The Dragon Ring is the proof that an individual has passed the Dragon Trial. These items figure prominently in Lunar: Dragon Song, and unlock within Jian the ability to cast Black Magic. It seems that the Dragon Rings either don't exist or are deemed unnecessary during Eternal Blue (either way, Hiro would not have been granted one as he did not undergo a Dragon Trial), but a White Dragon Ring was given to Alex in Silver Star Story. Alex was not granted any other rings, however, and the White Dragon Ring did not give him any magical ability.