How Did the Resurrection Stone Get in the Ring?
The Resurrection Stone Came into the Possession of Harry Potter via Dumbledore, Kept Hidden Inside the First Golden Snitch Harry Had Ever Caught in His First...
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Similarly, why was the resurrection stone in the ring?
The Resurrection Stone is one of the fabled Deathly Hallows. In "The Tale of the Three Brothers", it was the second Hallow created, supposedly by Death himself. It was bestowed upon Cadmus Peverell after he requested, as his bounty, something with the power to recall loved ones from Death.
Also, how did Harry know the Resurrection Stone was in the snitch? Placing the Stone within the Snitch Before his death at Snape's hands, Dumbledore magically sealed the cracked Resurrection Stone inside the Snitch that Harry had caught at his first ever Quidditch match by accidentally swallowing it.
In respect to this, how does Harry Potter come back to life without the Resurrection Stone?
Harry came back because Voldemort had used Harry's blood to regenerate. The protection that Lily's sacrifice provided to Harry got redoubled and now ran in both Harry's and Voldemort's veins. Thus, when Voldemort cast the killing curse on Harry, it didn't work properly. It only destroyed the horcrux inside Harry.
How does the resurrection stone work in Harry Potter?
It works for the one who possesses it (or at least who last possessed it) only. So, there would be a strong desire to keep it from a living Voldemort - hence he drops it. Harry didn't live just because he had the Resurrection Stone. It was because he was a horcrux and Voldemort killed the horcrux, not Harry.