# Noetherian, Artinian on graded ring and localization

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

Take a descending chain of homogeneous ideals in $R$ and localize at the homogeneous maximal ideal, $M$. Now use 2.10.
• @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 $R$. Jan 15, 2017 at 8:58