What are some simple examples of "ID-only" rings? integral domains, that are neither atomic nor Noether nor Prüfer... Most common examples in the literature are rings of "higher virtues", having finite decompositions into irreducibles (maybe non-unique) -> atomic rings (Cohn), or Noether, or Prüfer, or GCD rings, or their intersections (UFD, Bézout, Dedekind and even more refined objects like PIDs, Eucl., fields). Why is it so much harder to find an easy "ID-only" ring?
 A: Using this map at DaRT one would look for domains which are not N-1, not Goldman, not Mori, and not atomic.  By avoiding these we dodge more stringent conditions too.
Interestingly, I did not realize there was no non-Mori domain confirmed yet, so thank you for asking.
I suppose we could additionally ask for a domain that isn't any of these four things, or at least combinations of them. But that gets complicated.

Why is it so much harder to find an easy "ID-only" ring?

Of course, it is not really sensible to ask for an "ID-only ring" because the pool of conditions to avoid is potentially infinite. In general, every extra condition you have to dodge makes the problem that much harder. I think the best you could possibly hope to obtain is a domain that isn't the four things above, plus maybe some other conditions you are interested in that DaRT does not have.

Update:
It seems that I have been ignorant of another fact that will change the above map: Mori -> ACCP, so it actually is situated above atomic.  I will try to adapt this in the near future, and also include "noetherian" in there.
