Prime ideal correspondence for localization

Extension and contraction give inverse inclusion-preserving bijections between prime ideals of $S^{-1}R$ and prime ideals of $R$ disjoint from $S$:

For $S=1+I$, the prime ideal correspondence for localization identifies $\mathrm{Spec}{S}^{-1}R$ withIndeed, $\mathfrak{p}\cap (1+I)=\varnothing$ exactly when no element of $I$ is congruent to $-1$ modulo $\mathfrak{p}$, which is exactly $\mathfrak{p}+I\ne R$.