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?
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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. 

