Second isomorphism theorem for modules

For submodules A,B ≤ M, one has (A+B)/B ≅ A/(A∩B).

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

(A+B)/B \;\cong\; A/(A\cap B),

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 .