Sauron created the One Ring in Lord of the Rings, to be able to control all the other Rings of Power, which were made by the Elves. Of course, the Elves took off their Rings of Power once Sauron made the One Ring. Some of the Rings of Power that had been made by the Elves were then given to Dwarves and Men by Sauron. But why not Hobbits? Why were they left out of the gifting of the Rings? Did Sauron simply not see the Hobbits as important enough to control? Did they perhaps not exist on Middle Earth when Sauron was giving out the Rings? And how did this lead to Sauron's downfall? To understand the answers to these questions, it is crucial to know the history of Middle Earth. It is equally essential to understand Sauron as a character.
Sauron did not create the other Rings of Power, but he did play a role in their creation. During the Second Age of Middle Earth, Sauron disguised himself and called himself Annatar, Lord of Gifts. He then presented himself to the Elves. He played a role in guiding Celebrimbor's creation of the Rings of Power. And then later on in the Second Age, he created the One Ring in Mordor. The Elves, however, were too clever to fall for Sauron's trick entirely. Once they sensed that he had created and was using the One Ring, the Elves took their Rings off. The three Rings of the elves, however, were created solely by Celebrimbor and never touched by Sauron. All of this happened in the Second Age.
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Hobbits, however, did not pop up in the Shire until the Third Age. That is when they are first recorded, anyway. It is possible that they existed before then, though they might not have. Since the Shire Reckoning only starts in the Third Age. Of course, the Hobbits could've lived somewhere else before the Third Age and called their measure of time something else. It is also possible that, if the Hobbits did exist in the Second Age, and Sauron knew of their existence, he thought them beneath his notice. Indeed, Sauron does not seem to interact with Hobbits at all until he realizes that one of them might have the One Ring. And even then, he only has Gollum's words to go on about a Baggins in the Shire. Sauron probably had no idea what any of that meant, or that it referred to a Hobbit. Sauron might also have not known about Hobbits, or known very little.
Sauron probably did not view the Hobbits, if he did indeed know of their existence, as important enough to try to control. Most Hobbits do not have lofty aspirations, only wanting to live a peaceful life. Peace, of course, is no use to Sauron. He wants to dominate and control all of Middle Earth. He, therefore, would not see any point to trying to control a race of beings with no aspirations beyond wanting to live good lives. In fact, Sauron probably, if he knew of the existence of Hobbits, looked down on them for not wanting more than what they had. It's also possible that Sauron, after he failed at controlling the Dwarves, wondered if the Hobbits would give him similar problems and decided not to bother. Most likely, though, Sauron knew something of the existence of Hobbits at least by the Third Age. And he never thought of giving them Rings of Power, because he didn't think they'd be any good to him. Sauron also wanted to give the Rings of Power to beings who were already powerful, and Hobbits don't exactly embody power.
This probably led to Sauron's downfall. Sauron as a character is very arrogant and full of himself. Anyone who he feels he can control or who could help him to control others is important in his eyes. Anyone who can do neither is, therefore, unimportant. And Hobbits do not seem like a Race of people likely to want to dominate others. Because of that, Sauron would see them as not only unimportant but unable to oppose him. After all, what can a little Hobbit do against Sauron and his armies of Men and Orcs? Of course, this ended up leading to Sauron's downfall, when a Hobbit, Frodo Baggins, volunteers to take the One Ring into Mount Doom and destroy it. If Sauron had seen any use for Hobbits, he could have possibly given them Rings of Power and might have then been able to control them. Since he didn't, however, he couldn't control the Hobbits and several of them were able to oppose him, Frodo included.
Sauron also, if he had cared to notice the existence of Hobbits, would probably have figured out what Frodo and the other members of the Fellowship of the Ring were up to sooner. He would have known what Hobbits were like, their selflessness and loyalty, and how they were, in some ways, the ideal Race to try to destroy the Ring, even though it did affect them eventually. Even Frodo had difficulty at the end. However, Sauron did not know a thing about Hobbits and therefore they were in the ideal position to help destroy him and the Ring. Of course, not every Lord of the Rings fan will agree with this assessment of Sauron's reasoning for not giving the Hobbits Rings of Power. Some might feel that it made sense in a way that he did not give them Rings. However, if he had, he might have lasted longer on Middle Earth. And it definitely seems that he viewed the Hobbits as unimportant right until the end when they played a role in defeating him.