Let A be a commutative ring. A multiplicatevely closed subset S in A is called saturated if for all $a,b \in A$, $ab \in S$ implies $a\in S$ and $b\in S$.For a multiplicatively closed subset $S\subseteq A$, define $\bar{S} =${$a\in A$ | there exists $b\in A $ with $ab\in S$} is a multiplicatevely closed set.
Question:If S and T are multiplicatevely closed subset in A, then the A-algebras $S^{-1}A$ and $T^{-1} A$ are isomorphic iff $\bar{S} =\bar{T}$.
This question is from my commutative algebra assignment and I need help in solving it.
Work : Let $S^{-1} A$ and $T^{-1}A$ are isomorphic then it implies that there exists a function $f:S^{-1} A\to T^{-1} A$ , f is a ring hom. and f is an A-module hom. but how does it implies that $\bar{S}=\bar{T}$?
Conversly, let we have $\bar{S} =\bar{T}$ but how can I construct a A-algebra isomorphism f such that $f: S^{-1} A \to T^{-1} A$?
Unfortunately, I don't have much work to show for this and would appreciate help.