There's no Frodo here, folks: this ring of invisibility goes back way before Tolkien wrote his famous trilogy, or even before Tolkien's great-great-great-great-great grandparents were born.
This story belongs to Glaucon, who is trying to encourage Socrates to provide a better definition of justice. The story Glaucon tells is a thought experiment in which he attempts to demonstrate that people only act justly because they are worried about their reputations. They worry that if they act unjustly, other people will act unjustly to them.
Glaucon describes how, once upon a time, a man found a ring. When he turned it a certain direction, he suddenly became invisible. Now, what did this man do then? Save the world? Nope. He acted completely unjustly all the time. He knew that acting unfairly was actually a great way to get ahead, and now that no one would ever know what he was doing, he could do whatever he wanted.
Glaucon says that even if there were two of these rings, and they came into the possession of someone just and someone unjust, the just person would act in exactly the same way as the unjust person—meaning the just person would act unjustly, too. Bummer.
But remember: Socrates ultimately meets this challenge by showing that justice is actually something that can make you happier (check out our section on "Justice" for more). So this ring of invisibility story is only a temporary downer. Sweet.