I'm working with an $R$-module $M$ which is both noetherian and artinian. The problem at hand is showing that there is a finite chain of submodules $$M = M_0 \supseteq M_1 \supseteq M_2 \supseteq ... \supseteq M_{n-1} \supseteq M_n = 0$$ such that $M_{i}/M_{i+1}$ is simple for all $i=0,1,2,...,n-1$.
So let's assume that $M_{i}/M_{i+1}$ has the submodule $N$. Let $a,b \in N$. If $a-b \in M_{i+1}$ then $a \equiv b $ in $N$. Does this mean that $N$ "inherits" the entire coset $\bar{a}$ from $M_{i}/M_{i+1}$? If so, I'm curious about what happens with the other cosets in $M_{i}/M_{i+1}$ with regards to $N$. This is kind of where I stopped because I became very uncertain of the properties of $N$.
For some reason I'm having a really hard time understanding quotient modules properly, so please excuse me if this question is a very stupid one.