Hello,
It looks like contact dermatitis or allergic reaction. I suggest you that you should get it evaluated from a dermatologist. Till then, you can apply some calamine lotion at the rash as it will help in soothing the skin. You can take some OTC oral antihistamine medications like Benadryl or Claritin and see if it helps.
I hope it helps. Take care and regards.
You are probably allergic to the metal in the ring band. Try coating with clear nail polish or not wearing it at all. If you remove it, it will likely heal up and hit it with tea tree oil, the purest form you can find. Try a beauty supply in the nail section, it's a good fungicide among a thousand other things.