First of all, it isn't mold. Copper turns green when it reacts with different elements. Most metals are mixed with copper. To avoid the green colouring buy pure metals instead.
You can purchase a special jewelry cleaner which has a tray you use to dip the jewelry into a solution. Alternatively you can soak the ring in ammonia for 15-20 seconds, rinse with water and repeat if necessary.
First try cleaning with soapy water and a toothbrush, followed by a bleach rinse. Then purchase jewelry cleaner specific to the metal of the ring - ie: silver cleaner for silver. Apply a small amount of the cleaner at a time and rub into the metal until the ring is shiny again and the mold is gone.
This isn't mold, it's a chemical reaction between the metal and your skin, sometimes caused by hand lotion. Use jewlery cleaner or a buffing cloth to clean your ring.