I think juniorgrrl's idea is best. You will at least know that you tried to find the owner. Oh, and I'd just say "did you lose anything when you moved out?" rather than bring up the ring. If it's really theirs you will be able to tell by their reaction.
Side note: when we sold my grandparents' house, we were unable to find my grandma's engagement ring. We did tell the incoming owners and they never found it either. I have a feeling that there is a mouse (they had tons of them) running around wearing a very spiffy necklace.
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