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!