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.