Lucy has a little portrait of Edward in her pocket that she shows to Elinor.

She tells Elinor
"I have one other comfort in his picture; but poor Edward has not even that . If he had but my picture, he says he should be easy. I gave him a lock of my hair set in a ring when he was at Longstaple last, and that was some comfort to him, he said, but not equal to a picture."

I can imagine Lucy giving him the ring and saying something like "I keep your picture with me always, Edward. Promise to wear this always." Considering that they are engaged, and although he has met someone he prefers to her, she has done nothing wrong that he knows of--he could hardly refuse. If Edward was coerced into making such a promise--and Lucy is extremely manipuluative--I think he would be too honourable to take it off.