Why $P_i+N$ is stationnary when $P_i$ are submodules of $M$?

Question

Let $N\subset M$ be $R$-submodule. We suppose that $N$ and $M/N$ are Noetherian and we want to show that $M$ is also Noetherian.

The proof start by:

Let $P_i$ be a chain of submodule of $M$. Then $P_i+N$ is stationnary.

Question I don't understand why $P_i+N$ is stationary.

Answer 1

Submodules of $M/N$ correspond bijectively to submodules of $M$ that contain $N$. In particular, saying that $M/N$ is noetherian is the same as saying that every increasing chain of submodules of $M$ that contain $N$ is eventually stationary.

If $(M_i , i\geqslant 1)$ is a an increasing chain of submodules of $M$, then $(M_i+N, i\geqslant 1)$ is an increasing chain of submodules of $N$, hence by the previous remark, if $M/N$ is noetherian this chain is eventually stationary.