Getting back to Tolkien, I don't see any reason to have the ring boost stats. The One Ring doesn't seem to have that effect in the fiction. A simple ring of invisibility is already legendary-caliber in power.
Any legendary item that is made available at the start of the game is going to be supremely tempting to the players.
Specifically, it's the least-useful invisibility ever, because it makes you undetectable to everyone
except it's creator and undead minions. Who of course, want to take it back and potentially kill you in the process.
Technically, the ring has some other effects:
Life extension, understanding of the Black Speech, and you're not actually invisible. You're sort of between this reality and in D&D terms, the ethereal plane. It doesn't really imbue the user with any sort of unnatural strength, but a sort of unnatural drive and determination. You don't really need to sleep and you can go without eating and you are typically faster and stronger, but more because of a single-minded focus, than evil empowerment.
The Ring is also quasi-sentient, having a piece of Sauron's power in it, which is what gives it its evil will which is exerts over the bearer and anyone nearby.
Also as Galadriel sort of indirectly states, the Ring has other great power, but requires a powerful entity to use it, which is why she and Gandalf both refuse to touch it or even go near it, because they
could use all of it's terrible power. So it might be worth having some additional unlockable powers based on say, character level, and how long one has had to play with the ring.
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The biggest problem I see with The Ring and its associated mechanics is that even if you assign actual mechanical elements to it, you still need to get the players to properly role-play the mental side effects of being around it, using it, and how they use it. Because doing evil with the ring makes it easier for the Ring to corrupt you, while using the Ring for good makes it more difficult, but you get less power from the ring to do so, which is largely why all the Hobbits can do with it is turn invisible. Presumably really powerful and really evil folks could do more; and of course only the Lord of the Ring could use it to its full potential.