TITLE: Question on Noetherian/Artinian properties of a graded ring
QUESTION [7 upvotes]: Let $R$ be a non-negatively graded Noetherian ring such that $R_{0}$ is Artinian and $R_{+}$ is a nilpotent ideal. Prove that $R$ is Artinian. Give an example to show that this is false if the Noetherian property is removed.

This is an exercise from a note that I saw on the Internet. I can not solve it. Please help me. Thanks!

REPLY [2 votes]: By your assumption, $R_+$ is nilpotent, then of course the krull dimension of $R$ is the same as $R/R_+=R_0$ which is zero.  A commutative ring is Artinian if and only if it is Noetherian and of dimension zero.
So $R$ is Noetherian and of dimension zero, thus it is Artinian.  We are done!
Counter-example: omit