How did the death note end

In Death Note, what happens to Light Yagami after he dies?

Ryuk once commented that Death Note users cannot go to Heaven or Hell. (Though a source claimed this was just a wrong rule Ryuk wrote / spoke.)

No, the claim / source is wrong. Anyone who uses / writes in the death note cannot go to heaven or hell. This can be viewed as purgatory or 無 MU / Nothing

This suggests that light is becoming a shinigami.

Not correct. :) The rules of the death note have been specifically stated in both the anime and manga, as stated by Braiam who ...

The person who uses the notebook cannot go to heaven or hell and lead an unfulfilled and unhappy life.

If you touch the death note, you will not become a shinigami, although by touching you can see the shinigami.

Memor-X

While light does not become a shinigami, Ryuk states that he would make a good shinigami for using the death note. He makes this comment, I think the episode "Deal" when Ryuk mentions the Eye Deal after Light says, "If I keep doing Eyes and Wings, I'll probably become a Shinigami." I think that's where the misunderstanding comes from. Ryuk in no way says that light becomes one when he dies, so the rules of the death note apply, and the wrong rules were in the background of what I remember just before light was locked up.