Ideal correspondence for quotients

Ideal correspondence for quotients: Let $R$ be a ring and let $I\lhd R$ be a two-sided ideal, with quotient map $\pi:R\to R/I$ . The assignment

J\longmapsto J/I

is a bijection between two-sided ideals $J\lhd R$ with $I\subseteq J$ and two-sided ideals of $R/I$ , with inverse $K\mapsto \pi^{-1}(K)$ . This correspondence preserves inclusion and carries sums and intersections to sums and intersections.

Remarks

This is the ring form of the Correspondence Theorem applied to the canonical epimorphism $\pi$ , and it explains how ideals in a quotient ring lift and descend. In particular, prime ideals and maximal ideals of $R/I$ correspond to those of $R$ that contain $I$ (in the commutative setting).