Why didn't the One Ring go after Samwise Gamgee as it did to Frodo and Boromir?

Answer 1

Actually the One Ring did take over Frodo in the end, attracting Sauron's attention resulting in him sending a Nazgul to take the One Ring from Frodo. But before the Nazgul could arrive Gollum bit off Frodo's ring finger and fell into the "Cracks of Doom", destroying the ring, Sauron, and the Nazgul.

Answer 2

Well, the ring tried to get the weak men, but hobbits are stronger against the power of the ring. The ring tried to get Samwise Gamgee, but he was too strong as you could see, in the tower, after the orcs took Frodo out of the spider's cave. I don't know if you saw that scene in the normal version because I have the extended version. Anyway, in that scene, you see that Samwise kept the ring because he thought Frodo was dead, so he took the ring. When he found out that Frodo was alive, he followed the orcs and saved Frodo. Then Frodo panic's because he thinks that the orcs have the ring, and everything is lost. Then Samwise shows the ring out of his pocket and tries to give it to Frodo, but you can see it is hard for him to give it up. But he knows it is not his task.