I have heard somewhere in these four years of medical school that betadine tattoos onto gold (I think yellow), but not sure if that's true or not, and not sure if it only applies to yellow gold...
I usually just take it off and tie it (along with my watch) in the waistband of my scrubs (making sure to tie a double knot). Keeps them clean, the patients safe, and prevents glove failure.
I'm always afraid that I'll take off my scrubs to use the bathroom and lose my rings if I tie them on my pants, so I put the rings on my watch band, fasten the ends together, and put it in my scrub top pocket.