Exercise 9, § 2

Let $A$ be a ring, $S$ a multiplicative subset of $A$ and $q$ a prime ideal of $A$. Show that, if $T$ is the image of $A - q$ in $S^{-1}A$, the ring $T^{-1}(S^{-1}A)$ is isomorphic to $S'{}^{-1}A$, where $S'$ is the complement of the union of the prime ideals of $A$ contained in $q$ and not meeting $S$.