i'm guessing it is the green stuff that you're having trouble with?
if you've got some white vinegar, i'd say try a 50/50 mix of regular vinegar and hot tap water.
let it soak while you do dishes or make dinner, then try the scrub with a soft toothbrush and see if that helped at all?
as an extreme measure, and not something i'd normally suggest, but for odd cases like this... acetone. straight acetone, or some nail polish remove that is not the "non-acetone" kind.
as long as the center stone is a diamond, a Moissanite, a plain CZ, white topaz... something that isn't coated or treated in some way.
i bought a ring once, got a decent deal on it, but it had paint ... like latex-based wall paint ... underneath the stone wedged against a prong. a funky light blue blob.
dunno if the green is paint, but let me know if you have any luck getting at it. if you can pick up a wooden toothpick at a restaurant, that might be something to try to get at it with that's less likely to scratch either the ring or the stone.
I finally took it to the jeweler and with his tools he picked it out then ran it through the steam and it’s as good as new! I will remember the toothpick and vinegar for future cleanings if I need it. I didn’t even think of the toothpick LOL and I bet that would have worked!