In preparing to discuss the section of Plato's Republic in which Glaucon tells the story of the Ring of Gyges, I was struck by the abrupt way in which Glaucon stops talking about the ring: "Enough of this." What is the Ring of Gyges story supposed to prove? Glaucon's reaction sounds as if he's no longer convinced that the Gyges story is the right approach. This is significant since it's this story that is often mentioned--in ethics and philosophy classes (like mine)--in discussions of egoism.
Socrates and Glaucon had begun by considering what kind of good justice is, and Glaucon defends the common view that justice is only an instrumental good--or better, a necessary evil (because we must deal justly with others because we cannot get away with dealing unjustly with them).
People who think that way presumably would abandon any commitment to justice (or fairness) if they got hold of a Ring, and as Glaucon surmises, would think anyone who didn't was foolish. Importantly, the Ring of Gyges story does not prove that everyone would abandon the commitment to justice upon stumbling upon such a ring. (Glaucon seems aware that there is not much of a proof here that psychological egoism is true.) Perhaps he then abandons the story because the question he and Socrates are after is whether the people who think this way--in a moral and not merely a psychological sense--are right. Thus, he proposes the more powerful thought experiment in which we consider who has the better life: the perfectly unjust person who appears perfectly just, or the perfectly just person who appears perfectly unjust.
I don't have a grand point to make here, beyond the observation that, although it captures the imagination, the Ring of Gyges turns out not to be the main attraction. On the other hand, perhaps I'm selling it short: what's important about it is that it helps Glaucon clarify the idea that perhaps all that really matters--or all we actually (if mistakenly) care about--are appearances. And one way to get at that is to consider how we would act (or think) if we could disappear.