You're right. If you say...
Have you found the right ring?
No, I haven't.
... this means that you haven't found it YET. You haven't found it SO FAR, but you are still hoping to find it and you are still looking. This is the Present Perfect, used to talk about unfinished - or 'open' - periods of time.
This is different from the Past Simple. which we use for finished - or 'closed' - periods of time.
If you say....
Did you find the right ring?
No, I didn't.
... it might mean that you gave up and stopped looking. Or it you might say it in this situation:
I went to town today to look for a ring.
So did you find one?
No, I didn't. I'll go again tomorrow.
In that conversation, the exchange 'Did you..? No, I didn't' refers to the specific context of the shopping trip which happened today. That shopping trip is now over (a finished period) of time, but you are still looking for the ring.