Here's the technical answer. Let's assume the ring exceeds the value thresholds that your state has, for their lost & found laws. Typically that will $100 (or $250, or whatever). That is, if you find something of that value or more, then technically, you were required to have turned it in to the police, for proper lost & found procedures. The police, in turn, will hold it for 30 days. If no one comes in to claim it, THEN it belongs to you.
And thus, in that case, the answer to your question would be: "
After the police return it to you".
Oh, and not sure about your particular state, but in CA, once the item exceeds $250 in value, the police not only hold it for that 30 day period, but they also run an ad in a local newspaper "found property, come in to describe", blah blah, in the classifieds. And if no one claims it, you're more than welcome to get the item after that 30 days. But if you elect to, guess who pays the cost of the newspaper classified posting? YOU DO (if you want to get the item, that is).
Or were you asking for the "realistic" answer?