The Twelfth Doctor wore a gold, dual banded wedding ring with a green stone, in honour of his marriage to River Song. (PROSE: Twice Upon a Time)
When he regenerated into the Thirteenth Doctor, the ring was too big for her finger and fell off her hand. (TV: Twice Upon a Time)
Behind the scenes
The ring was, in actuality, two bands, which comprised of Peter Capaldi's own wedding ring, and a separate band with the green stone, which backed up against Capaldi's own ring. This was done so Capaldi would not have to remove his ring during filming. In the final scenes of TV: Twice Upon a Time, when the ring fell of the Thirteenth Doctor's finger, Jodie Whittaker was only wearing the secondary band for the shot.