Spectrum of localization away from one plus an ideal

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 $\mathfrak{p}+I\ne R$.