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In this question, I gave an example of a ring whose lattice of two-sided ideals is order-isomorphic to $\omega+1$. I've been playing a bit with trying to find rings with a given lattice of ideals since, and one case I found interesting and difficult at the same time. (Difficult for me of course. I've learned here that many questions I can't answer turn out to be trivial.)

Is there a unital ring whose lattice of two-sided ideals is order-isomorphic to $(\omega+1)^{\operatorname{op}}?$

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    That's how it normally is, with a few technical tricks accumulated through experience. The problem in general of realizing a given latice (not every lattice, of course) as an ideal lattice is a very hard one.2012-06-15

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Maybe you can tell me if this is what you're looking for.

What about the power series $R=\mathbb{F}[[x]]$? Doesn't $R\supset(x)\supset(x^2)\supset\dots$ make the chain you are looking for?

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    I know nothing about valuation theory except some random facts without proofs. I understand your argument though. Thanks again!2012-06-15