Is $R= A[T] / (T^3-a)$ where $A$ is a left artinian ring and $a \in A$, left noetherian?

Question

Is $R= A[T] / (T^3-a)$ where $A$ is a left artinian ring and $a \in A$, left noetherian?

I know $A[T]$ does NOT have to be a left artinian ring.

Any hints?

Maybe I can use the theorem if $N$ (submodule of $M$) and $M/N$ are artinian then so is $M$.

Answer 1

Yes.

By the Hopkins-Levitzki Theorem, if $A$ is left Artinian, then it is left Noetherian. Next, by Hilbert's basis theorem, $A[T]$ is left Noetherian. Finally, any quotient of a left Noetherian ring is left Noetherian.