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There's an exercise in Tom Marley's Graded Rings and Modules making me confused, stating

$R$ is a nonnegatively graded local ring with $R_0$ being local. Let $M$ be the unique homogeneous maximal ideal Prove $R_M$ Artinian (Noetherian) implies $R$ Artinian (Noetherian).

This may be an easy question. But I can't solve it since I still don't see the connection between localization and Artinity of $R$. Any ideas

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For Artinian case, It is result of Exercises 4.7, and 2.10:
Take a descending chain of homogeneous ideals in $R$ and localize at the homogeneous maximal ideal, $M$. Now use 2.10.

For Noetherian case replace 4.7 with 4.3

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  • $\begingroup$ What is the idea here? $\endgroup$
    – T C
    Jan 14, 2017 at 16:18
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    $\begingroup$ @chítrungchâu The idea is very simple, but you have to run within the Marley notes in order to understand this terse answer: The ring $R$ is Artinian/Noetherian iff satisfies DCC/ACC on homogeneous ideals (see the exercises 4.7/4.3). Now such a chain of homogeneous ideals must stop in $R_M$ (since it is Artinian/Noetherian), so you got some homogeneous ideals $I\subseteq J$ such that $I_M=J_M$. Now conclude $I=J$ from exercise 2.10, so the chains also stop in $R$. $\endgroup$
    – user26857
    Jan 15, 2017 at 8:58
  • $\begingroup$ @user26857 Since op have not noted any try, I only post a terse answer (as hint). It is common in the site to do so, isnt it? $\endgroup$
    – user 1
    Jan 15, 2017 at 10:31
  • $\begingroup$ @chí trung châu didnt get the idea? $\endgroup$
    – user 1
    Jan 23, 2017 at 13:08

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