Death Note Relight and Shinigami
In Death Note Relight, Light did turn into a Shinigami. However, if you take this into consideration, all the people who touched the death note will become shinigamis.
Unnamed Shinigami
There is a popular theory that the unnamed Shinigami featured in the anime OVA, Death Note Relight 1, is the reincarnation of Light Yagami. However, since a similar Shinigami appears in the manga while Light is still in school, this is debatable.
Light Yagami and Shinigami Eyes
Light never got the Shinigami Eyes because he did not want to surrender half of his remaining lifespan in exchange for them.
Shinigami King
The King of Death or the Shinigami King (死神大王, Shinigami Daiō), is the ruler of all Shinigami in the Shinigami Realm.
Shinigami Concept
Shinigami (死神, literally "death god" or "death spirit") are gods or supernatural spirits that invite humans toward death in certain aspects of Japanese religion and culture. They have been described as monsters, helpers, and creatures of darkness.
Creation of Shinigami
Shinigamis are created when individuals commit suicide. They collect souls as punishment. If a shinigami fully remembers his or her past life before becoming a full shinigami, he or she disappears and becomes a ghost. They become full shinigamis by collecting their first soul.
Ryuk’s Action
Ryuk wrote Light’s name in the Death Note because he saw that Light had lost and was doomed to go to prison. Ryuk didn’t have the patience to wait for Light’s demise, so he decided to end it by himself.