Warning: contains spoilers for Crime Syndicate #4
The evil version of Lord of the Rings' Sauron, and they could get worse.
As part of his masterful plot to conquer Middle-earth, the eponymous Dark Lord of the Rings forges the One Ring to control all of the 19 Rings of Power that were already in the possession of Middle-earth's great rulers: the three Elven-kings, the seven Dwarf-lords, and the nine mortal men doomed to die. Sauron the Deceiver crafts the One Ring so these lesser rings would be dependent on his, allowing him to control the wearers. The most famous of Sauron's servants that came about as a result of this phenomenon are the Ringwraiths or Nazgul.
In DC, the Oans of Earth-3's upside-down version of Earth-Prime use their Rings in a similar fashion, to subjugate their wearers, as revealed in Crime Syndicate #4 by Andy Schmidt, Bryan Hitch, Kieran McKeown, Dexter Vines, Alex Sinclair, Steve Oliff, and Rob Leigh. Earth-3's Emerald Knight, John Stewart, has so far been able to resist the power of his Ring, but both the feel of it and the constant reminder that the ring could change him still weigh heavily on him.
Unlike Lord of the Rings, the rings of Johnny Quick and Atomica for their crimes against humanity. John is under the impression that only those metahumans will be hunted. But his ring says that it won't stop with them: "They'll label you all weapons of mass destruction. They'll come for you...and your precious city." At another point, the ring goads John to destroy some jet fighters that he had been able to quickly render harmless, to which John responds that he will make them his messengers. But the ring quickly recovers, saying that this was even better.
Although the success of Earth-3's rings is not contingent on the well-being of a master ring, it appears as though their influence might not be as powerful or effective as Sauron's. Upon learning of Sauron's treachery, the Elven-kings removed their rings, protecting themselves from his dominion, at least, in that regard. The fate of the dwarven-lords might mostly be unknown, but of the Ringwraiths who did fall under the Lord of the Rings' control, none of them ever escaped their servitude. Meanwhile, Earth-3's versions of Thaal Sinestro and Green Lantern beat or have so far fought off the influence of their ring, respectively.
