Lemma 10.123.5. In Situation 10.123.4. Suppose $u \in S$, $a_0, \ldots , a_ k \in R$, $u \varphi (a_0 + a_1x + \ldots + a_ k x^ k) \in J$. Then there exists an $m \geq 0$ such that $u \varphi (a_ k)^ m \in J$.
Proof. Assume that $S$ is generated by $t_1, \ldots , t_ n$ as an $R[x]$-module. In this case $J = \{ g \in S \mid gt_ i \in \mathop{\mathrm{Im}}(\varphi )\text{ for all }i\} $. Note that each element $u t_ i$ is integral over $R[x]$, see Lemma 10.36.3. We have $\varphi (a_0 + a_1x + \ldots + a_ k x^ k) u t_ i \in \mathop{\mathrm{Im}}(\varphi )$. By Lemma 10.123.3, for each $i$ there exists an integer $n_ i$ and an element $q_ i \in R[x]$ such that $\varphi (a_ k^{n_ i}) u t_ i - \varphi (q_ i)$ is integral over $R$. By assumption this element is in $\varphi (R)$ and hence $\varphi (a_ k^{n_ i}) u t_ i \in \mathop{\mathrm{Im}}(\varphi )$. It follows that $m = \max \{ n_1, \ldots , n_ n\} $ works. $\square$
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