* The rivalry between the two teams was fierce, with both sides determined to win.
* The rivalry between Coke and Pepsi has been going on for decades.
* The rivalry between the two siblings was a constant source of tension in the family.
* The rivalry between the two companies has been a driving force behind innovation in the industry.
* The rivalry between the two countries was rooted in historical conflicts.
You can also use "rivalry" in more figurative ways:
* The rivalry between the two chefs was evident in their elaborate dishes.
* The rivalry between the two ideas was a battle of minds.
* The rivalry between the two artists was a source of creative inspiration.