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It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following statement.

If a ring satisfies the DCC on two-sided ideals, then it also satisfies the ACC on two-sided ideals.

The best I could come up with is not really good. My example is of large cardinality and since I don't know much about set theory, I can't be sure if it's correct. I use this. I take the ring $R$ of endomorphisms of an $\aleph_\omega$-dimensional vector space over a field $\mathbb F,$

$$R=\operatorname{End}\left(\bigoplus_{i\in\aleph{\omega}}\mathbb F\right).$$

From the linked answer, I know that the two-sided ideals in $R$ are the sets of endomorphisms of rank $\kappa$, for each infinite cardinal $\kappa \leq \aleph_\omega.$ In ZFC, these ideals form a lattice isomorphic to $\mathbb N\cup \{\infty\}$ with the natural order. This lattice satisfies the DCC, but not the ACC.

This is a silly example, and (a) I'm not sure it's correct, (b) I have never seen the proofs of the facts relevant to it.

I have three questions.

(1) Is the above correct?

(2) (Changed) Is there an example of a smaller cardinality (at most $\mathfrak c$), and preferably uncomplicated? I'm quite sure there must be, but I can't think of one.

(3) Is it possible to construct a simple example of a ring whose lattice of two-sided ideals is isomorphic to $\mathbb N\cup\{\infty\}?$

EDIT After the the discussion in comments in which my ignorance in set theory became obvious, I would like to add a restriction in (2) and (3) that the examples can proven to be examples without the use of the axiom of choice. I'm not sure it's a good restriction, but I don't see one that would fit my needs better.

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The infinite cardinals $\le\aleph_{\omega}$ are ordered in type $\omega+1$ in ZFC; you don’t need GCH for that. – Brian M. Scott May 19 '12 at 19:11
@BrianM.Scott Oh, I'm not really sure now why I wanted to use the GCH... Thanks, I'll edit the question. – user23211 May 19 '12 at 19:16
@BrianM.Scott So this example works in ZFC? Or is there a mistake? – user23211 May 19 '12 at 19:18
You can drop question (2) now, I think, or at least the first part of it. As far as (3) goes, I think that this example is pretty simple. As for (1), I’ve not really thought about the answer to the other question, but modulo that it appears to be right. – Brian M. Scott May 19 '12 at 19:20
@BrianM.Scott OK, thank you! I've changed (2) to ask for examples of smaller cardinality. This one seems very unnatural to me, because I have never really encountered any rings of cardinality greater than $\mathfrak c$ in ring theory. – user23211 May 19 '12 at 19:26

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