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?
There are three possible reasons for this. The first and the most obvious explanation is that he never knew it was there at all. Rings of power crafty after all, and are usually imbued with magical properties by their creator. For example, the one ring itself, created by Sauron, has a mind of its own. It has the ability to twist and manipulate its wearer, to call to them and corrupt their very core. It can betray its master in favor of another who will get it closer to Sauron.
The magical properties of the three elven rings aren’t as toxic as this, because they were made by the elven creator Calebrimbor, and never touched by Sauron's maliciously tainting hands. Nonetheless, Celebrimbor did give the rings some enchanted gifts, in order to keep them safe from they tyrants. And the one that is the best known is that the rings cannot be seen unless their wearer chooses them to be. Galadriel proves this in the Fellowship of the Ring, when she reveals Nenya, her water ring, to Frodo, but to Sam it simply appears as starlight twinkling through her fingers. Similarly, Frodo only sees that Gandalf has a ring of power very late into their journey:
"As he turned and came towards them Frodo saw that Gandalf now wore openly upon his hand the third ring, Narya the great, and the stone upon it was red as fire."
Therefore, it is quite plausible that Saruman didn’t take the ring when he had Gandalf captive, simply because he didn’t know it was there.
Another explanation could be that Saruma knew Gandalf could possess the ring, but he didn’t feel that it was worth taking. This seems implausible, because all the original rings of power possess amazing abilities and rare magic. But it could be that Saruman didn’t feel any of the three elven rings would be able to help him in his quest to achieve dominion over Middle Earth before Sauron could. The elven rings, as mentioned, were never tainted by the dark lord. Therefor, their magic is still designed for hope, for healing, and to help bring love and light into the world, rather than for nefarious purposes. In fact, the elven rings openly resist evil, which is why Galadriel, one of the most powerful beings to have ever lived, trusts her ring to guide her and to help her bring light into the beautiful realm of Lothlorien. Therefore, potentially, Saruman didn’t feel that Gandalf’s elven ring would have the right kind of power to help his evil scheme. If anything, perhaps he felt it could even make him more of a target to Sauron if the dark lord knew that he possessed it.
The final reason is that Saruman was in the process of trying to create a ring of power of his own. Having spent years delving into the secrets of the rings, their creation, their properties, and their whereabouts, Saruman believed that he would be capable of making his own ring, and using it to boost his status in Middle Earth. When he captures Gandalf, he describes himself as “Saruman the wise, Saruman Ring-maker, Saruman of many colors!”
Saruman ring-maker would explain why he doesn’t need to take Gandalf’s elven ring of power. He is arrogant enough to believe that he already has a better one, one that he has made for himself, and one that he hopes will help him achieve the impossible.