Game #76859066, rated.
I played Prince and attached Research to it. The next turn I trashed an Estate with the Research from Prince and it put two cards aside. The next turn, it put the two cards into my hand, but Research was discarded and Prince remained in exile.
Is this correct?
Yes, from the Official FAQ (can see it here http://wiki.dominionstrategy.com/index.php/Prince)
"If you don't discard the Action then you stop playing it with Prince; Prince at that point is just set aside doing nothing for the rest of the game. That won't normally happen but will happen for example if the Action is a Feast or Mining Village and you trashed it, or if it's a Duration card and so it stayed in play, or if it's a Madman and was returned to its pile, or if it's an Island and was set aside, or if it's a card you put back on your deck with Scheme."