This exercise is given in the lecture notes of Homological Algebra and Algebraic Topology written by Wojciech Chach´olski & Roy Skjelnes (1.8.16). Hence, there might be a typo.
The sum of two submodules $M$ and $N$ is defined similarly as in vector spaces. ((P,+) is an abelian group.) \begin{align} M+N=\{m+n|m\in M, n\in N\}\leq P. \end{align} I can see that $M+N$ and $M\cap N$ are submodules of P and there is an inclusion relation as $M\cap N\subseteq M, N\subseteq M+N\subseteq P$. I had two attempts that I couldn't conclude.
My First Attempt: I tried to define a map $\psi:(M+N)\rightarrow N/(M\cap N)$ whose kernel is $N$. It seems not possible.
My Second Attempt: I tried to use something I already proved that is $(M/L)/(N/L)=M/N$ where the inclusion of A-modules $L\subseteq N\subseteq M$ is given. However, this doesn't provide any help.
Although I don't think the identity is true, any hint or help is appreciated.
