Let R be a Noetherian local ring with maximal ideal M and there exists a non-maximal prime ideal of R, then Prove that $M^{n+1} \subset M^n$ for every $n \in \mathbb{N}$(strict subset) i.e. R is not artinian.
How can you use existence of non-maximal prime ideal.
By Nakayama, if there is some $n \geq 0$ such that $M^{n+1}=M^n$, then $M^n=0$. In particular, for every prime ideal $P$ of $A$, if $x \in M$, $x^n=0 \in P$ so $x \in P$, thus $P=M$.