Thrain did have a Ring of Power, which he inherited from his father Thror. It was taken from Thrain when he was imprisoned in Dol Guldur.
The Arkenstone was an heirloom of the Kings of the Longbeards. That is why Thorin was outraged to learn that Bilbo had stolen it and given it to Bard and Thranduil as a bargaining chip.
As eldariontdk wrote, Thrain (through his father Thror) possessed one of the Seven Rings of the Dwarves, specifically the one that was said to have been given to his ancestor Durin III directly from Celebrimbor.
Peter Jackson expanded on the role of the Arkenstone to make it a symbol to all Dwarves, not just the Longbeards. With it, Thorin could apparently have commanded the armies of all seven of the dwarven clans as a sort of High King. But that only applies to the films.