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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!

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    What have you tried so far? Of course, you should use the statement in your other question http://math.stackexchange.com/questions/1474662012-05-20
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    Sorry, how can I use it ? I am not good at algebra.2012-05-21

2 Answers 2