The main point of the arm-ring is that it's a piece of treasure Eustace is trying to hoard for himself. He puts it on because he's selfish and grasping, like a dragon. That's why he turns into a dragon, not any curse attached to the ring in particular.
Sleeping on a dragon’s hoard with greedy, dragonish thoughts in his heart, he had become a dragon himself.
But an arm-ring in particular resembles manacles or chains, here one that he chose to put on (even though it didn't in fact fit). The thing he thought he wanted ends up imprisoning him:
What woke him was a pain in his arm. ... [P]resently it occurred to him that the bracelet which he had shoved up above his elbow had become strangely tight.
Being caught by the bracelet and trapped in the body of a dragon amount to the same thing here, since they both stem from Eustace's poor character.
He wanted to get back among humans and talk and laugh and share things. He realised that he was a monster cut off from the whole human race. An appalling loneliness came over him.
He was already cut off from the others by his own selfishness; now his outside matches his inside.
The arm-ring stays on his arm when he turns back into a human, but it no longer traps him. It's not mentioned during his description of his transformation, but when he gets back to camp and starts to tell his story to Edmund, he says
"...And that beastly arm-ring was hurting like anything—"
"Is that all right now?" [said Edmund.]
Eustace laughed—a different laugh from any Edmund had heard him give before—and slipped the bracelet easily off his arm. "There it is," he said, "and anyone who likes can have it as far as I'm concerned."
As it turns out, no one does like. Even if they might otherwise have wanted to bring it home as a memento, everyone seems to be thoroughly put off treasure-hoarding by this point.
The Lord Octesian's arm ring had a curious fate. Eustace did not want it and offered it to Caspian and Caspian offered it to Lucy. She did not care about having it. “Very well, then, catch as catch can,” said Caspian and flung it up in the air. This was when they were all standing looking at the inscription. Up went the ring, flashing in the sunlight, and caught, and hung, as neatly as a well-thrown quoit, on a little projection on the rock. No one could climb up to get it from below and no one could climb down to get it from above. And there, for all I know, it is hanging still and may hang till that world ends.