In the Death Note anime, Rem kills Ginzo Kaneboshi in episode 21 to help Misa convince Higuchi that she is the second Kira. But why didn't Rem die after this? If she died after killing Watari/L, but why not after killing Kaneboshi?
That is because a Shinigami can die only if they try to extend or prolong the life of a human. In this case, Misa was not in any danger, and killing Ginzo would in no way extend Misa's life directly.
This excerpt from Wikipedia explains that Shinigami can die only by saving a human life (emphasis mine):
Like humans, Shinigami also can die, of sorts by extending the life of a human they care about (by saving them from certain death): the purpose of a Shinigami is to end life, not give it, and hence saving a human is contrary to their nature. Shinigami who die in this manner are reduced to dust. Their Death Note is left behind. The other way that a Shinigami can die is that they stop writing names in their Death Note- since Shinigami take human lifespans for their own when they write a name in their Death Note, if they stop they will eventually die when their lifespan runs out.
Hope this helps :D