Correspondence theorem for rings

Ideals of a quotient ring correspond to ideals of the original ring containing the kernel.

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

J\ \longmapsto\ J/I

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 $\pi^{-1}(K)$ .

Under this correspondence, prime ideals (and likewise maximal ideals in the commutative case) correspond to prime (respectively maximal) ideals containing $I$ .