Correspondence theorem for rings

Correspondence theorem (rings): Let $R$ be a ring, let $I\triangleleft R$ be an ideal, and let $\pi:R\to R/I$ be the quotient map. Then the assignment

is an inclusion-preserving bijection between ideals $J$ of $R$ with $I\subseteq J$ and ideals of the quotient ring $R/I$. The inverse bijection sends an ideal $K\triangleleft R/I$ to the preimage $\pi^{-1}(K)$.