If $f: A\to B$ is a ring homomorphism between commutative rings and $S$ any multiplicative closed subset of $R$. Let $T=f(S)$. Then it is well known that there exists an $S^{-1}R$-module isomorphism $h:S^{-1}B\to T^{-1}B$, $b/s\mapsto b/f(s)$, where $b\in B$ and $s\in S$.
My question is "Does $h$ is a ring isomorphism?"
Here is my attempt: One can define a multiplication in $S^{-1}B$ as $(b/s)\cdot(b_1/s_1)=bb_1/ss_1$. Under this multiplication $h$ becomes a ring isomorphism.
Is it true or not?