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The problem is "Let $R$ be a $\mathbb{Z}$-graded ring. Prove that $R$ is Noetherian if and only if $R$ satisfies the ascending condition on homogeneous ideals."

I know it sounds a lot like Ascending chain conditions on homogeneous ideals. But here is a $\mathbb{Z}$-graded ring and clearly we can't use the same technique. Help me

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    I suppose it's clear that ACC on homogeneous ideals implies that every homogeneous ideal is finitely generated. Then use Theorem 1.5.5.from Bruns and Herzog. – 2017-01-12

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