noun, plural rivalries.
the action, position, or relation of a rival or rivals; competition: rivalry between Yale and Harvard.
an instance of this.

1590–1600; rival + -ry

1. opposition, antagonism; jealousy.
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World English Dictionary
rivalry (ˈraɪvəlrɪ)
n , pl -ries
1.  the act of rivalling; competition
2.  the state of being a rival or rivals

Collins English Dictionary - Complete & Unabridged 10th Edition
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American Heritage
Medical Dictionary

rivalry ri·val·ry (rī'vəl-rē)
The state or condition of competition or antagonism.

The American Heritage® Stedman's Medical Dictionary
Copyright © 2002, 2001, 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company.
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Example sentences
There is a fierce rivalry here over who exactly dropped the first burrito into
  a vat of hot oil and thus invented the chimichanga.
Grand theories are always open to correction, and other grand theories stand in
It was built on the notion that economic integration would overcome war and
  national rivalry.
It wasn't cut throats or any rivalry among competing teams.
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