TITLE: Noetherian Local Ring
QUESTION [6 upvotes]: I came across this old exam problem. If $R$ is a local Noetherian ring and $I$ is an ideal in $R$ such that $I^2=I$ then $I =0$.
Any hint would be appreciated. I'm only familiar with what the definition of local and Noetherian mean. I'm not sure why these properties are useful.

REPLY [4 votes]: If you don't get how Zev's hint goes here are three more hints to apply Nakayama's Lemma:

Since $R$ is Noetherian it follows that $I$ is finitely generated as an $R$ - module.
Since $R$ is a local ring, $\operatorname{Jac}( R) = \mathfrak{m}$, the maximal ideal of $R$.
Every ideal is contained in some maximal ideal. The containment could be proper or we can have equality.