Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 2 / b / i / Solution

Extension and contraction give the prime ideal correspondence for localizationExplicitly, $\mathfrak{p}$ maps to $S^{-1}\mathfrak{p}$, while $\mathfrak{q}\subseteq S^{-1}R$ maps to $\{r:r/1\in \mathfrak{q}\}$. Primality follows by clearing denominators, and the two operations are inverse because an ideal in a localization contains $r/s$ exactly when it contains $r/1$.