In the proof of Proposition 1.9 in Chapter VII of Algebra by Serge Lang, it seems to me that the following property is used.
Let $A$ be a commutative entire ring, $S$ a multiplicative subset of $A$, $0 \not \in S$. Let $\alpha$ be an element of the quotient field of $A$. If $s \alpha \in A$ for some $s \in S$, then $\alpha \in S^{-1}A$.
(in a proof that $A$ integrally closed implies $S^{-1}A$ integrally closed).
Is this bold statement true ?
Since $\alpha$ is expressed as $\alpha=a/s'$, where $a, s' \in A, s' \neq 0$, $s \alpha \in A$ implies that $s a = s' b$ for some $b \in A$.
From this, how can I prove that $s' \in S$ ?
Any help would be appreciated.