So, my 13 yo son is reading the LotR's and we were having a discussion on the Rings and the histories and such and I got smacked up side the head with a thought.
I could find no mention that Saruman had knowledge that Elrond and Galadriel had rings or that he knew that Cirdan had one, and what had become of the three. Saruman knew the number of the rings, knew about the Nine, and the Seven, and that the Three were in possession of the Elves.
As the head member of the White Council, did he surmise that his colleagues on the Council were Ring Bearers? Did he not care about possessing them as they were "weaker" and subject to The One, so, in turn, why bother to take them by force, hoping instead to find The One? If he had had such suspicions, would he have reported this to Sauron under duress in the Palantir?
Not that Saruman totally didn't care - in the Unfinished Tales, The Istari chapter, this is what is said (Grey Messenger being Gandalf and White Saruman), about the fact that Cirdan gave Narya to Gandalf: "And the Grey Messenger took the Ring, and kept it ever secret; yet the White Messenger (who was skilled to uncover all secrets) after a time became aware of this gift, and begrudged it, and it was the beginning of the hidden ill-will that he bore to the Grey, which afterwards became manifest." It is also said that Cirdan chose the Grey Messenger because 'he divined in him the greatest spirit and the wisest'. In this version of the story about the Istari Gandalf comes the last, so Cirdan could have given the Ring to any of the Wizards that came earlier, including Saruman, but he didn't. Interesting.
I agree that Saruman was preoccupied with The One Ring anyways so maybe he was not after the 'lesser' great rings, just annoyed that Gandalf was found more worthy than he? And he made some ring for himself, we are just not told whether it had any magic power to it.
So then, when Saruman took Gandalf prisoner, was Gandalf not in possession of the ring at that time? Or was it not possible for Saruman to take the ring from Gandalf, even though he took Gandalf's staff? Interesting!