I'm trying to understand a proof that $f:A\to B$ is integral implies $S^{-1}f:S^{-1}A\to S^{-1}B$ is integral. Here $S$ is a multiplicative subset of $A$.
Take $\alpha\in B$, since $B$ is integral over $A$, then there is a relation $ \alpha^n+a_{n-1}\alpha^{n-1}+\cdots+a_1\alpha+a_0=0 $ for $a_i\in A$, by writing $a\beta$ for $f(a)\beta$ for $a\in A$ and $\beta\in B$.
Canonically projecting this into $S^{-1}A$ and $S^{-1}B$ yields $ (\alpha/1)^n+[a_{n-1}/1](\alpha/1)^{n-1}+\cdots+[a_1/1](\alpha/1)+[a_0/1]=0/1 $ so $\alpha/1\in S^{-1}B$ is integral over $S^{-1}A$. I don't see how this proves all of $S^{-1}B$ is integral over $S^{-1}A$, it seems to only prove that the canonical image of $B$ in $S^{-1}B$ is integral.