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human
[LOTR] If Shelob killed Frodo, would she have tried to claim the Ring?

Gollum's plan was for Shelob to kill the hobbits so then he could take the Ring. But would Shelob have allowed that? If Frodo was dead and the Ring was right there for her to take, wouldn't she try to claim it for herself?
gpt
The ring was made by Sauron, and his malice and will to rule over others (which in itself was but the inherited will to rule of Morgoth) was imbued in the ring.  
This is important because for the ring to give power to any creature, the creature must have a will (i.e. a soul or something alike). If a fox, a fish, algae or a crab were to touch the ring, it would not affect those creatures. As such, when the ring was lost in the Anduin, certainly it was exposed to life as there are fish and all sort of creature in the river. Only when Smeagle saw it that it was awakened and started exerting its power.

The children of Iluvatar have will and are affected by the ring. The dwarves were given will by  Iluvatar and are also affected. Supposedly the ents were also given will by Iluvatar, as to fulfil the wishes of Yavanna. The Maiar as Gandalf, Sauron and Durin's Bane (the balrog) also are affected by the ring. The great powers of Arda (the Valar) are above Sauron himself and most likely are unaffected by his malice and, therefore, the ring.  


Dragons are a hard one to guess. Morgoth could not create life (i.e. create a creature with a soul. He could create a working body as Aule and Yavanna also did) because to do that he needed the Flame Imperishable with is within Iluvatar and his alone to use. So the dragons, as they are not maia, are made in mockery (i.e. Morgoth made artificial selection like we do with dogs) of some existing creature.  Depending of the creature that was corrupted, it could or not affect dragons.

  
And now, what about Shellob? Shellob was of the line of Ungolianth. Ungolianth was a primordial creature. We do not know what those creatures are and where they come from. They were not created by Morgoth nor are they servant, but they serve only themselves. If the Valar know what they are, they did not share with the Noldor or the Maiar, as none knows for sure what they are. So we can't be sure whether they have a soul or whether they would be affected. Knowing that even Morgoth could not bend those creatures to his will, Sauron would not stand a chance. Therefore Saurons creations, like the ring, would be incapable of influencing them as much as it was incapable of influencing Tom Bombadil.