Third isomorphism theorem for rings

Quotienting by an intermediate ideal is the same as quotienting in one step.

Third isomorphism theorem for rings

Third isomorphism theorem (rings): Let $R$ be a ring and let $I\subseteq J$ be ideals of $R$ . Then $J/I$ is an ideal of $R/I$ , and there is a natural ring isomorphism

(R/I)/(J/I)\ \cong\ R/J

of quotient rings .

This formalizes the idea that “modding out in stages” is equivalent to modding out by the total relation.