Shinigami seen in the film Death Note Relight 1: Visions of a God is a reincarnation ofLight Yagami.  Some people say, however, that the Unnamed Shinigami only appears in the anime Relight films, which take place in their own continuity. The Relight films are essentially rewrites of the original anime series, and make several alterations to the plot. There are many notable similarities between the Unnamed Shinigami and Light, as well as some suggestive dialogue by Ryuk. Because of this, it is plausible that the Unnamed Shinigami is meant to be a reincarnation of Light.  

The Shinigami Realm is Purgatory

In Death Note, Purgatory can be acknowledged as being analogous with the Shinigami Realm. Both are grey realms where beings exist in a cold world, filled with grunge and grime, absent of any light. It is not known how Shinigami are created but they still all seem to mainly be humanoid, leading to the belief that they could have once been human. However, it has never been mentioned in the manga or anime that the Shinigami Realm serves as purgatory. In fact, it is specifically stated that humans simply cease to exist upon dying. The only exception to this possibly occurs in Death Note Relight 1: Visions of a God.