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

[Thinlómien]

Originally Posted by Inziladun

I think I recall a quote from UT (don't have the book handy) that implies Sauron could be certain of their loyalty because they were enslaved by their rings, which he held. I think that was from the chapter The Hunt for the Ring.

Now phrased that way, it sounds very familiar.

But hmm... I wonder if Tolkien came up with that explanation merely because the problem of the disappearing rings would otherwise be too difficult...