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.

Quotienting in two steps or in one step gives the same result, so the square below commutes:

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