Seeing that Light has finally lost, he is killed when Ryuk writes his name in his own Death Note, just as the Shinigami had warned when they first met.

Does L or light die in Death Note?

After Ryuk kills Light, Ryuk offers L the Death Note. ... Twenty-three days after writing his name in the Death Note, and after burning all the remaining Death Notes and having a conversation with Soichiro Yagami, L dies peacefully while eating a chocolate bar, with a picture of Watari lying right beside him.

Does Ryuk die for light?

That's why Ryuk didn't die when he write Light's name in the deathnote. actually its because rem liked misa and that ryuk had no feelings for light, and rem used the death note to prolong misa's life, while shinigami are supposed to shorten life.