[Math] A Noetherian Ring with Discrete Spectrum is Artinian

Question

I'm trying to solve an exercise.

I should prove that if $R$ is a notherian ring and $\operatorname{Spec}(R)$ is discrete then $R$ is artinian.

I think it is enough to show that $\dim R=0$ since I can use that $R$ is artinian if and only if $R$ is noetherian and $\dim R=0$. Can you help me?

Thank you!

Answer 1

Hint

Suppose $\mathfrak{P}$ is prime, show that $\mathfrak{P}$ is maximal (Hint: $\mathfrak{P}$ is closed due to the topology on $\operatorname{Spec}(R)$).

What does that say about $\operatorname{Dim}(R)$?