An act of courage
Somehow this escaped me completely, but a review of this movie says that all that was needed for Ringo to lose the ring was to display courage - at movies' end of course he tries to warn the others of an ambush and the ring flies off his finger. He then tells his mates about the courage thing, but that's the first I'd heard of it - or so I thought.
Earlier on, at the end of Part One, Ahme, preparing to inject Ringo with a shrinking drug, turns to us and says "Alas - if he were brave, this would not be necessary!" I always thought that was a cryptic comment suggesting he should have bucked up and been a sacrifice, but apparently not.
Ages ago my sibs had the novelization of the film in our possession (a good long time before I saw the movie) - can't recall it now, but I wonder if there was a more direct reference to it there?