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Loneliness was just a fact of life for the Shinigami. If they fell in love, that typically resulted in death. Love was, as far as they were concerned, only for mortals and fools.
Many had been tempted over the years but that hardly mattered to Ryuk. He'd never had those urges. Even better, he'd even managed to befriend another Shinigami.
Rem was always so curious about the love that humans shared for each other and Ryuk was always quick to remind her that they were mortals and that she wasn't. He didn't see what the big deal was or why Rem was so invested in their silly human romances, but he eventually learned to accept it as just one of her quirks.
But then she fell in love with Misa.
He knew it before she did. As little as he cared about the concept of romance, he was very observant and was able to spot it a mile away, even as Rem was still navigating her confusion over the emotion.
Still, he thought she knew better than to act on it. But, as it would turn out, she didn't.
With Rem gone, he was left only with Light, but that clearly wasn't anything sustainable. Not only was it largely Light's fault that Rem had died in the first place, but the boy was also getting awfully boring.
After Light was gone, Ryuk found himself alone again, just as he'd been for countless years before Rem started hanging around him. He felt a void form in his heart (not literally) as he realized that he was probably doomed to be alone forever. He could never form those irrational connections and be granted the sweet release of death as punishment for acting on them in a beautiful sacrifice.
No, as it turned out, his fate was to just endlessly watch as his friends died around him for reasons he couldn't understand.
As it would turn out, loneliness was not simply a fact of life for Shinigami. Love was for mortals and fools, and loneliness was for everyone else.