If the acetone suggested earlier doesn't work, put the ring into paint thinner to soak (or mineral spirits, but paint thinner is somewhat less expensive and simply a less refined version of mineral spirits). Let it soak at least 15 minutes, then brush with an old stiff toothbrush. If the varnish comes off, put the ring into fresh paint thinner to soak a few minutes, in order to remove the final residue, wiping the ring between rinsing soaks. Let dry.
If the varnish does not come off with a 15 minute soak in the paint thinner, then soak for increasingly long periods of time, up to an hour, followed each time by vigorous brushing. Whenever it does come off, finish up with several rinsing soaks in fresh paint thinner.
If after longer soaks the 'varnish' still isn't attacked by the paint thinner, then it is probably some sort of epoxy resin coating, or enameling that has been fired on, and only mechanical methods, such as scraping, will safely take it off. (Paint thinner will soften epoxy resins but scraping is still necessary to remove it--brushing will simply spread it around. Also there are very aggressive approaches such as hydrofluoric acid that would attack enameling, but the safety risks are not worth it.)