Rivalry, n.; pl. Rivalries (.
The act of rivaling, or the state of being a rival; a competition. "Keen contention and eager rivalries." Jeffrey.
Syn. -- Emulation; competition. See Emulation.
Shades of meaning
Fernald has no entry of its own for rivalry; he lists it under ambition.
Where he writes about it:
Emulation regards the abstract, competition the concrete; rivalry is the same in essential meaning with competition, but differs in the nature of the objects contested for, which, in the case of rivalry, are usually of the nobler sort and less subject to direct gaging, measurement, and rule. — from ambition
We speak of competition in business, emulation in scholarship, rivalry in love, politics, etc.; emulation of excellence, success, achievement; competition for a prize; rivalry between persons or nations. — from ambition
Competition may be friendly, rivalry is commonly hostile. — from ambition