It seems to me that Gandalf never considered Saruman's ring to be of big significance, as he did not care about it after Saruman's defeat. He would have never left Sauron with The Ring on his own in Barad Dur in similar circumstances, I am sure.
Gandalf might consider Saruman's ring as a lesser one, which was able to amplify Saruman's senses and also his control over his army or to make him invincible in Orthanc. He could also think that with the demise of The One Saruman's ring looses its power. It looks at least probable if Saruman's ring was made with the use of the same knowledge that created older rings of power. Can we guess that the failure of Saruman's ring was the reason why he decided to leave Orthanc one day?
But we can also make up a different story. What if Saruman's ring was a ring of power, much lesser than The One but not attached to Sauron's and elven stock, so the Wielders of The Three had no true knowledge about it.
Lalwendė's reasoning gave me an idea that if Sauron's goal was craft, evil and weird indeed, but very personal in its core, Saruman was the Spirit of Technology. Apparently, he built this main skill and interest into his ring and thus it had not failed after Sauron's end but functioned in a pretty different way. Shall we imagine it was initially weak but was able to accumulate its strength in the way technology establishes its power over people (iPhone and iCloud, for instance
). And if we than try to think of our world as a successor of ME (Tolkien himself was inclined to think at some point), it means, Sarumans ring is still around and at work... )))