What happens to the gollum?

He leads frodo an Sam towards mordor and slowly turns frodo against Sam. Then when he falls off a cliff after a fight with frodo, we don't see his for a while until Sam and frodo are climbing up the slopes of mount doom. He then attacks them and tries to take the ring. Eventually Sam and frodo escape and are standing on the edge of the lava about to drop the ring into the volcano when gollum hits Sam on the head with a rock and then bites frodos finger off and claims the ring for himsefl, jumping around with joy. THEN frodo attacks gollum an they fight for a bit before they both fall off the edge. Frodo manages to hold onto the rock but gollum, who is holding the ring, falls into the lava and dies.