If it is a first floor tub, go into the basement and look at the ceiling under the tub. You’ll see a large trap that looks like a small coffee can. If you can (they generally freeze shut if they’re iron) open the bottom and you’ll find your ring in there. Otherwise, it could be an “S” or “P” type trap and you would either have to open up the small cleanout and fishout your ring, or if there is no small cleanout access, take the plumbing apart.
If it’s a second floor tub, and you don’t have an access panel to the plumbing, you may have to tear out some drywall to get at the plumbing under the tub.
But in any case, if the trap did stop the ring from flowing downstream (ie: you turned the water off right away) your ring will be very close to the tub.
If it did get washed downstream and you’re on city sewer and water, contact the city waste management people, they strain these things out of the rest of the effluent.
