Why did Ryuk write Light's name on the death note?

Answer 1

Well as it is the first human who touches the note when it is dropped in the human world becomes the new owner. It didnt really matter to Ryuk because the note that he dropped wasn't in fact his. At the time it was it belonged to Sidoh and towards the end of the series Sidoh come to take the notebook back witch leaves two notebooks in the human world (not including ryuks).

Getting to the point when it is time for the current owner of the death note it is the job of the shinigami possesssing that person to write his/her name into his/her notebook. In a sense it was eaither Light's time to die anyway or Ryuk was fed up of waiting for light to die so he wrote his name in the notebook early.

If you mean why did Light HAVE to die then it's obvious because Light or rather Kira was an evil mass murderer who didnt really create the "perfect world" the bad people were still out there they are just too afraid to commit crimes. Light was the bad guy but i kinda liked his personality.

Answer 2

Yes, as Light's time was over.