$A$ be a domain and $S \subset A$ be an mcs, for $\alpha$ integral over $A$ is it true $S^{-1}A[\alpha]\cong(S^{-1}A)[\alpha]$? Let $A$ be an integral domain and $S \subset A$ be an multiplicative closed set. Let $\alpha$ be an element in some extension of $A$ which is integral over $A$. Then I want to know whether $S^{-1}A[\alpha]\cong(S^{-1}A)[\alpha]$ holds or not. Actually the natural map $\phi:S^{-1}A[\alpha]\to(S^{-1}A)[\alpha]$ is surjective ring homomorphism, but can't prove that it is injective.
This question comes from here. Makoto Kato used this in the proof. I failed to prove it. Need some help. Thanks.
 A: There's a subtle but important point of ambiguity in the way this question is phrased; suppose that $B\supseteq A$ is the extension that contains $\alpha$. Talking about $(S^{-1}A)[\alpha]$ makes an assumption that $S^{-1}A$ is a subring of $B$, or at least that the two rings can be "amalgamated" in a coherent way: ie that there exists a ring $C$ containing both $S^{-1}A$ and $B$ as subrings. But this may not possible in general; for instance, take $A=\mathbb{Z}$, fix a prime $p\in A$, and let $S=\{p^n:n\in\mathbb{N}\}$. Now take $B$ to be the ring $\mathbb{Z}\times\mathbb{F}_p$. This ring has characteristic $0$, so it indeed contains $A$ as a subring. Furthermore, the element $\alpha=(0,1)\in B$ is integral over $A$, since it satisfies the monic polynomial $t^2-t\in A[t]$. But it is impossible for a ring $C$ to contain both $B$ and $S^{-1}A$ as subrings; indeed, if $C$ contains $S^{-1}A$, then $p$ is invertible in $C$, so $C$ contains no elements of additive order $p$.
So, it is possible for the question to not even make sense. However, this kind of pathological behavior only arises because $B$ is not an integral domain. If we do demand that $B$ be an integral domain, which is the case in the post that you link to, then the theorem as stated is correct. In that case, we can work in the field of fractions $K=Q(B)$. We can then safely consider everything as taking place in $K$, and both $(S^{-1}A)[\alpha]$ and $S^{-1}(A[\alpha])$ will be isomorphic to the subring $A[\alpha,1/s:s\in S]$ of $K$, as desired.
