Use a vise grip wrench
While unscratchable, tungsten rings are, however, very brittle and not malleable. Gently apply firm compression using an adjustable vise grip (borrow from your hospital's electrical or engineering department). This will fracture the ring. Carefully collect the pieces, which may have sharp edges. The YouTube videos below interestingly were made by tungsten ring retailers. Thanks to Dr. Gregg Miller (San Joaquin Community Hospital in Bakersfield, CA) for this tip.
(start at the 1:00 mark to skip ahead to the important part)