It's not immediately clear why cutting off the One Ring effectively destroys Sauron in The Lord of the Rings. The premise of The Lord of the Rings hinges on the fact that the One Ring needs to be destroyed to defeat Sauron once and for all. The Fellowship, and later Frodo (Elijah Wood) and Sam (Sean Astin) alone, set out on a journey to Mordor to throw the Ring into the fires of Mount Doom and thus end Sauron’s threat to Middle-earth. At the very start of Peter Jackson’s trilogy, however, another moment is revealed in which Sauron seems to have been destroyed, simply by having the Ring cut off his finger.
The Lord of the Rings: The Fellowship of the Ring begins with a prologue, explaining the history behind the Rings of Power. Three were kept by the Elves, seven were given to the Dwarf-lords, and nine were given to the race of Men. Sauron forged one final ring, the One Ring, to enhance his power and exercise control over the other Rings. The prologue in The Fellowship of the Ring recreates the final battle of the War of the Last Alliance, which shows Isildur (Aragorn’s ancestor) cutting off the ring from Sauron’s hand, and Sauron is seemingly destroyed. Isildur ultimately fell to the One Ring’s power, which corrupted him and led to his death. The Ring was later stolen by Smeagol (Andy Serkis) before it made its way to Bilbo (Ian Holm) and eventually to Frodo, who became the final ring-bearer.
The prologue, as narrated by Galadriel (Cate Blanchett), states that “…into this Ring, he poured his cruelty, his malice, and his will to dominate all life. One Ring to rule them all.” The reason that being cut off from the Ring nearly destroys Sauron, however, is a little more complex than that. The books by J.R.R. Tolkien delve deeper into what the One Ring in The Lord of the Rings is and why Sauron’s life is bound to it. In order that the One Ring was capable of exerting the influence he needed it to, Sauron knew that an extraordinary amount of power would need to be stored within it. So, to make it as strong as possible, Sauron imbued it with a substantial part of his soul. As a result, his fate was bound to the One Ring, and if it were ever to be damaged or destroyed, Sauron’s power and strength would be destroyed along with it. This "destruction" is what is seen during the opening of The Fellowship of the Ring.
In Tolkien’s work, the soul (fëa) and the body (hröa) are two parts that make a whole. The soul is powerless without the body, and the body will die without a soul. In Sauron’s case, as he imbued the One Ring with a significant part of his soul, his spirit would remain in Middle-earth until the Ring was properly destroyed. As a result, while cutting off the Ring did destroy Sauron’s body, it didn’t destroy his spirit, and thus paved the way for Sauron’s eventual return, even without a proper physical presence. This is why his will is carried out by those who serve him in The Lord of the Rings, like the Ring-Wraiths and Saruman.
Sauron’s quest for power dominates much of Middle-earth’s history. It is his greed, however, that eventually leads to his ultimate destruction in The Lord of the Rings. By imbuing the One Ring with a part of his soul, Sauron sealed his fate by giving Middle-earth’s heroes a way to destroy him once and for all.
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