Actually, the ring and power battery have passed from:
Alan Scott, who operates as a mentor and backup under the name Sentinel
Hal Jordan (the early JLA GL), who is now thought dead
John Stewart, who still occasionally operates under the name Darkstar,
Kyle Rayner (an identity unknown to most of the world)
The interesting thing is that the ring (1) eventually imbues its wearer with the green energy so the ring is not necessary and (2) usually allows the possessor of the Green Flame to have children with the power or a variation thereof. Why? Is it just a quirk or a planned phenomenon? Good questions. As far as can be told, bearers of the ring are not metahumans when they put the ring on.
So all the above still have GL powers in some degree or another. Also having GL powers are Scott's daughter Jade, his son Obsidian, and Hal's illegitimate Brazilian daughter, Fire. The green flame does not always manifest the same way with each bearer.
It is known that once a bearer manifests the powers, the ring seems to want to move on...