By Hopkins Theorem it is wellknown 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 twosided ideals. Does R satisfy the ascending chain condition on twosided ideals?
I'm somewhat surprised this question hasn't been answered previously. It turns out DCC on twosided ideals does not imply ACC on twosided 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 twosided 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. 

