Saruman's death

[Azrakhor Akallabeth]

As for falling under the power of the One, I differ in opinion on the issue because it says in the books that the Elf Lords perceived the mind of Sauron when he donned the One RIng and hastily removed their own. Yet, after the fall of Sauron by the hand of Isildur, they began wearing them again.

I can't recall if it states flatly that they removed their Elven rings for fear of falling under the dominion of the One or not, but I think it's true that they did so for that reason. Once again, the arcane knowledge that gave birth to the ability to forge the rings were said to have come from Sauron and his influence nonetheless, and had enough of a link to his own being to constitute a threat.

Having said that, I reiterate that Saruman was probably the "source" of his own missing knowledge, apart from that discovered by the Elves, and therefore wouldn't have necessarily fallen under the power of the One. But even if Sauron was powerful enough to instantly infuse himself into every ring subsequently made, if any, he could only do so while in possession of the One Ring. So whether Sarumans ring was subject to the One (which I doubt) or not, until Sauron recovered the One Ring, he and the Elf Lords were quite safe from the hazard of becoming wraiths or Nazgul. And given their magically potent and immortal nature, becoming rather, dark and twisted servants of a higher order than the Nazgul would be liklier in any event.