Exercise 2.17 in Miles Reid's Commutative Algebra

Question

In exercise 2.17 in Reid's book I need to prove the third isomorphism theorem for modules:

If $L \subseteq M \subseteq N$ are modules, then $N/M \cong (N/L)/(M/L)$

using Snake lemma, but I don't see how I can apply it.

Answer 1

If $p$, $q$, $r$ are the three vertical morphisms, the Snake lemma gives a long exact sequence:$\DeclareMathOperator\coker{coker}$ $$0\to\ker p\to \ker q\to\ker r \xrightarrow{\partial} \coker p\to\coker q\to\coker r\to 0.$$ From the diagram you have, you deduce $\ker r=0$, and as $q$  is surjective, and the diagram is commutative, $r$ is surjective too. Whence $r$ is an isomorphism.