I have a white gold set of rings that are rhodium plated. I had the same problem, and was trying to find out what to do. I would take them off, clean them good, put triple antibiotic on my finger until it seemed to heal, then put the rings back on and the same thing would happen (a reddish rash under where the rings touched my skin). I have a ring on my right hand that this did not happen with, and I noticed that the rash was confined to the area where the diamonds were (so bacteria could have accumulated there due to moisture and caused a form of athletes foot). To test this possibility, I took off my rings (cleaned them and put them on the dresser to dry all night), put Lotrimin (athletes foot medication) around my ring finger, and put a band aid on it overnight. The next morning the rash was gone, I put my clean and dry rings, and haven't had a problem since. I hope people read this and I hope it helps.
Kristi
I just posted a question which has similarities to what u have but not sure if the same. Was diagnosed with Erythema Annulare Centrifugum EAC. No known cure though so am desperate to hear what others do to get rid of it or what the causes were. Not much research on it. My Dad was in Vietnam war before I was born and has horrible skin problems as a result of malaria tablets over prolonged time (and agent orange which no-one admits was used of course) so I wonder if it is genetic from what you say.