LOTR: Why Didn't Saruman Take Gandalf's Ring Of Power On Orthanc?

gamerant.com:

On the surface, Saruman seems like one of the simplest Lord of the Rings characters to define. At first glance, it is easy to assess that he is selfish, power-hungry, and driven by his own betrayal and greed. However, he was once a much more pure and enlightened being as a member of the Istari alongside Gandalf the Grey. There is a reason that he was made head of the white council, and a reason that he was well trusted by the other wizards, and indeed many of the creatures of Middle Earth. Saruman once used his vast wealth of knowledge to help the creatures of the world, guide its peoples, and foster peace among those who needed it most. It is difficult to say exactly when and why he was corrupted. Perhaps it was because he obtained a palantir and was slowly indoctrinated by Sauron through it, or because his delving into the rings of power lead him to jealousy and covetousness.

Either way, by the time the War of the Rings comes into motion, Saruman is already obsessed with his quest for dominion. This is why he has already begun creating his army of Uruk-hai orcs, in order to defeat his enemies and capture the hobbits so that he can take the one ring and use it for his own dastardly means. Interestingly though, when Saruman captures Gandalf and holds him hostage on top of the tower of Orthanc, he has another of the rings of power right there under his very nose. Narya, the ring of fire and one of the three elven rings of power, is in Gandalf's possession, as he was left to be it’s safe-keeper after Cirdan the shipwright sailed from Middle Earth. So why doesn't Saruman take it?

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There are three possible reasons for this. The first and the most obvious