Artinian ring with unbounded descending chain length

Question

The ascending chain condition does not imply the existence of an integer $n$ such that all ascending chains stop after $n$ steps. For a simple example consider $\mathbb{Z}$.

Is there a nice example of this for the descending chain condition?

Answer 1

A theorem of Hopkins states that each left Artinian unital ring $R$ is also left Noetherian. So $R$ as a left $R$-module satisfies both chain conditions, and so has finite length $m$: it has a composition series (with finitely many terms). By the Jordan-Holder theorem, every composition series has length $m$ and every chain of left ideals in $I$ can be refined to a composition series, and so has length $\le m$.

Of course the same is true for right ideals if $R$ is right Artinian, and if $R$ is commutative, the proofs are a bit simpler.