Rivalry - Competition for the same objective or for superiority in the same field.
The two people in the rivalry have to be after the same thing, have the same objective, and the same goal. Pokemon trainer Red vs Blue is a great rivalry. They both want to be the best Pokemon trainer in the world. Same objective. Link vs Ganondorf is NOT a rivalry. They don't have the same objective at all. Ganon wants to take over the world and Link is trying to stop him. They don't have the same objective. If Link was trying to take over the world then it would be a rivalry. There, I'm done venting.