Elves are makers, and builders. It is their nature, especially the Noldor. They just don’t like leaving things the way they find them. The Noldor made gems by the myriads in Aman, when they came there. The Teleri, who were given many of them, strew them about the beaches of the undying lands, thinking them beautiful, and wanting to make the land more beautiful.
But the Noldor were proud, and sought to make things beyond their own reach into things they could hold, and control. The most egregious example of this was Feanor, who took the light of the two trees of Aman, and blended them, and captured them in the Silmarils. The nature of such things is fraught with doom, in the world of J. R. R. Tolkien. Everyone who touched the Silmarils was either killed, or maimed as a direct result, except the few who were utterly banished from the world, to carry them.
The rings were yet another foray into the realm of power taken by the elves after Morgoth was banished from the world. They were encouraged in this by Sauron. It was not said that the nine, the seven and the three were the only rings of power ever made, only that they were the greatest, and Sauron made the one to bind them to his will. But even Sauron was touched by the peril that comes to those who use device and artifice to reach beyond themselves. He must give his own power into the ring, to make it strong enough to bind the lesser rings. Then he is bound to the ring, as much as any of the lesser beings were.
Why do it? To gain stature, and power in the world. The rings granted great enhancement to the way a wearer was perceived by friend or foe. In the case of the three, it also gave the wearer the ability to preserve and protect those who were close and places near at hand. Healing and preservation were benign intentions of Celebrimbor. The nine gave long existence, but only a pseudo life. They gave great power to dominate the hearts of masses of other beings, with fear, and despair. The rings given to the dwarves enhanced their already strong abilities over rock and stone, and metal in making mighty weapons, and armor, and jewelry. But the rings were still tools of power, and had in them the doom of those who seek power.
By the desire for greater power than was native to them men became enslaved to the rings, and through the rings, made at first pawns to the plans of Sauron, and finally slaves to his will. Dwarves were not easily subjected to control, but greed and selfishness engendered in them did great harm to their people. Hobbits were of no interest to either Elves, or Sauron in the second age, when the rings were made. The nine were not made for men, but were given to men by Sauron. The rings were probably each made for the elf that created them, and each intended to enhance that elf’s power in the world. Those elves were unable or were unwilling to keep their rings.
It is interesting to note that very few of the Valar are associated with objects of any sort. There are no great weapons of the Valar, or Crowns, or thrones. The power of the Valar is inherent in themselves, in their voices, or their senses. Even the Maiar tend to do things without props, as Melian protected the Hidden Realm of Doriath with a veil of confusion and misdirection, rather than a physical barrier. Morgoth, on the other hand built towers, and dungeons, and wielded a hammer. He desired the Silmarils, and put them into a crown.
Tris