A relation between prime ideals and ring of fraction.

Question

Let $A$ be a commutative ring with unity, $S\subseteq A$ a multiplicatively closed subset of $A$ and $q$ a prime ideal which doesn't meet $S$. Are the $S^{-1}A$-modules $(S^{-1}A)_{S^{-1}q}$ and $S^{-1}(A_q)$ isomorphic? If so, how can I prove this?

Answer 1

They are isomorphic.

We can recall an exercise in Atiyah's commutative algebra 3.3: If R is commutative ring, S,T are two multiplicative closed subsets, U is the image of T in $S^{-1}R$, then $U^{-1}S^{-1}R$ is isomorphic to $(ST)^{-1}R$. (This is not difficult).

For your question: T=A-q, since S and q has empty intersection we can show that $S^{-1}A-S^{-1}q=S^{-1}T$. So $(ST)^{-1}A\cong (S^{-1}A)_{S^{-1}q}$ by the exercise.

It is also not difficult to see $S^{-1}A_q\cong (ST)^{-1}A$.