The Stacks project

Lemma 115.5.3. Let $\varphi : R \to S$ be a ring map. Suppose $t \in S$ satisfies the relation $\varphi (a_0) + \varphi (a_1)t + \ldots + \varphi (a_ n) t^ n = 0$. Let $J \subset S$ be an ideal such that for at least one $i$ we have $\varphi (a_ i) \not\in J$. Then there exists a $u \in S$, $u \not\in J$ such that both $u$ and $ut$ are integral over $R$.

Proof. This is immediate from Lemma 115.5.2 since one of the elements $u_ i$ will not be in $J$. $\square$

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