Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question

1 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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.