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In the Foreword to the second edition of The Lord of the Rings, Tolkien implied that Saruman's ring-making ability had gotten pretty advanced. He wrote that if the War of the Ring had resembled World War II,

Saruman, failing to get possession of the Ring, would in the confusion and treacheries of the time have found in Mordor the missing links in his own researches into Ring-lore, and before long he would have made a Great Ring of his own...

It shows Saruman's ambitions to be on a level with Sauron as a controller of peoples, a maker of Rings of Power.
Him actively using the Palantir and thinking he is not controlled by Sauron through it is another sign of this.

Which of course shows Saruman's corruption and folly.

I thought that an interesting aspect of this question was that Gandalf had possession of Narya, one of the Elven Rings of Power.

As an aside, although Gandalf the Grey appeared subordinate to Saruman the White, Cirdan gave Gandalf this ring.
(Gandalf himself says Saruman is greatest of their order; Cirdan sensed a greater good in Gandalf.)

However, as stated, Saruman at least temporarily defeated and imprisoned Gandalf, so Saruman's newly forged Ring must have been significantly powerful.

Saruman was also considered a Lore-master, especially about the history of the The Rings of Power (one reason Gandalf visited Saruman in the first place), so I wonder if Saruman knew where the Elven rings were being kept and was jealous of Gandalf.

It seems likely the Saruman's ring was made by him as he implied. There is however the small chance that it could be a lesser ring made by the Elves of Eregion in the second age, presumably given to him by Sauron as a gift (or a trap), or found by him during his researches.

In Eregion long ago many Elven-rings were made, magic rings as you call them, and they were, course, of various kinds: some more potent and some less. The lesser rings were only essays in the craft before it was full-grown, and to the Elven-smiths they were but trifles — yet still to my mind dangerous for mortals. But the Great Rings, the Rings of Power, they were perilous.

_{LotR I 2 (60)}

You're right - nothing more is ever said of this ring, even when we see Saruman later in Isengard or in the Scouring of the Shire.

Saruman definitely lays claim to making the ring in the Fellowship of the Ring:

He wore a ring on his finger. ... 'For I am Saruman the Wise, Saruman Ring-maker...'

Gandalf never calls the claim into doubt so presumably he believed him, and we do have the word of the author from the foreword of the Lord of the Rings that Saruman had been researching Ring-lore:

Saruman, failing to get possession of the Ring, would ... have found in Mordor the missing links in his own researches into Ring-lore, and before long he would have made a Great Ring of his own with which to challenge the self-styled Ruler of Middle-earth.

We now definitely know that Saruman's ring was not a Great Ring but that Saruman had been researching rings, so it should be reasonable to assume that the ring is of Saruman's making rather than a gift from Sauron. It is never revealed what powers it may or may not have - Saruman definitely ascribed some potency to the ring (otherwise why call attention to it), but that may have been a bluff on his behalf.

One reason that we may not have seen the ring again was because of the nature of the later appearances by Saruman - in Isengard he was up in Orthanc, not seen closely enough to identify the ring. The remaining appearances all occur after the destruction of the One Ring and given all Ring-lore is ultimately derived from Sauron's influence on the Elven-smiths of Eregion, even the Three, Saruman's ring (if it had power) may have been rendered useless and hence discarded once that destruction occurred.

Sauron (nee: Mairon) as well as Saruman were Maia of Aule the Smith. It is safe to say that both of them studied and learned the craft of magic rings from Aule. Sauron, however might have used some powers that Melkor lent him or gave him in the making of the One, as he was Melkor's chief lieutenant, and Melkor taught him many things and gave him additional power to do his evil works. Saruman never had the knowledge that Sauron apparently was granted from his master, and therefore though he could make magic rings probably due to the knowledge they had both received from Aule, he didn't know how to make one to enslave other people, as of course they would never learn such things from Aule, as doing so would be against the will of Eru.

The Silmarillion doesn't say any of that but I think he is on the right track. The lesser rings were not like this. And Saruman's powers after he became plainly corrupted differ greatly from Gandalf and Radagast's powers. It could be simply his ego leading to further demonstration of his powers, but the idea that he has manifested this power in an object could explain things like his "voice" something known to have powerful influential effects, something that he had in common with Sauron.

I think it is probably a case of Saruman imitating people- he is frequently imitating Gandalf, (example- he smokes pipeweed because Gandalf does, he pays attention to the Shire because Gandalf does, etc) and he also aspires to be the "Lord of the Rings" like Sauron. It also might have served a practical purpose, maybe it allowed him to communicate with his captains or something, I'm just speculating here. He is trying to raise himself up to the level of Sauron here- previously he had been trying to get Saurons ring, but this is implying he is inferior to Sauron, since he can't make his own ring. He tries to defy this by making his own ring, then he will have a ring of power like Gandalf and have made a ring of power like Saruman

Could it be that Saruman's creation, his ring, has something to do with the "will" that lends speed to the Orcs and tires Aragorn, Legolas, and Gimli during their pursuit of Merry and Pippin?