How to remove coating from metal ring?
Start with acetone (nail polish remover) and see if that does the trick. Agitating with a Q-tip can speed things along. It all depends on what the coating is, but acetone won't hurt the copper so it's a good place to start. If it works but isn't fast, you might try soaking it for an hour or so.

A more aggressive approach would be oven cleaner, but it's likely to remove any patina from the copper and shouldn't be left on there too long. Also make sure to rinse thoroughly with water afterwards to remove any caustic residue.