Let $R$ be a local Noetherian ring and $\mathfrak{m}$ its maximal ideal. I have shown that either $\mathfrak{m}^n\neq\mathfrak{m}^{n+1}$ or $\mathfrak{m}^n = 0$ and supposedly in this latter case $R$ is Artinian. But I am having a hard time proving this.
The easiest way seemed to me to create a composition series $R\supsetneq\mathfrak{m}\supsetneq\mathfrak{m}^2\supsetneq\cdots\supsetneq\mathfrak{m}^{n+1}=0$ since an $R$-module is Noetherian and Artinian iff it has a composition series. I can show $R/\mathfrak{m}$ is simple but not the other quotients. I am not really sure if they have to be simple. I also tried to prove they are isomorphic with $R/\mathfrak{m}$ but I struggle to prove the injectivity of such a morphism. Is the way I am trying to prove this possible or do I need to use something else? (e.g. this)
