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

[Eönwë]

Originally Posted by Morthoron

'One Ring to rule them all, One ring to find them...'

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.

But wait. I think that the One Ring was made just to control the Nine and their owners. It was just the first stage. Once he had control of them they could do whatever he wanted. The true power of the ring was himself, after all. Maybe he wasn't as stupid as we tend to think. But then this theory does sort of devalue the power of the ring...

Or, what about his other nine fingers? Isildur might have cut one off, but he could have been wearing the others. Then, later when he returned, he could have searched for the remains of his body and found the rings. Or did his body disappear when he "died"?

On the other (third, and therefore not normally possible) hand, he may have just kept them safely somewhere in Mordor where no-one, not even the Nazgul, could find them.