If $N$ and $P$ are submodules of the $A$-module $M$ (where $A$ is a commutative ring with unity), why is there a difference between $(N+P)/N$ and $P/N$?
If $x\in (N+P)/N$ then $x=n+p+N=p+N$ for some $n \in N$ and $p\in P$. So actually $x \in P/N$ right? Hope somebody can help me.
In the general case, there is a canonical isomorphism: \begin{align*}P/N\cap P &\longrightarrow(P+N)/N\\ x+N\cap P & \longmapsto x+N\end{align*}
Also, there's a bijection between submodules of $M/N$ and submodules of $M$ that contain $N$, and $P$ is not supposed to contain $N$. However, by the canonical homomorphism $\,p\colon M\longrightarrow M/N$, $p(P)$ is a submodule of $M/N$, and $$p^{-1}\bigl(p(P)\bigr)=P+N. $$