Peter says, "they are not drunk for it is only the third hour." Acts of the Apostles 2:15.
Amethyst, (a-methusko in Greek), means 'not to be intoxicated'. Therefore bishops wearing the traditional amethyst ring are stating they are successors of the Apostles.
Just to horrify a Bishop or two, remember you get a partial indulgence from kissing a Bishop's ring, amethyst or not.