magical items - What is the significance of Saruman's Ring?

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.