Lemma 29.46.3. Let $A \subset B$ be a ring extension. Let $S \subset A$ be a multiplicative subset. Let $n \geq 1$ and $b_ i \in B$ for $1 \leq i \leq n$. If the set

$\{ x \in S^{-1}B \mid x \not\in S^{-1}A\text{ and } b_ i x^ i \in S^{-1}A\text{ for }i = 1, \ldots , n\}$

is nonempty, then so is

$\{ x \in B \mid x \not\in A\text{ and } b_ i x^ i \in A\text{ for }i = 1, \ldots , n\}$

Proof. Omitted. Hint: clear denominators. $\square$

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