The Collaborative International Dictionary of English v.0.48:

Rivalry \Ri"val*ry\, n.; pl. Rivalries.
   The act of rivaling, or the state of being a rival; a
   competition. "Keen contention and eager rivalries."
   --Jeffrey.
   [1913 Webster]

   Syn: Emulation; competition. See Emulation.
        [1913 Webster]

WordNet (r) 3.0 (2006):

rivalry
    n 1: the act of competing as for profit or a prize; "the teams
         were in fierce contention for first place" [syn:
         competition, contention, rivalry] [ant:
         cooperation]

Moby Thesaurus II by Grady Ward, 1.0:

52 Moby Thesaurus words for "rivalry":
   antagonism, antipathy, bad blood, clashing, collision, competition,
   competitiveness, conflict, contention, contrariety, contrariness,
   controversy, covetousness, cross-purposes, cutthroat competition,
   disaccord, discord, dispute, dissension, emulation, enmity,
   enviousness, envy, feud, fractiousness, friction, gamesmanship,
   grudging, grudgingness, hostility, inimicalness, invidia,
   invidiousness, jealousy, jockeying, lifemanship, negativeness,
   noncooperation, obstinacy, one-upmanship, oppugnancy, perverseness,
   recalcitrance, refractoriness, repugnance, strife, striving,
   struggle, tug-of-war, uncooperativeness, vying, warfare