I haven't seen Ringu, but I find it difficult to believe that it could be better than The Ring.
And, having seen both, I feel quite confident that The Departed is an improvement on Infernal Affairs.
Granted, in both instances you have to give more credit to the film that originated each idea, but I'm putting originality aside and assuming you're viewing each remake as standalone and without any knowledge of the original.
Anyway, here's my list of great remakes (I don't know if I'd say all are as good as or better than their source material, but they're all still worth a look. Don't ignore them because they're remakes).
The Departed--Infernal Affairs
Dawn of the Dead
Vanilla Sky--Abre los ojos (Open Your Eyes)
The Fly
3:10 to Yuma
Cape Fear
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"I want a film I watch to express either the joy of making cinema or the anguish of making cinema" -Francois Truffaut
"I want a film I watch to express either the joy of making cinema or the anguish of making cinema" -Francois Truffaut