[Vanya]

Well, as for Grima, we had a small discusion, and came to conclusion that it is not likely that Grima wore a ring made for him by Saruman, in order to have strong influence on Theoden. In that case not only Theoden would be afected, but all the other rohirrim as well would be. And he was not called the Wormtongue for nothing - he was seen through. And besides, Saruman's Voice was mighty wheapon, it could influence Nazguls, an wiser persons than Grima, he didn't need the Ring for that purpose.

Ther is another point that you should reconsider. In the VII book HoME, the same scene as in Council of Elrond is told, but Ring-making is not mentioned.

“I came to the great gate in the wall of rock and they said that Saruman
expected me,(34) and I rode in, and the gate closed behind, and a
sudden fear came on me.
'Saruman was there but he had changed. He wore a ring on
his finger. "So you have come, Gandalf," he said to me, and I
seemed to see a deadly laughter in his eyes. "Yes, I have come
for your aid, Saruman the White." But that title seemed to fill
him with anger. "For aid?" he said coldly. "It is seldom heard
that Gandalf the Grey sought for aid, one so cunning and so
wise, wandering about the lands, and concerning himself in
every business, be it his own or others".'
'"But now matters are afoot," I said, "that need all our
strengths [?in union]. The Chief of the Nine is guised as a Rider
in Black and his companions likewise. This Radagast told me."
'"Radagast the Brown," he said, and shook with laughter.
"Radagast the Simple, Radagast the Fool. [Added in pencil: Yet
he had just the wits to play the part that I set him.] He must
have played his part well nonetheless. For here you are [added
in pencil: and that is the purpose of the message]. And, Gandalf
the Grey, here you will stay. FOR I AM SARUMAN: SARUMAN THE WISE, SARUMAN OF MANY COLOURS. For white cloth may be dyed,
and the white page overwritten, and the white light broken."
[Pencilled in margin without direction for insertion: And I
looked then and saw that his robes were not white as had been
his custom, but were of many hues, and with every movement
he changed hue.]

And in the Foreword of the LOTR, Tolkien says:
"Saruman, failing to get possesion 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 with which to challenge the self-styled Ruler ofMiddle-earth."

Since this never happend, we can make a conclusion that he NEVER DID FIND those missing links he needed, and therefore was not able to make one (or he could make some insignificant ring)