I'm working on the following problem and I'm stuck. Any hints or solutions would be appreciated
Let $R$ be a left Artinian ring with Jacobson radical $J(R)$. If $R \neq J(R)$. show that $R$ is a left Noetherian ring.
Here are my thoughts: I think we have to use the ascending/descending chain definitions for artinian and noetherian since I dont see how we can show that all ideals are finitely generated. Besides that, maybe we can somehow use the condition that $J(R)\neq R$ by considering an element in $R$ that is not in $J(R)$... but I'm not sure how that's useful.