Third isomorphism theorem for modules

Third isomorphism theorem (modules): Let $M$ be an $R$-module and let $A\subseteq B\subseteq M$ be submodules . Then there is a natural isomorphism

of $R$-modules, where each quotient is a quotient module .

This theorem expresses the compatibility of iterated quotients and is fundamental in organizing “modding out step by step” in module theory.

Proof sketch (optional): Use the natural surjection $M/A\to M/B$ induced from $M\to M/B$; its kernel is $B/A$. Apply the first isomorphism theorem to that map.