Such a wonderful book from my favorite author!
I'm pulling from Classical Traditions in Modern Fantasy, but in short, the ring (similar to Tolkien's ring in the LOTR trilogy) is a painful, constant reminder to Eustace of his greed and his magical transformation. Remember, up to this point, Eustace was very reliant on logic and refused to believe Lucy and Edmund on some of the more "magical" aspects of Narnia. This armband constantly sank into his skin. He was without relief. The very item that transformed him was now an instrument of torture. There are allusions to other authors (the aforementioned Tolkien; Socrates) but the armband became an instrument of not only pain, but of change. After his transformation back to Eustace, he was immediately more helpful, more humble. Caspian tossed the armband on an unreachable rock, so that other travelers would be unable to repeat his trial.
For a deeper review (and further Socratic information), I refer you to the excellent Classical Traditions in Modern Fantasy, available on Google Books, Brett M. Rogers, Benjamin Eldon Stevens
Oxford University Press, 2017. Particularly helpful are pages 149-151. Brett M. Rogers, Benjamin Eldon Stevens
Oxford University Press, 2017
Please feel free to get in touch with me if you'd like to pursue this information and analyze further! I'd be happy to help!