Second isomorphism theorem for modules

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

where $A\cap B$ is the intersection and each quotient is a quotient module .

This isomorphism is obtained by restricting the quotient map $M\to M/B$ to $A$, and it is a standard application of the first isomorphism theorem .

Proof sketch: Consider the homomorphism $A\to (A+B)/B$ induced by inclusion $A\hookrightarrow A+B$ followed by the quotient map. Its kernel is exactly $A\cap B$, and its image is all of $(A+B)/B$. Apply the first isomorphism theorem.