Forget the ring. Why do you want it? As a souvenir? As a bargaining chip for later? Let her keep it, pawn it, whatever she wants to do with it. You're trying to hold on to something of "hers".
You are the pursuer right now. She is the distacer. You have to work on yourself to change that dymamic. Read the distancer/pursuer homework from Cadet and it will start to become clear.
Start to accept that there is nothing you can do at this point to change your W's fellings, actions, opinions, thoughts, etc.
