:
: : : How's this for an analogy:
: : : I offer people a machine that will provide electricity to their homes at no cost. Unknown to those people I make a machine that allows me to take not only total control the their machines, but also control every electical device in their home. Three of the people discover this and immediately disconnect their machines. They reconnect and use their machines only when they know I am no where near my machine. My machine is destroyed, none of their machines no longer work, and I am killed because my machine also powered my pace-maker.

: : Nope, doesn't work...
: : Sauron had much to do with the physics of the rings, but gravity will hardly shut off if Newton dies! He may have discovered the way to channel the powers into rings, but i doubt if all ring lore will die just because Sauron does.

: : Gandalf

: The main problem with the above analogy is that the machines worked before the One machine was made, so there's no reason why they wouldn't work when the One is destroyed. It works better if we say that the One was plugged to the same network, and when it was destroyed, it caused a power surge which burned out the other machines.
: However, I can't fathom what the conection between the Rings was that Sauron was able to plug his Ring into, nor what exactly would have cause this 'power surge'.
: -Foradan

Yes, the machines worked when I gave them away. Had to let the people think they had something special, didn't I? When I turned on my master machine, it not only took control the other machines, but also became the sole supply of power to operate those machines. Oh, I didn't tell you that, did I?
Seriously, the One Ring was the link between other rings and the power,lore or whatever that made them function. When the One was destroyed the link was broken.