d + \sum _{i < d} \varphi (f_ i)t^ i = 0$ with $f_ i \in R[x]$. Let $\ell $ be the maximum degree in $x$ of all the $f_ i$. Multiply the equation by $\varphi (a)^\ell $ to get $\varphi (a)^\ell t^ d + \sum _{i < d} \varphi (a^\ell f_ i)t^ i = 0$. Note that each $\varphi (a^\ell f_ i)$ is in the image of $\varphi _ a$. The result follows from Algebra, Lemma 10.123.1. $\square$
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