Before Ryuk killed Light, was it possible that Light could have relinquished ownership of the Death Note so Ryuk wouldn't write his name?
Light actually relinquished ownership of the one Death Note he had back in Episode 29. Right before that point, Misa was the owner of Gelus' Death Note, Light was the owner of Rem's and Mello's mafia underling was the owner of Sidoh's. After giving up ownership, Light has Misa lend him her DN so that he doesn't lose his memories. She is still the owner since she still has her Shinigami eyes. Light never takes back ownership of any DN for the rest of the anime. I guess we can assume he keeps some sliver of a DN page on his body somewhere to make sure he has all the memories.
The fact that Light is not currently an owner of a DN is not relevant. At the very end of the last episode of the anime, no human is actually in ownership of any of the Death Notes. However, Ryuk still has to write the names of the original owner in his DN. according to Rule LXIV:
If nobody claims the ownership of the Death Note [...] gods of death are [still] obliged to confirm the death of the first owner and write down that humans name in his/her Death Note even when he/she is in the world of gods of death.
in the box thing below there are massive spoilers for death note only hover over it if you have watched the anime
It would not matter as he only had a small snippet of the book (corner i belive) and the police have the book as they caught the person that had the real book and swapped it with a fake book as it broke his schedual and as @trim24 has stated Ryuk stated when light died (or when he time was up) he would write his name in the book, I think you forget that there are 2 books one is given to Ryuk (wont go in to much detail) the police only have 1 book