Film chemistry is based on silver so virtually all chemicals used will have some effect on silver jewelry. Gold and platinum are not affected. But as The Traveller says, it's not good practice to wear jewelry in the darkroom.
Once its been through dev stop n fix you should have a nice black ring I would think, though never tried it personally. Working with chems in the darkroom its best to wear only what you can chuck out afterwards and always an apron (plastic) and eye protection when mixing.