When the Nazgûl lose their bodily form, what happens to their rings?

[obloquy]

Originally Posted by Morthoron

Odd, isn't it, that Sauron can demand the Nine Rings back without the Ruling Ring? Seriously, it is a flaw in the plot. If Sauron has the ability to control the Nazgul without the aid of the Ring, then the Ring is unnecessary, isn't it? The Rings of the Elves proved more beneficial to the Free People than the One Ring to its master. If it weren't for mistakes by Sauron and his allies, the Dark Lord would have won the war without the Ring. Had it not been found, he certainly would have won a war of attrition.

I hate to sound heretical, but there is a disconnect there.

It is not a flaw. Sauron could have dominated Middle-earth even without his Ring; he was supposed to be in "rapport" with the power that he had invested in it. The Nazgul were his through and through, rings or none. Retrieving The One remained critical, however, for while it was not in his possession, he remained vulnerable. There were several individuals still in Middle-earth who might have been capable of usurping mastery of the Ring, and even an incomplete claim on the Ring (like Isildur's) would have been enough to greatly inconvenience him.

Sauron's actions suggest that he expected a gambit of this kind rather than the madness of destroying the Ring. Any victory he could have had before recovering The One could be undone by a powerful claimant, and the likelihood of such a claim would increase as his realm and power grew. Step one, the most essential step, was to obviate this potentiality, preferably while his enemies believed they still had some slim window of time.

Here's a little discussion of Ring-claiming.