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
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