Cheap rings are often made of copper with a thin layer of silver or gold over them. Copper turns green when it is oxidized (look at a copper water fountain, for example), which occurs whenever it is in contact with air, but speeds up considerably in the presense of water. The sweat from your finger, as well as water from washing your hands, etc., causes the oxidation reaction to occur relatively quickly. Some of the oxidized copper from the ring is then transfered to you finger, turning it green.