Descending chain condition in noncommutative rings

Question

By Hopkins Theorem it is well-known that every right (resp. left) artinian unitary ring is right (left) noetherian. Suppose that a noncommutative unitary ring R satisfies the descending chain condition on its two-sided ideals. Does R satisfy the ascending chain condition on two-sided ideals?

Answer 1

I'm somewhat surprised this question hasn't been answered previously. It turns out DCC on two-sided ideals does not imply ACC on two-sided ideals.

Let $V$ be a $k$-vector space with a basis of size $\aleph_{\omega}$, and put $R={\rm End}_k(V)$. Then, by Exercise 3.16 in Lam's "A First Course in Noncommutative Rings" the two-sided ideals are of the form $0$, $R$, and $I_{\aleph}=\{x\in R\ :\ \text{the rank of $x$ is }<\aleph\}$ for each infinite cardinal $\aleph\leq \aleph_{\omega}$. These ideals are linearly ordered and do not satisfy ACC since we have an increasing chain $\aleph_0<\aleph_1<\cdots< \aleph_{\omega}$, but these ideals do have DCC since the cardinals do.