TITLE: Is a local ring Artinian, if the maximal ideal is nilpotent?
QUESTION [5 upvotes]: This answer suggests the idea, that a local ring $(R, \mathfrak{m})$ whose maximal ideal is nilpotent is in fact an Artinian ring.
Is this true? If so, how is it proven?

REPLY [3 votes]: There is a version of this theorem even for noncommutative rings. It is called the Hopkins-Levitzki theorem.
It says, in a nutshell, that if $R/J(R)$ is semisimple and $J(R)$ is nilpotent, then $R$ is right Artinian iff right Noetherian.  Yours is a special case where $R/J(R)$ is a field.
Here is the DaRT search which currently yields two examples of local, non-Artinian rings with nilpotent Jacobson radicals.